Shorten axpr and reduce axiom usage #5019
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LegionMammal978
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icecream17
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I've revised bm1.3ii into a new theorem sepexi, which generalizes the conclusion of bm1.3ii while still avoiding ax-9. All users are adjusted for the extra syntax step. It is proved from a new closed-form sepex, which is proved from a weaker version sepexlem. |
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This PR shortens axpr by collapsing axprlem4 and axprlem5 into a common lemma, and also eliminates its usage of ax-9, ax-10, ax-11, and ax-12. In service of this, new theorems axrep4v (axrep4 with a DV condition in place of a not-free hypothesis, avoiding ax-12) and bm1.3iiv (extracted from bm1.3ii, avoiding ax-9) are added. The existing theorems axrep4, axrep6, and bm1.3ii are also shortened.
(Note that the non-usage of ax-8 could be considered a bit of a fiction, considering how either ax-8 or ax-12 is seemingly necessary in the proof of axsep. But it is what it is.)