Theorem oaur 910

Description: Transformation lemma for studying the orthoarguesian law.

Hypothesis

Ref	Expression
oaur.1	b =< (a ->1 c)

Assertion

Ref	Expression
oaur	(a ^ ((a ->1 c) v b)) =< c

Proof of Theorem oaur

Step	Hyp	Ref	Expression
1		leid 140	. . . . 5 (a ->1 c) =< (a ->1 c)
2		oaur.1	. . . . 5 b =< (a ->1 c)
3	1, 2	lel2or 162	. . . 4 ((a ->1 c) v b) =< (a ->1 c)
4	3	lelan 159	. . 3 (a ^ ((a ->1 c) v b)) =< (a ^ (a ->1 c))
5		ancom 68	. . . 4 (a ^ (a ->1 c)) = ((a ->1 c) ^ a)
6		u1lemaa 582	. . . 4 ((a ->1 c) ^ a) = (a ^ c)
7	5, 6	ax-r2 35	. . 3 (a ^ (a ->1 c)) = (a ^ c)
8	4, 7	lbtr 131	. 2 (a ^ ((a ->1 c) v b)) =< (a ^ c)
9		lear 153	. 2 (a ^ c) =< c
10	8, 9	letr 129	1 (a ^ ((a ->1 c) v b)) =< c

Syntax hints:   =< wle 2   v wo 6   ^ wa 7   ->1 wi1 13

This theorem is referenced by:  oa4gto4u 956

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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