Theorem oau 909

Description: Transformation lemma for studying the orthoarguesian law.

Hypothesis

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

Assertion

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

Proof of Theorem oau

Step	Hyp	Ref	Expression
1		ax-a2 30	. . 3 (b v (a ->1 c)) = ((a ->1 c) v b)
2		lea 152	. . . . . . . 8 (a ^ ((a ->1 c) v b)) =< a
3		oau.1	. . . . . . . 8 (a ^ ((a ->1 c) v b)) =< c
4	2, 3	ler2an 165	. . . . . . 7 (a ^ ((a ->1 c) v b)) =< (a ^ c)
5		u1lemaa 582	. . . . . . . 8 ((a ->1 c) ^ a) = (a ^ c)
6	5	ax-r1 34	. . . . . . 7 (a ^ c) = ((a ->1 c) ^ a)
7	4, 6	lbtr 131	. . . . . 6 (a ^ ((a ->1 c) v b)) =< ((a ->1 c) ^ a)
8	7	lelor 158	. . . . 5 ((a ->1 c) v (a ^ ((a ->1 c) v b))) =< ((a ->1 c) v ((a ->1 c) ^ a))
9		u1lemc1 662	. . . . . . . 8 a C (a ->1 c)
10	9	comcom 435	. . . . . . 7 (a ->1 c) C a
11		comorr 176	. . . . . . 7 (a ->1 c) C ((a ->1 c) v b)
12	10, 11	fh3 453	. . . . . 6 ((a ->1 c) v (a ^ ((a ->1 c) v b))) = (((a ->1 c) v a) ^ ((a ->1 c) v ((a ->1 c) v b)))
13		u1lemoa 602	. . . . . . 7 ((a ->1 c) v a) = 1
14		ax-a3 31	. . . . . . . . 9 (((a ->1 c) v (a ->1 c)) v b) = ((a ->1 c) v ((a ->1 c) v b))
15	14	ax-r1 34	. . . . . . . 8 ((a ->1 c) v ((a ->1 c) v b)) = (((a ->1 c) v (a ->1 c)) v b)
16		oridm 102	. . . . . . . . 9 ((a ->1 c) v (a ->1 c)) = (a ->1 c)
17	16	ax-r5 37	. . . . . . . 8 (((a ->1 c) v (a ->1 c)) v b) = ((a ->1 c) v b)
18	15, 17	ax-r2 35	. . . . . . 7 ((a ->1 c) v ((a ->1 c) v b)) = ((a ->1 c) v b)
19	13, 18	2an 72	. . . . . 6 (((a ->1 c) v a) ^ ((a ->1 c) v ((a ->1 c) v b))) = (1 ^ ((a ->1 c) v b))
20		ancom 68	. . . . . . 7 (1 ^ ((a ->1 c) v b)) = (((a ->1 c) v b) ^ 1)
21		an1 98	. . . . . . 7 (((a ->1 c) v b) ^ 1) = ((a ->1 c) v b)
22	20, 21	ax-r2 35	. . . . . 6 (1 ^ ((a ->1 c) v b)) = ((a ->1 c) v b)
23	12, 19, 22	3tr 62	. . . . 5 ((a ->1 c) v (a ^ ((a ->1 c) v b))) = ((a ->1 c) v b)
24		a5b 112	. . . . 5 ((a ->1 c) v ((a ->1 c) ^ a)) = (a ->1 c)
25	8, 23, 24	le3tr2 133	. . . 4 ((a ->1 c) v b) =< (a ->1 c)
26		leo 150	. . . 4 (a ->1 c) =< ((a ->1 c) v b)
27	25, 26	lebi 137	. . 3 ((a ->1 c) v b) = (a ->1 c)
28	1, 27	ax-r2 35	. 2 (b v (a ->1 c)) = (a ->1 c)
29	28	df-le1 122	1 b =< (a ->1 c)

Syntax hints:   =< wle 2   v wo 6   ^ wa 7  1wt 9   ->1 wi1 13

This theorem is referenced by:  oa4uto4g 955

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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