Theorem oas 905

Description: "Strengthening" lemma for studying the orthoarguesian law.

Hypothesis

Ref	Expression
oas.1	(a_|_ ^ (a v b)) =< c

Assertion

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

Proof of Theorem oas

Step	Hyp	Ref	Expression
1		oml 427	. . . . . 6 (a v (a_|_ ^ (a v b))) = (a v b)
2	1	ax-r1 34	. . . . 5 (a v b) = (a v (a_|_ ^ (a v b)))
3		lea 152	. . . . . . 7 (a_|_ ^ (a v b)) =< a_|_
4		oas.1	. . . . . . 7 (a_|_ ^ (a v b)) =< c
5	3, 4	ler2an 165	. . . . . 6 (a_|_ ^ (a v b)) =< (a_|_ ^ c)
6	5	lelor 158	. . . . 5 (a v (a_|_ ^ (a v b))) =< (a v (a_|_ ^ c))
7	2, 6	bltr 130	. . . 4 (a v b) =< (a v (a_|_ ^ c))
8	7	lelan 159	. . 3 ((a ->1 c) ^ (a v b)) =< ((a ->1 c) ^ (a v (a_|_ ^ c)))
9		u1lemc1 662	. . . . 5 a C (a ->1 c)
10		comanr1 446	. . . . . 6 a_|_ C (a_|_ ^ c)
11	10	comcom6 441	. . . . 5 a C (a_|_ ^ c)
12	9, 11	fh2 452	. . . 4 ((a ->1 c) ^ (a v (a_|_ ^ c))) = (((a ->1 c) ^ a) v ((a ->1 c) ^ (a_|_ ^ c)))
13		u1lemaa 582	. . . . 5 ((a ->1 c) ^ a) = (a ^ c)
14		ancom 68	. . . . . 6 ((a ->1 c) ^ (a_|_ ^ c)) = ((a_|_ ^ c) ^ (a ->1 c))
15		lea 152	. . . . . . . 8 (a_|_ ^ c) =< a_|_
16		leo 150	. . . . . . . . 9 a_|_ =< (a_|_ v (a ^ c))
17		df-i1 43	. . . . . . . . . 10 (a ->1 c) = (a_|_ v (a ^ c))
18	17	ax-r1 34	. . . . . . . . 9 (a_|_ v (a ^ c)) = (a ->1 c)
19	16, 18	lbtr 131	. . . . . . . 8 a_|_ =< (a ->1 c)
20	15, 19	letr 129	. . . . . . 7 (a_|_ ^ c) =< (a ->1 c)
21	20	df2le2 128	. . . . . 6 ((a_|_ ^ c) ^ (a ->1 c)) = (a_|_ ^ c)
22	14, 21	ax-r2 35	. . . . 5 ((a ->1 c) ^ (a_|_ ^ c)) = (a_|_ ^ c)
23	13, 22	2or 67	. . . 4 (((a ->1 c) ^ a) v ((a ->1 c) ^ (a_|_ ^ c))) = ((a ^ c) v (a_|_ ^ c))
24	12, 23	ax-r2 35	. . 3 ((a ->1 c) ^ (a v (a_|_ ^ c))) = ((a ^ c) v (a_|_ ^ c))
25	8, 24	lbtr 131	. 2 ((a ->1 c) ^ (a v b)) =< ((a ^ c) v (a_|_ ^ c))
26		lear 153	. . 3 (a ^ c) =< c
27		lear 153	. . 3 (a_|_ ^ c) =< c
28	26, 27	lel2or 162	. 2 ((a ^ c) v (a_|_ ^ c)) =< c
29	25, 28	letr 129	1 ((a ->1 c) ^ (a v b)) =< c

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13

This theorem is referenced by:  oa4ctob 947  oa3-2wto2 969

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

metamath.org