Theorem u1lem8 758

Description: Lemma used in study of orthoarguesian law.

Assertion

Ref	Expression
u1lem8	((a ->1 b) ^ (a_|_ ->1 b)) = ((a ^ b) v (a_|_ ^ b))

Proof of Theorem u1lem8

Step	Hyp	Ref	Expression
1		df-i1 43	. . 3 (a ->1 b) = (a_|_ v (a ^ b))
2		df-i1 43	. . . 4 (a_|_ ->1 b) = (a_|__|_ v (a_|_ ^ b))
3		ax-a1 29	. . . . . 6 a = a_|__|_
4	3	ax-r5 37	. . . . 5 (a v (a_|_ ^ b)) = (a_|__|_ v (a_|_ ^ b))
5	4	ax-r1 34	. . . 4 (a_|__|_ v (a_|_ ^ b)) = (a v (a_|_ ^ b))
6	2, 5	ax-r2 35	. . 3 (a_|_ ->1 b) = (a v (a_|_ ^ b))
7	1, 6	2an 72	. 2 ((a ->1 b) ^ (a_|_ ->1 b)) = ((a_|_ v (a ^ b)) ^ (a v (a_|_ ^ b)))
8		comor1 443	. . . 4 (a v (a_|_ ^ b)) C a
9	8	comcom2 175	. . 3 (a v (a_|_ ^ b)) C a_|_
10		coman1 177	. . . . 5 (a ^ b) C a
11	10	comcom2 175	. . . . . 6 (a ^ b) C a_|_
12		coman2 178	. . . . . 6 (a ^ b) C b
13	11, 12	com2an 466	. . . . 5 (a ^ b) C (a_|_ ^ b)
14	10, 13	com2or 465	. . . 4 (a ^ b) C (a v (a_|_ ^ b))
15	14	comcom 435	. . 3 (a v (a_|_ ^ b)) C (a ^ b)
16	9, 15	fh1r 455	. 2 ((a_|_ v (a ^ b)) ^ (a v (a_|_ ^ b))) = ((a_|_ ^ (a v (a_|_ ^ b))) v ((a ^ b) ^ (a v (a_|_ ^ b))))
17		omlan 430	. . . 4 (a_|_ ^ (a v (a_|_ ^ b))) = (a_|_ ^ b)
18		lea 152	. . . . . 6 (a ^ b) =< a
19		leo 150	. . . . . 6 a =< (a v (a_|_ ^ b))
20	18, 19	letr 129	. . . . 5 (a ^ b) =< (a v (a_|_ ^ b))
21	20	df2le2 128	. . . 4 ((a ^ b) ^ (a v (a_|_ ^ b))) = (a ^ b)
22	17, 21	2or 67	. . 3 ((a_|_ ^ (a v (a_|_ ^ b))) v ((a ^ b) ^ (a v (a_|_ ^ b)))) = ((a_|_ ^ b) v (a ^ b))
23		ax-a2 30	. . 3 ((a_|_ ^ b) v (a ^ b)) = ((a ^ b) v (a_|_ ^ b))
24	22, 23	ax-r2 35	. 2 ((a_|_ ^ (a v (a_|_ ^ b))) v ((a ^ b) ^ (a v (a_|_ ^ b)))) = ((a ^ b) v (a_|_ ^ b))
25	7, 16, 24	3tr 62	1 ((a ->1 b) ^ (a_|_ ->1 b)) = ((a ^ b) v (a_|_ ^ b))

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

This theorem is referenced by:  oa4to4u 953  oa4to4u2 954  oa3-u1 971  oa3-u2 972

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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