Theorem mlaconjolem 867

Description: Lemma for OML proof of Mladen's conjecture,

Assertion

Ref	Expression
mlaconjolem	((a == c) v (b == c)) =< ((c ^ (a v b)) v (c_|_ ^ (a_|_ v b_|_)))

Proof of Theorem mlaconjolem

Step	Hyp	Ref	Expression
1		orbile 825	. 2 ((a == c) v (b == c)) =< (((a ^ b) ->2 c) ^ (c ->1 (a v b)))
2		df-i2 44	. . . . 5 ((a ^ b) ->2 c) = (c v ((a ^ b)_|_ ^ c_|_))
3		oran3 85	. . . . . . . 8 (a_|_ v b_|_) = (a ^ b)_|_
4	3	ran 71	. . . . . . 7 ((a_|_ v b_|_) ^ c_|_) = ((a ^ b)_|_ ^ c_|_)
5	4	lor 66	. . . . . 6 (c v ((a_|_ v b_|_) ^ c_|_)) = (c v ((a ^ b)_|_ ^ c_|_))
6	5	ax-r1 34	. . . . 5 (c v ((a ^ b)_|_ ^ c_|_)) = (c v ((a_|_ v b_|_) ^ c_|_))
7	2, 6	ax-r2 35	. . . 4 ((a ^ b) ->2 c) = (c v ((a_|_ v b_|_) ^ c_|_))
8		df-i1 43	. . . 4 (c ->1 (a v b)) = (c_|_ v (c ^ (a v b)))
9	7, 8	2an 72	. . 3 (((a ^ b) ->2 c) ^ (c ->1 (a v b))) = ((c v ((a_|_ v b_|_) ^ c_|_)) ^ (c_|_ v (c ^ (a v b))))
10		comor1 443	. . . . 5 (c v ((a_|_ v b_|_) ^ c_|_)) C c
11	10	comcom2 175	. . . 4 (c v ((a_|_ v b_|_) ^ c_|_)) C c_|_
12		leao1 154	. . . . . 6 (c ^ (a v b)) =< (c v ((a_|_ v b_|_) ^ c_|_))
13	12	lecom 172	. . . . 5 (c ^ (a v b)) C (c v ((a_|_ v b_|_) ^ c_|_))
14	13	comcom 435	. . . 4 (c v ((a_|_ v b_|_) ^ c_|_)) C (c ^ (a v b))
15	11, 14	fh1 451	. . 3 ((c v ((a_|_ v b_|_) ^ c_|_)) ^ (c_|_ v (c ^ (a v b)))) = (((c v ((a_|_ v b_|_) ^ c_|_)) ^ c_|_) v ((c v ((a_|_ v b_|_) ^ c_|_)) ^ (c ^ (a v b))))
16		ancom 68	. . . . . . . 8 ((a_|_ v b_|_) ^ c_|_) = (c_|_ ^ (a_|_ v b_|_))
17	16	lor 66	. . . . . . 7 (c v ((a_|_ v b_|_) ^ c_|_)) = (c v (c_|_ ^ (a_|_ v b_|_)))
18	17	ran 71	. . . . . 6 ((c v ((a_|_ v b_|_) ^ c_|_)) ^ c_|_) = ((c v (c_|_ ^ (a_|_ v b_|_))) ^ c_|_)
19		ancom 68	. . . . . 6 ((c v (c_|_ ^ (a_|_ v b_|_))) ^ c_|_) = (c_|_ ^ (c v (c_|_ ^ (a_|_ v b_|_))))
20		omlan 430	. . . . . 6 (c_|_ ^ (c v (c_|_ ^ (a_|_ v b_|_)))) = (c_|_ ^ (a_|_ v b_|_))
21	18, 19, 20	3tr 62	. . . . 5 ((c v ((a_|_ v b_|_) ^ c_|_)) ^ c_|_) = (c_|_ ^ (a_|_ v b_|_))
22		ancom 68	. . . . . 6 ((c v ((a_|_ v b_|_) ^ c_|_)) ^ (c ^ (a v b))) = ((c ^ (a v b)) ^ (c v ((a_|_ v b_|_) ^ c_|_)))
23	12	df2le2 128	. . . . . 6 ((c ^ (a v b)) ^ (c v ((a_|_ v b_|_) ^ c_|_))) = (c ^ (a v b))
24	22, 23	ax-r2 35	. . . . 5 ((c v ((a_|_ v b_|_) ^ c_|_)) ^ (c ^ (a v b))) = (c ^ (a v b))
25	21, 24	2or 67	. . . 4 (((c v ((a_|_ v b_|_) ^ c_|_)) ^ c_|_) v ((c v ((a_|_ v b_|_) ^ c_|_)) ^ (c ^ (a v b)))) = ((c_|_ ^ (a_|_ v b_|_)) v (c ^ (a v b)))
26		ax-a2 30	. . . 4 ((c_|_ ^ (a_|_ v b_|_)) v (c ^ (a v b))) = ((c ^ (a v b)) v (c_|_ ^ (a_|_ v b_|_)))
27	25, 26	ax-r2 35	. . 3 (((c v ((a_|_ v b_|_) ^ c_|_)) ^ c_|_) v ((c v ((a_|_ v b_|_) ^ c_|_)) ^ (c ^ (a v b)))) = ((c ^ (a v b)) v (c_|_ ^ (a_|_ v b_|_)))
28	9, 15, 27	3tr 62	. 2 (((a ^ b) ->2 c) ^ (c ->1 (a v b))) = ((c ^ (a v b)) v (c_|_ ^ (a_|_ v b_|_)))
29	1, 28	lbtr 131	1 ((a == c) v (b == c)) =< ((c ^ (a v b)) v (c_|_ ^ (a_|_ v b_|_)))

Syntax hints:   =< wle 2  _|_wn 4   == tb 5   v wo 6   ^ wa 7   ->1 wi1 13   ->2 wi2 14

This theorem is referenced by:  mlaconjo 868

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-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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