Theorem wom2 416

Description: Weak orthomodular law for study of weakly orthomodular lattices.

Assertion

Ref	Expression
wom2	a =< ((a == b)_|_ v ((a v c) == (b v c)))

Proof of Theorem wom2

Step	Hyp	Ref	Expression
1		le1 138	. 2 a =< 1
2		conb 114	. . . . . 6 (a == b) = (a_|_ == b_|_)
3	2	ax-r4 36	. . . . 5 (a == b)_|_ = (a_|_ == b_|_)_|_
4		oran 79	. . . . . . 7 (a v c) = (a_|_ ^ c_|_)_|_
5		oran 79	. . . . . . 7 (b v c) = (b_|_ ^ c_|_)_|_
6	4, 5	2bi 91	. . . . . 6 ((a v c) == (b v c)) = ((a_|_ ^ c_|_)_|_ == (b_|_ ^ c_|_)_|_)
7		conb 114	. . . . . . 7 ((a_|_ ^ c_|_) == (b_|_ ^ c_|_)) = ((a_|_ ^ c_|_)_|_ == (b_|_ ^ c_|_)_|_)
8	7	ax-r1 34	. . . . . 6 ((a_|_ ^ c_|_)_|_ == (b_|_ ^ c_|_)_|_) = ((a_|_ ^ c_|_) == (b_|_ ^ c_|_))
9	6, 8	ax-r2 35	. . . . 5 ((a v c) == (b v c)) = ((a_|_ ^ c_|_) == (b_|_ ^ c_|_))
10	3, 9	2or 67	. . . 4 ((a == b)_|_ v ((a v c) == (b v c))) = ((a_|_ == b_|_)_|_ v ((a_|_ ^ c_|_) == (b_|_ ^ c_|_)))
11		ska4 415	. . . 4 ((a_|_ == b_|_)_|_ v ((a_|_ ^ c_|_) == (b_|_ ^ c_|_))) = 1
12	10, 11	ax-r2 35	. . 3 ((a == b)_|_ v ((a v c) == (b v c))) = 1
13	12	ax-r1 34	. 2 1 = ((a == b)_|_ v ((a v c) == (b v c)))
14	1, 13	lbtr 131	1 a =< ((a == b)_|_ v ((a v c) == (b v c)))

Syntax hints:   =< wle 2  _|_wn 4   == tb 5   v wo 6   ^ wa 7  1wt 9

This theorem is referenced by:  ka4ot 417

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-wom 343

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le 121  df-le1 122  df-le2 123  df-cmtr 126

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