Theorem sadm3 820

Description: Weak DeMorgan's law for attempt at Sasaki algebra.

Assertion

Ref	Expression
sadm3	(((a_|_ ->1 c) ^ (b_|_ ->1 c)) ->1 c) =< ((a ->1 c) v (b ->1 c))

Proof of Theorem sadm3

Step	Hyp	Ref	Expression
1		oran3 85	. . . . . . 7 ((a_|_ ->1 c)_|_ v (b_|_ ->1 c)_|_) = ((a_|_ ->1 c) ^ (b_|_ ->1 c))_|_
2	1	ax-r1 34	. . . . . 6 ((a_|_ ->1 c) ^ (b_|_ ->1 c))_|_ = ((a_|_ ->1 c)_|_ v (b_|_ ->1 c)_|_)
3		u1lem9a 759	. . . . . . 7 (a_|_ ->1 c)_|_ =< a_|_
4		u1lem9a 759	. . . . . . 7 (b_|_ ->1 c)_|_ =< b_|_
5	3, 4	le2or 160	. . . . . 6 ((a_|_ ->1 c)_|_ v (b_|_ ->1 c)_|_) =< (a_|_ v b_|_)
6	2, 5	bltr 130	. . . . 5 ((a_|_ ->1 c) ^ (b_|_ ->1 c))_|_ =< (a_|_ v b_|_)
7		an32 76	. . . . . 6 (((a_|_ ->1 c) ^ (b_|_ ->1 c)) ^ c) = (((a_|_ ->1 c) ^ c) ^ (b_|_ ->1 c))
8		lea 152	. . . . . 6 (((a_|_ ->1 c) ^ c) ^ (b_|_ ->1 c)) =< ((a_|_ ->1 c) ^ c)
9	7, 8	bltr 130	. . . . 5 (((a_|_ ->1 c) ^ (b_|_ ->1 c)) ^ c) =< ((a_|_ ->1 c) ^ c)
10	6, 9	le2or 160	. . . 4 (((a_|_ ->1 c) ^ (b_|_ ->1 c))_|_ v (((a_|_ ->1 c) ^ (b_|_ ->1 c)) ^ c)) =< ((a_|_ v b_|_) v ((a_|_ ->1 c) ^ c))
11		leo 150	. . . . . 6 (a_|_ v b_|_) =< ((a_|_ v b_|_) v (a ^ c))
12		or32 75	. . . . . 6 ((a_|_ v b_|_) v (a ^ c)) = ((a_|_ v (a ^ c)) v b_|_)
13	11, 12	lbtr 131	. . . . 5 (a_|_ v b_|_) =< ((a_|_ v (a ^ c)) v b_|_)
14		u1lemab 592	. . . . . . 7 ((a_|_ ->1 c) ^ c) = ((a_|_ ^ c) v (a_|__|_ ^ c))
15		lea 152	. . . . . . . 8 (a_|_ ^ c) =< a_|_
16		ax-a1 29	. . . . . . . . . . 11 a = a_|__|_
17	16	ax-r1 34	. . . . . . . . . 10 a_|__|_ = a
18	17	bile 134	. . . . . . . . 9 a_|__|_ =< a
19	18	leran 145	. . . . . . . 8 (a_|__|_ ^ c) =< (a ^ c)
20	15, 19	le2or 160	. . . . . . 7 ((a_|_ ^ c) v (a_|__|_ ^ c)) =< (a_|_ v (a ^ c))
21	14, 20	bltr 130	. . . . . 6 ((a_|_ ->1 c) ^ c) =< (a_|_ v (a ^ c))
22		leo 150	. . . . . 6 (a_|_ v (a ^ c)) =< ((a_|_ v (a ^ c)) v b_|_)
23	21, 22	letr 129	. . . . 5 ((a_|_ ->1 c) ^ c) =< ((a_|_ v (a ^ c)) v b_|_)
24	13, 23	lel2or 162	. . . 4 ((a_|_ v b_|_) v ((a_|_ ->1 c) ^ c)) =< ((a_|_ v (a ^ c)) v b_|_)
25	10, 24	letr 129	. . 3 (((a_|_ ->1 c) ^ (b_|_ ->1 c))_|_ v (((a_|_ ->1 c) ^ (b_|_ ->1 c)) ^ c)) =< ((a_|_ v (a ^ c)) v b_|_)
26		leo 150	. . . 4 b_|_ =< (b_|_ v (b ^ c))
27	26	lelor 158	. . 3 ((a_|_ v (a ^ c)) v b_|_) =< ((a_|_ v (a ^ c)) v (b_|_ v (b ^ c)))
28	25, 27	letr 129	. 2 (((a_|_ ->1 c) ^ (b_|_ ->1 c))_|_ v (((a_|_ ->1 c) ^ (b_|_ ->1 c)) ^ c)) =< ((a_|_ v (a ^ c)) v (b_|_ v (b ^ c)))
29		df-i1 43	. 2 (((a_|_ ->1 c) ^ (b_|_ ->1 c)) ->1 c) = (((a_|_ ->1 c) ^ (b_|_ ->1 c))_|_ v (((a_|_ ->1 c) ^ (b_|_ ->1 c)) ^ c))
30		df-i1 43	. . 3 (a ->1 c) = (a_|_ v (a ^ c))
31		df-i1 43	. . 3 (b ->1 c) = (b_|_ v (b ^ c))
32	30, 31	2or 67	. 2 ((a ->1 c) v (b ->1 c)) = ((a_|_ v (a ^ c)) v (b_|_ v (b ^ c)))
33	28, 29, 32	le3tr1 132	1 (((a_|_ ->1 c) ^ (b_|_ ->1 c)) ->1 c) =< ((a ->1 c) v (b ->1 c))

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

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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