Theorem sadm3 838

Description: Weak DeMorgan's law for attempt at Sasaki algebra. (Contributed by NM, 4-Jan-1999.)

Assertion

Ref	Expression
sadm3	(((a⊥ →1 c) ∩ (b⊥ →1 c)) →1 c) ≤ ((a →1 c) ∪ (b →1 c))

Proof of Theorem sadm3

Step	Hyp	Ref	Expression
1		oran3 93	. . . . . . 7 ((a⊥ →1 c)⊥ ∪ (b⊥ →1 c)⊥ ) = ((a⊥ →1 c) ∩ (b⊥ →1 c))⊥
2	1	ax-r1 35	. . . . . 6 ((a⊥ →1 c) ∩ (b⊥ →1 c))⊥ = ((a⊥ →1 c)⊥ ∪ (b⊥ →1 c)⊥ )
3		u1lem9a 777	. . . . . . 7 (a⊥ →1 c)⊥ ≤ a⊥
4		u1lem9a 777	. . . . . . 7 (b⊥ →1 c)⊥ ≤ b⊥
5	3, 4	le2or 168	. . . . . 6 ((a⊥ →1 c)⊥ ∪ (b⊥ →1 c)⊥ ) ≤ (a⊥ ∪ b⊥ )
6	2, 5	bltr 138	. . . . 5 ((a⊥ →1 c) ∩ (b⊥ →1 c))⊥ ≤ (a⊥ ∪ b⊥ )
7		an32 83	. . . . . 6 (((a⊥ →1 c) ∩ (b⊥ →1 c)) ∩ c) = (((a⊥ →1 c) ∩ c) ∩ (b⊥ →1 c))
8		lea 160	. . . . . 6 (((a⊥ →1 c) ∩ c) ∩ (b⊥ →1 c)) ≤ ((a⊥ →1 c) ∩ c)
9	7, 8	bltr 138	. . . . 5 (((a⊥ →1 c) ∩ (b⊥ →1 c)) ∩ c) ≤ ((a⊥ →1 c) ∩ c)
10	6, 9	le2or 168	. . . 4 (((a⊥ →1 c) ∩ (b⊥ →1 c))⊥ ∪ (((a⊥ →1 c) ∩ (b⊥ →1 c)) ∩ c)) ≤ ((a⊥ ∪ b⊥ ) ∪ ((a⊥ →1 c) ∩ c))
11		leo 158	. . . . . 6 (a⊥ ∪ b⊥ ) ≤ ((a⊥ ∪ b⊥ ) ∪ (a ∩ c))
12		or32 82	. . . . . 6 ((a⊥ ∪ b⊥ ) ∪ (a ∩ c)) = ((a⊥ ∪ (a ∩ c)) ∪ b⊥ )
13	11, 12	lbtr 139	. . . . 5 (a⊥ ∪ b⊥ ) ≤ ((a⊥ ∪ (a ∩ c)) ∪ b⊥ )
14		u1lemab 610	. . . . . . 7 ((a⊥ →1 c) ∩ c) = ((a⊥ ∩ c) ∪ (a⊥ ⊥ ∩ c))
15		lea 160	. . . . . . . 8 (a⊥ ∩ c) ≤ a⊥
16		ax-a1 30	. . . . . . . . . . 11 a = a⊥ ⊥
17	16	ax-r1 35	. . . . . . . . . 10 a⊥ ⊥ = a
18	17	bile 142	. . . . . . . . 9 a⊥ ⊥ ≤ a
19	18	leran 153	. . . . . . . 8 (a⊥ ⊥ ∩ c) ≤ (a ∩ c)
20	15, 19	le2or 168	. . . . . . 7 ((a⊥ ∩ c) ∪ (a⊥ ⊥ ∩ c)) ≤ (a⊥ ∪ (a ∩ c))
21	14, 20	bltr 138	. . . . . 6 ((a⊥ →1 c) ∩ c) ≤ (a⊥ ∪ (a ∩ c))
22		leo 158	. . . . . 6 (a⊥ ∪ (a ∩ c)) ≤ ((a⊥ ∪ (a ∩ c)) ∪ b⊥ )
23	21, 22	letr 137	. . . . 5 ((a⊥ →1 c) ∩ c) ≤ ((a⊥ ∪ (a ∩ c)) ∪ b⊥ )
24	13, 23	lel2or 170	. . . 4 ((a⊥ ∪ b⊥ ) ∪ ((a⊥ →1 c) ∩ c)) ≤ ((a⊥ ∪ (a ∩ c)) ∪ b⊥ )
25	10, 24	letr 137	. . 3 (((a⊥ →1 c) ∩ (b⊥ →1 c))⊥ ∪ (((a⊥ →1 c) ∩ (b⊥ →1 c)) ∩ c)) ≤ ((a⊥ ∪ (a ∩ c)) ∪ b⊥ )
26		leo 158	. . . 4 b⊥ ≤ (b⊥ ∪ (b ∩ c))
27	26	lelor 166	. . 3 ((a⊥ ∪ (a ∩ c)) ∪ b⊥ ) ≤ ((a⊥ ∪ (a ∩ c)) ∪ (b⊥ ∪ (b ∩ c)))
28	25, 27	letr 137	. 2 (((a⊥ →1 c) ∩ (b⊥ →1 c))⊥ ∪ (((a⊥ →1 c) ∩ (b⊥ →1 c)) ∩ c)) ≤ ((a⊥ ∪ (a ∩ c)) ∪ (b⊥ ∪ (b ∩ c)))
29		df-i1 44	. 2 (((a⊥ →1 c) ∩ (b⊥ →1 c)) →1 c) = (((a⊥ →1 c) ∩ (b⊥ →1 c))⊥ ∪ (((a⊥ →1 c) ∩ (b⊥ →1 c)) ∩ c))
30		df-i1 44	. . 3 (a →1 c) = (a⊥ ∪ (a ∩ c))
31		df-i1 44	. . 3 (b →1 c) = (b⊥ ∪ (b ∩ c))
32	30, 31	2or 72	. 2 ((a →1 c) ∪ (b →1 c)) = ((a⊥ ∪ (a ∩ c)) ∪ (b⊥ ∪ (b ∩ c)))
33	28, 29, 32	le3tr1 140	1 (((a⊥ →1 c) ∩ (b⊥ →1 c)) →1 c) ≤ ((a →1 c) ∪ (b →1 c))

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)