Theorem sa5 818

Description: Possible axiom for a "Sasaki algebra" for orthoarguesian lattices.

Hypothesis

Ref	Expression
sa5.1	(a →1 c) ≤ (b →1 c)

Assertion

Ref	Expression
sa5	(b⊥ →1 c) ≤ ((a⊥ →1 c) ∪ c)

Proof of Theorem sa5

Step	Hyp	Ref	Expression
1		leor 151	. . . . 5 b ≤ (c ∪ b)
2		ax-a2 30	. . . . . . . . . 10 (b⊥ ∪ c⊥ ) = (c⊥ ∪ b⊥ )
3	2	lan 70	. . . . . . . . 9 (b ∩ (b⊥ ∪ c⊥ )) = (b ∩ (c⊥ ∪ b⊥ ))
4	3	ax-r5 37	. . . . . . . 8 ((b ∩ (b⊥ ∪ c⊥ )) ∪ c) = ((b ∩ (c⊥ ∪ b⊥ )) ∪ c)
5		ax-a2 30	. . . . . . . 8 ((b ∩ (c⊥ ∪ b⊥ )) ∪ c) = (c ∪ (b ∩ (c⊥ ∪ b⊥ )))
6		oml6 470	. . . . . . . 8 (c ∪ (b ∩ (c⊥ ∪ b⊥ ))) = (c ∪ b)
7	4, 5, 6	3tr 62	. . . . . . 7 ((b ∩ (b⊥ ∪ c⊥ )) ∪ c) = (c ∪ b)
8	7	ax-r1 34	. . . . . 6 (c ∪ b) = ((b ∩ (b⊥ ∪ c⊥ )) ∪ c)
9		sa5.1	. . . . . . . . . 10 (a →1 c) ≤ (b →1 c)
10	9	lecon 146	. . . . . . . . 9 (b →1 c)⊥ ≤ (a →1 c)⊥
11		ud1lem0c 269	. . . . . . . . 9 (b →1 c)⊥ = (b ∩ (b⊥ ∪ c⊥ ))
12		ud1lem0c 269	. . . . . . . . 9 (a →1 c)⊥ = (a ∩ (a⊥ ∪ c⊥ ))
13	10, 11, 12	le3tr2 133	. . . . . . . 8 (b ∩ (b⊥ ∪ c⊥ )) ≤ (a ∩ (a⊥ ∪ c⊥ ))
14		lea 152	. . . . . . . 8 (a ∩ (a⊥ ∪ c⊥ )) ≤ a
15	13, 14	letr 129	. . . . . . 7 (b ∩ (b⊥ ∪ c⊥ )) ≤ a
16	15	leror 144	. . . . . 6 ((b ∩ (b⊥ ∪ c⊥ )) ∪ c) ≤ (a ∪ c)
17	8, 16	bltr 130	. . . . 5 (c ∪ b) ≤ (a ∪ c)
18	1, 17	letr 129	. . . 4 b ≤ (a ∪ c)
19		ax-a1 29	. . . 4 b = b⊥ ⊥
20		ax-a1 29	. . . . . 6 a = a⊥ ⊥
21		ax-a2 30	. . . . . . 7 (c ∪ (c ∩ a⊥ )) = ((c ∩ a⊥ ) ∪ c)
22		a5b 112	. . . . . . 7 (c ∪ (c ∩ a⊥ )) = c
23		ancom 68	. . . . . . . 8 (c ∩ a⊥ ) = (a⊥ ∩ c)
24	23	ax-r5 37	. . . . . . 7 ((c ∩ a⊥ ) ∪ c) = ((a⊥ ∩ c) ∪ c)
25	21, 22, 24	3tr2 61	. . . . . 6 c = ((a⊥ ∩ c) ∪ c)
26	20, 25	2or 67	. . . . 5 (a ∪ c) = (a⊥ ⊥ ∪ ((a⊥ ∩ c) ∪ c))
27		ax-a3 31	. . . . . 6 ((a⊥ ⊥ ∪ (a⊥ ∩ c)) ∪ c) = (a⊥ ⊥ ∪ ((a⊥ ∩ c) ∪ c))
28	27	ax-r1 34	. . . . 5 (a⊥ ⊥ ∪ ((a⊥ ∩ c) ∪ c)) = ((a⊥ ⊥ ∪ (a⊥ ∩ c)) ∪ c)
29	26, 28	ax-r2 35	. . . 4 (a ∪ c) = ((a⊥ ⊥ ∪ (a⊥ ∩ c)) ∪ c)
30	18, 19, 29	le3tr2 133	. . 3 b⊥ ⊥ ≤ ((a⊥ ⊥ ∪ (a⊥ ∩ c)) ∪ c)
31		lear 153	. . . 4 (b⊥ ∩ c) ≤ c
32		leor 151	. . . 4 c ≤ ((a⊥ ⊥ ∪ (a⊥ ∩ c)) ∪ c)
33	31, 32	letr 129	. . 3 (b⊥ ∩ c) ≤ ((a⊥ ⊥ ∪ (a⊥ ∩ c)) ∪ c)
34	30, 33	lel2or 162	. 2 (b⊥ ⊥ ∪ (b⊥ ∩ c)) ≤ ((a⊥ ⊥ ∪ (a⊥ ∩ c)) ∪ c)
35		df-i1 43	. 2 (b⊥ →1 c) = (b⊥ ⊥ ∪ (b⊥ ∩ c))
36		df-i1 43	. . 3 (a⊥ →1 c) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
37	36	ax-r5 37	. 2 ((a⊥ →1 c) ∪ c) = ((a⊥ ⊥ ∪ (a⊥ ∩ c)) ∪ c)
38	34, 35, 37	le3tr1 132	1 (b⊥ →1 c) ≤ ((a⊥ →1 c) ∪ c)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ 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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