Theorem ka4ot 417

Description: 3-variable version of weakly orthomodular law. It is proved from a weaker-looking equivalent, wom2 416, which in turn is proved from ax-wom 343.

Assertion

Ref	Expression
ka4ot	((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c))) = 1

Proof of Theorem ka4ot

Step	Hyp	Ref	Expression
1		le1 138	. 2 ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c))) ≤ 1
2		wom2 416	. . . . . 6 a ≤ ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c)))
3		wom2 416	. . . . . . 7 b ≤ ((b ≡ a)⊥ ∪ ((b ∪ c) ≡ (a ∪ c)))
4		bicom 88	. . . . . . . . 9 (b ≡ a) = (a ≡ b)
5	4	ax-r4 36	. . . . . . . 8 (b ≡ a)⊥ = (a ≡ b)⊥
6		bicom 88	. . . . . . . 8 ((b ∪ c) ≡ (a ∪ c)) = ((a ∪ c) ≡ (b ∪ c))
7	5, 6	2or 67	. . . . . . 7 ((b ≡ a)⊥ ∪ ((b ∪ c) ≡ (a ∪ c))) = ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c)))
8	3, 7	lbtr 131	. . . . . 6 b ≤ ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c)))
9	2, 8	le2or 160	. . . . 5 (a ∪ b) ≤ (((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c))) ∪ ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c))))
10		oridm 102	. . . . 5 (((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c))) ∪ ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c)))) = ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c)))
11	9, 10	lbtr 131	. . . 4 (a ∪ b) ≤ ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c)))
12	11	leror 144	. . 3 ((a ∪ b) ∪ ((a ∪ c) ≡ (b ∪ c))) ≤ (((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c))) ∪ ((a ∪ c) ≡ (b ∪ c)))
13		ka4lemo 220	. . 3 ((a ∪ b) ∪ ((a ∪ c) ≡ (b ∪ c))) = 1
14		ax-a3 31	. . . 4 (((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c))) ∪ ((a ∪ c) ≡ (b ∪ c))) = ((a ≡ b)⊥ ∪ (((a ∪ c) ≡ (b ∪ c)) ∪ ((a ∪ c) ≡ (b ∪ c))))
15		oridm 102	. . . . 5 (((a ∪ c) ≡ (b ∪ c)) ∪ ((a ∪ c) ≡ (b ∪ c))) = ((a ∪ c) ≡ (b ∪ c))
16	15	lor 66	. . . 4 ((a ≡ b)⊥ ∪ (((a ∪ c) ≡ (b ∪ c)) ∪ ((a ∪ c) ≡ (b ∪ c)))) = ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c)))
17	14, 16	ax-r2 35	. . 3 (((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c))) ∪ ((a ∪ c) ≡ (b ∪ c))) = ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c)))
18	12, 13, 17	le3tr2 133	. 2 1 ≤ ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c)))
19	1, 18	lebi 137	1 ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c))) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6  1wt 9

This theorem is referenced by:  i3or 479

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

metamath.org