Theorem wom5 381

Description: Orthomodular law. Kalmbach 83 p. 22. (Contributed by NM, 13-Oct-1997.)

Hypotheses

Ref	Expression
wom5.1	(a ≤2 b) = 1
wom5.2	((b ∩ a⊥ ) ≡ 0) = 1

Assertion

Ref	Expression
wom5	(a ≡ b) = 1

Proof of Theorem wom5

Step	Hyp	Ref	Expression
1		wom5.2	. . . . 5 ((b ∩ a⊥ ) ≡ 0) = 1
2	1	wr1 197	. . . 4 (0 ≡ (b ∩ a⊥ )) = 1
3		ancom 74	. . . . 5 (b ∩ a⊥ ) = (a⊥ ∩ b)
4	3	bi1 118	. . . 4 ((b ∩ a⊥ ) ≡ (a⊥ ∩ b)) = 1
5	2, 4	wr2 371	. . 3 (0 ≡ (a⊥ ∩ b)) = 1
6	5	wlor 368	. 2 ((a ∪ 0) ≡ (a ∪ (a⊥ ∩ b))) = 1
7		or0 102	. . 3 (a ∪ 0) = a
8	7	bi1 118	. 2 ((a ∪ 0) ≡ a) = 1
9		wom5.1	. . 3 (a ≤2 b) = 1
10	9	wom4 380	. 2 ((a ∪ (a⊥ ∩ b)) ≡ b) = 1
11	6, 8, 10	w3tr2 375	1 (a ≡ b) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   ≤₂ wle2 10

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-wom 361

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131

This theorem is referenced by:  wfh1  423  wfh2  424