Theorem wom4 380

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

Hypothesis

Ref	Expression
wom4.1	(a ≤2 b) = 1

Assertion

Ref	Expression
wom4	((a ∪ (a⊥ ∩ b)) ≡ b) = 1

Proof of Theorem wom4

Step	Hyp	Ref	Expression
1		woml 211	. 2 ((a ∪ (a⊥ ∩ (a ∪ b))) ≡ (a ∪ b)) = 1
2		wom4.1	. . . . 5 (a ≤2 b) = 1
3	2	wdf-le2 379	. . . 4 ((a ∪ b) ≡ b) = 1
4	3	wlan 370	. . 3 ((a⊥ ∩ (a ∪ b)) ≡ (a⊥ ∩ b)) = 1
5	4	wlor 368	. 2 ((a ∪ (a⊥ ∩ (a ∪ b))) ≡ (a ∪ (a⊥ ∩ b))) = 1
6	1, 5, 3	w3tr2 375	1 ((a ∪ (a⊥ ∩ b)) ≡ b) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   ≤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:  wom5  381  wcomlem  382  wcom3i  422