Theorem wom4 362

Description: Orthomodular law. Kalmbach 83 p. 22.

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 203	. 2 ((a ∪ (a⊥ ∩ (a ∪ b))) ≡ (a ∪ b)) = 1
2		wom4.1	. . . . 5 (a ≤2 b) = 1
3	2	wdf-le2 361	. . . 4 ((a ∪ b) ≡ b) = 1
4	3	wlan 352	. . 3 ((a⊥ ∩ (a ∪ b)) ≡ (a⊥ ∩ b)) = 1
5	4	wlor 350	. 2 ((a ∪ (a⊥ ∩ (a ∪ b))) ≡ (a ∪ (a⊥ ∩ b))) = 1
6	1, 5, 3	w3tr2 357	1 ((a ∪ (a⊥ ∩ b)) ≡ b) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 9   ≤₂ wle2 11

This theorem is referenced by:  wom5 363  wcomlem 364  wcom3i 404

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

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