Theorem wcom3i 404

Description: Lemma 3(i) of Kalmbach 83 p. 23.

Hypothesis

Ref	Expression
wcom3i.1	((a ∩ (a⊥ ∪ b)) ≡ (a ∩ b)) = 1

Assertion

Ref	Expression
wcom3i	C (a, b) = 1

Proof of Theorem wcom3i

Step	Hyp	Ref	Expression
1		anor1 80	. . . . . . . . 9 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
2	1	bi1 110	. . . . . . . 8 ((a ∩ b⊥ ) ≡ (a⊥ ∪ b)⊥ ) = 1
3	2	wcon2 200	. . . . . . 7 ((a ∩ b⊥ )⊥ ≡ (a⊥ ∪ b)) = 1
4	3	wran 351	. . . . . 6 (((a ∩ b⊥ )⊥ ∩ a) ≡ ((a⊥ ∪ b) ∩ a)) = 1
5		ancom 68	. . . . . . 7 ((a⊥ ∪ b) ∩ a) = (a ∩ (a⊥ ∪ b))
6	5	bi1 110	. . . . . 6 (((a⊥ ∪ b) ∩ a) ≡ (a ∩ (a⊥ ∪ b))) = 1
7	4, 6	wr2 353	. . . . 5 (((a ∩ b⊥ )⊥ ∩ a) ≡ (a ∩ (a⊥ ∪ b))) = 1
8		wcom3i.1	. . . . 5 ((a ∩ (a⊥ ∪ b)) ≡ (a ∩ b)) = 1
9	7, 8	wr2 353	. . . 4 (((a ∩ b⊥ )⊥ ∩ a) ≡ (a ∩ b)) = 1
10	9	wlor 350	. . 3 (((a ∩ b⊥ ) ∪ ((a ∩ b⊥ )⊥ ∩ a)) ≡ ((a ∩ b⊥ ) ∪ (a ∩ b))) = 1
11		wlea 370	. . . 4 ((a ∩ b⊥ ) ≤2 a) = 1
12	11	wom4 362	. . 3 (((a ∩ b⊥ ) ∪ ((a ∩ b⊥ )⊥ ∩ a)) ≡ a) = 1
13		ax-a2 30	. . . 4 ((a ∩ b⊥ ) ∪ (a ∩ b)) = ((a ∩ b) ∪ (a ∩ b⊥ ))
14	13	bi1 110	. . 3 (((a ∩ b⊥ ) ∪ (a ∩ b)) ≡ ((a ∩ b) ∪ (a ∩ b⊥ ))) = 1
15	10, 12, 14	w3tr2 357	. 2 (a ≡ ((a ∩ b) ∪ (a ∩ b⊥ ))) = 1
16	15	wdf-c1 365	1 C (a, b) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 9   C wcmtr 28

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

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