Theorem u3lemona 627

Description: Lemma for Kalmbach implication study. (Contributed by NM, 15-Dec-1997.)

Assertion

Ref	Expression
u3lemona	((a →3 b) ∪ a⊥ ) = (a⊥ ∪ b)

Proof of Theorem u3lemona

Step	Hyp	Ref	Expression
1		df-i3 46	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2	1	ax-r5 38	. 2 ((a →3 b) ∪ a⊥ ) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∪ a⊥ )
3		or32 82	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∪ a⊥ ) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ ) ∪ (a ∩ (a⊥ ∪ b)))
4		lea 160	. . . . . . 7 (a⊥ ∩ b) ≤ a⊥
5		lea 160	. . . . . . 7 (a⊥ ∩ b⊥ ) ≤ a⊥
6	4, 5	lel2or 170	. . . . . 6 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ a⊥
7	6	df-le2 131	. . . . 5 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ ) = a⊥
8	7	ax-r5 38	. . . 4 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ ) ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ (a ∩ (a⊥ ∪ b)))
9		omln 446	. . . 4 (a⊥ ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ b)
10	8, 9	ax-r2 36	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ ) ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ b)
11	3, 10	ax-r2 36	. 2 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∪ a⊥ ) = (a⊥ ∪ b)
12	2, 11	ax-r2 36	1 ((a →3 b) ∪ a⊥ ) = (a⊥ ∪ b)

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 14

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This proof depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131

This theorem is used by:  u3lemnaa  642  u3lem5  763