Theorem oal42 935

Description: Derivation of Godowski/Greechie Eq. II from Eq. IV. (Contributed by NM, 25-Nov-1998.)

Hypothesis

Ref	Expression
oal42.1	(b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ ((b⊥ ∩ (a →2 b)) ∪ (c⊥ ∩ (a →2 c)))

Assertion

Ref	Expression
oal42	(b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ a⊥

Proof of Theorem oal42

Step	Hyp	Ref	Expression
1		oal42.1	. . 3 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ ((b⊥ ∩ (a →2 b)) ∪ (c⊥ ∩ (a →2 c)))
2		ancom 74	. . . . 5 (b⊥ ∩ (a →2 b)) = ((a →2 b) ∩ b⊥ )
3		u2lemanb 616	. . . . 5 ((a →2 b) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
4	2, 3	ax-r2 36	. . . 4 (b⊥ ∩ (a →2 b)) = (a⊥ ∩ b⊥ )
5		ancom 74	. . . . 5 (c⊥ ∩ (a →2 c)) = ((a →2 c) ∩ c⊥ )
6		u2lemanb 616	. . . . 5 ((a →2 c) ∩ c⊥ ) = (a⊥ ∩ c⊥ )
7	5, 6	ax-r2 36	. . . 4 (c⊥ ∩ (a →2 c)) = (a⊥ ∩ c⊥ )
8	4, 7	2or 72	. . 3 ((b⊥ ∩ (a →2 b)) ∪ (c⊥ ∩ (a →2 c))) = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ c⊥ ))
9	1, 8	lbtr 139	. 2 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ c⊥ ))
10		lea 160	. . 3 (a⊥ ∩ b⊥ ) ≤ a⊥
11		lea 160	. . 3 (a⊥ ∩ c⊥ ) ≤ a⊥
12	10, 11	lel2or 170	. 2 ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ c⊥ )) ≤ a⊥
13	9, 12	letr 137	1 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ a⊥

This proof depends on syntax axioms:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

This proof depends on 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-r3 439

This proof depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is used by:  oa43v  1028  oa63v  1032