Theorem oal42 915

Description: Derivation of Godowski/Greechie Eq. II from Eq. IV.

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 68	. . . . 5 (b⊥ ∩ (a →2 b)) = ((a →2 b) ∩ b⊥ )
3		u2lemanb 598	. . . . 5 ((a →2 b) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
4	2, 3	ax-r2 35	. . . 4 (b⊥ ∩ (a →2 b)) = (a⊥ ∩ b⊥ )
5		ancom 68	. . . . 5 (c⊥ ∩ (a →2 c)) = ((a →2 c) ∩ c⊥ )
6		u2lemanb 598	. . . . 5 ((a →2 c) ∩ c⊥ ) = (a⊥ ∩ c⊥ )
7	5, 6	ax-r2 35	. . . 4 (c⊥ ∩ (a →2 c)) = (a⊥ ∩ c⊥ )
8	4, 7	2or 67	. . 3 ((b⊥ ∩ (a →2 b)) ∪ (c⊥ ∩ (a →2 c))) = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ c⊥ ))
9	1, 8	lbtr 131	. 2 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ c⊥ ))
10		lea 152	. . 3 (a⊥ ∩ b⊥ ) ≤ a⊥
11		lea 152	. . 3 (a⊥ ∩ c⊥ ) ≤ a⊥
12	10, 11	lel2or 162	. 2 ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ c⊥ )) ≤ a⊥
13	9, 12	letr 129	1 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ a⊥

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 14

This theorem is referenced by:  oa43v 1008  oa63v 1011

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-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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