Theorem gomaex3lem1 894

Description: Lemma for Godowski 6-var -> Mayet Example 3.

Hypothesis

Ref	Expression
gomaex3lem1.3	c ≤ d⊥

Assertion

Ref	Expression
gomaex3lem1	(c ∪ (c ∪ d)⊥ ) = d⊥

Proof of Theorem gomaex3lem1

Step	Hyp	Ref	Expression
1		comid 179	. . . 4 c C c
2	1	comcom2 175	. . 3 c C c⊥
3		gomaex3lem1.3	. . . 4 c ≤ d⊥
4	3	lecom 172	. . 3 c C d⊥
5	2, 4	fh3 453	. 2 (c ∪ (c⊥ ∩ d⊥ )) = ((c ∪ c⊥ ) ∩ (c ∪ d⊥ ))
6		anor3 82	. . 3 (c⊥ ∩ d⊥ ) = (c ∪ d)⊥
7	6	lor 66	. 2 (c ∪ (c⊥ ∩ d⊥ )) = (c ∪ (c ∪ d)⊥ )
8		ancom 68	. . 3 ((c ∪ c⊥ ) ∩ (c ∪ d⊥ )) = ((c ∪ d⊥ ) ∩ (c ∪ c⊥ ))
9	3	df-le2 123	. . . . . 6 (c ∪ d⊥ ) = d⊥
10	9	ax-r1 34	. . . . 5 d⊥ = (c ∪ d⊥ )
11		df-t 40	. . . . 5 1 = (c ∪ c⊥ )
12	10, 11	2an 72	. . . 4 (d⊥ ∩ 1) = ((c ∪ d⊥ ) ∩ (c ∪ c⊥ ))
13	12	ax-r1 34	. . 3 ((c ∪ d⊥ ) ∩ (c ∪ c⊥ )) = (d⊥ ∩ 1)
14		an1 98	. . 3 (d⊥ ∩ 1) = d⊥
15	8, 13, 14	3tr 62	. 2 ((c ∪ c⊥ ) ∩ (c ∪ d⊥ )) = d⊥
16	5, 7, 15	3tr2 61	1 (c ∪ (c ∪ d)⊥ ) = d⊥

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9

This theorem is referenced by:  gomaex3lem7 900

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-le1 122  df-le2 123  df-c1 124  df-c2 125

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