Theorem marsdenlem1 862

Description: Lemma for Marsden-Herman distributive law.

Hypotheses

Ref	Expression
marsden.1	a C b
marsden.2	b C c
marsden.3	c C d
marsden.4	d C a

Assertion

Ref	Expression
marsdenlem1	((a ∪ b) ∩ (a⊥ ∪ d⊥ )) = ((a⊥ ∩ (a ∪ b)) ∪ (d⊥ ∩ (a ∪ b)))

Proof of Theorem marsdenlem1

Step	Hyp	Ref	Expression
1		ancom 68	. 2 ((a ∪ b) ∩ (a⊥ ∪ d⊥ )) = ((a⊥ ∪ d⊥ ) ∩ (a ∪ b))
2		comorr 176	. . . 4 a C (a ∪ b)
3	2	comcom3 436	. . 3 a⊥ C (a ∪ b)
4		marsden.4	. . . . 5 d C a
5	4	comcom4 437	. . . 4 d⊥ C a⊥
6	5	comcom 435	. . 3 a⊥ C d⊥
7	3, 6	fh2r 456	. 2 ((a⊥ ∪ d⊥ ) ∩ (a ∪ b)) = ((a⊥ ∩ (a ∪ b)) ∪ (d⊥ ∩ (a ∪ b)))
8	1, 7	ax-r2 35	1 ((a ∪ b) ∩ (a⊥ ∪ d⊥ )) = ((a⊥ ∩ (a ∪ b)) ∪ (d⊥ ∩ (a ∪ b)))

Syntax hints:   = wb 1   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7

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

metamath.org