Theorem fh1rc 461

Description: Foulis-Holland Theorem.

Hypotheses

Ref	Expression
fh.1	a C b
fh.2	a C c

Assertion

Ref	Expression
fh1rc	((c ∪ b) ∩ a) = ((c ∩ a) ∪ (b ∩ a))

Proof of Theorem fh1rc

Step	Hyp	Ref	Expression
1		fh.1	. . 3 a C b
2		fh.2	. . 3 a C c
3	1, 2	fh1r 455	. 2 ((b ∪ c) ∩ a) = ((b ∩ a) ∪ (c ∩ a))
4		ax-a2 30	. . 3 (c ∪ b) = (b ∪ c)
5	4	ran 71	. 2 ((c ∪ b) ∩ a) = ((b ∪ c) ∩ a)
6		ax-a2 30	. 2 ((c ∩ a) ∪ (b ∩ a)) = ((b ∩ a) ∪ (c ∩ a))
7	3, 5, 6	3tr1 60	1 ((c ∪ b) ∩ a) = ((c ∩ a) ∪ (b ∩ a))

Syntax hints:   = wb 1   C wc 3   ∪ wo 6   ∩ wa 7

This theorem is referenced by:  mhlemlem1 856

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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