Theorem fh2rc 480

Description: Foulis-Holland Theorem.

Hypotheses

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

Assertion

Ref	Expression
fh2rc	((c ∪ a) ∩ b) = ((c ∩ b) ∪ (a ∩ b))

Proof of Theorem fh2rc

Step	Hyp	Ref	Expression
1		fh.1	. . 3 a C b
2		fh.2	. . 3 a C c
3	1, 2	fh2r 474	. 2 ((a ∪ c) ∩ b) = ((a ∩ b) ∪ (c ∩ b))
4		ax-a2 31	. . 3 (c ∪ a) = (a ∪ c)
5	4	ran 78	. 2 ((c ∪ a) ∩ b) = ((a ∪ c) ∩ b)
6		ax-a2 31	. 2 ((c ∩ b) ∪ (a ∩ b)) = ((a ∩ b) ∪ (c ∩ b))
7	3, 5, 6	3tr1 63	1 ((c ∪ a) ∩ b) = ((c ∩ b) ∪ (a ∩ b))

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

This theorem was proved from 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 theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  mlalem  832  bi3  839  bi4  840  mlaconj4  844  comanblem1  870  mhlem  876  mhlem1  877  marsdenlem2  881  cancellem  891