Theorem wwfh1 216

Description: Foulis-Holland Theorem (weak). (Contributed by NM, 3-Sep-1997.)

Hypotheses

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

Assertion

Ref	Expression
wwfh1	((a ∩ (b ∪ c)) ≡ ((a ∩ b) ∪ (a ∩ c))) = 1

Proof of Theorem wwfh1

Step	Hyp	Ref	Expression
1		bicom 96	. 2 ((a ∩ (b ∪ c)) ≡ ((a ∩ b) ∪ (a ∩ c))) = (((a ∩ b) ∪ (a ∩ c)) ≡ (a ∩ (b ∪ c)))
2		ledi 174	. . 3 ((a ∩ b) ∪ (a ∩ c)) ≤ (a ∩ (b ∪ c))
3		ancom 74	. . . . . 6 (a ∩ (b ∪ c)) = ((b ∪ c) ∩ a)
4		df-a 40	. . . . . . . . 9 (a ∩ b) = (a⊥ ∪ b⊥ )⊥
5		df-a 40	. . . . . . . . 9 (a ∩ c) = (a⊥ ∪ c⊥ )⊥
6	4, 5	2or 72	. . . . . . . 8 ((a ∩ b) ∪ (a ∩ c)) = ((a⊥ ∪ b⊥ )⊥ ∪ (a⊥ ∪ c⊥ )⊥ )
7		df-a 40	. . . . . . . . . 10 ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ )) = ((a⊥ ∪ b⊥ )⊥ ∪ (a⊥ ∪ c⊥ )⊥ )⊥
8	7	ax-r1 35	. . . . . . . . 9 ((a⊥ ∪ b⊥ )⊥ ∪ (a⊥ ∪ c⊥ )⊥ )⊥ = ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))
9	8	con3 68	. . . . . . . 8 ((a⊥ ∪ b⊥ )⊥ ∪ (a⊥ ∪ c⊥ )⊥ ) = ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))⊥
10	6, 9	ax-r2 36	. . . . . . 7 ((a ∩ b) ∪ (a ∩ c)) = ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))⊥
11	10	con2 67	. . . . . 6 ((a ∩ b) ∪ (a ∩ c))⊥ = ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))
12	3, 11	2an 79	. . . . 5 ((a ∩ (b ∪ c)) ∩ ((a ∩ b) ∪ (a ∩ c))⊥ ) = (((b ∪ c) ∩ a) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ )))
13		anass 76	. . . . . . 7 (((b ∪ c) ∩ a) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))) = ((b ∪ c) ∩ (a ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))))
14		ax-a1 30	. . . . . . . . . . . . 13 b = b⊥ ⊥
15	14	ax-r1 35	. . . . . . . . . . . 12 b⊥ ⊥ = b
16		wwfh.1	. . . . . . . . . . . 12 b C a
17	15, 16	bctr 181	. . . . . . . . . . 11 b⊥ ⊥ C a
18	17	wwcom3ii 215	. . . . . . . . . 10 (a ∩ (a⊥ ∪ b⊥ )) = (a ∩ b⊥ )
19		ax-a1 30	. . . . . . . . . . . . 13 c = c⊥ ⊥
20	19	ax-r1 35	. . . . . . . . . . . 12 c⊥ ⊥ = c
21		wwfh.2	. . . . . . . . . . . 12 c C a
22	20, 21	bctr 181	. . . . . . . . . . 11 c⊥ ⊥ C a
23	22	wwcom3ii 215	. . . . . . . . . 10 (a ∩ (a⊥ ∪ c⊥ )) = (a ∩ c⊥ )
24	18, 23	2an 79	. . . . . . . . 9 ((a ∩ (a⊥ ∪ b⊥ )) ∩ (a ∩ (a⊥ ∪ c⊥ ))) = ((a ∩ b⊥ ) ∩ (a ∩ c⊥ ))
25		anandi 114	. . . . . . . . 9 (a ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))) = ((a ∩ (a⊥ ∪ b⊥ )) ∩ (a ∩ (a⊥ ∪ c⊥ )))
26		anandi 114	. . . . . . . . 9 (a ∩ (b⊥ ∩ c⊥ )) = ((a ∩ b⊥ ) ∩ (a ∩ c⊥ ))
27	24, 25, 26	3tr1 63	. . . . . . . 8 (a ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))) = (a ∩ (b⊥ ∩ c⊥ ))
28	27	lan 77	. . . . . . 7 ((b ∪ c) ∩ (a ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ )))) = ((b ∪ c) ∩ (a ∩ (b⊥ ∩ c⊥ )))
29	13, 28	ax-r2 36	. . . . . 6 (((b ∪ c) ∩ a) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))) = ((b ∪ c) ∩ (a ∩ (b⊥ ∩ c⊥ )))
30		an12 81	. . . . . 6 ((b ∪ c) ∩ (a ∩ (b⊥ ∩ c⊥ ))) = (a ∩ ((b ∪ c) ∩ (b⊥ ∩ c⊥ )))
31	29, 30	ax-r2 36	. . . . 5 (((b ∪ c) ∩ a) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))) = (a ∩ ((b ∪ c) ∩ (b⊥ ∩ c⊥ )))
32	12, 31	ax-r2 36	. . . 4 ((a ∩ (b ∪ c)) ∩ ((a ∩ b) ∪ (a ∩ c))⊥ ) = (a ∩ ((b ∪ c) ∩ (b⊥ ∩ c⊥ )))
33		oran 87	. . . . . . . . . 10 (b ∪ c) = (b⊥ ∩ c⊥ )⊥
34	33	ax-r1 35	. . . . . . . . 9 (b⊥ ∩ c⊥ )⊥ = (b ∪ c)
35	34	con3 68	. . . . . . . 8 (b⊥ ∩ c⊥ ) = (b ∪ c)⊥
36	35	lan 77	. . . . . . 7 ((b ∪ c) ∩ (b⊥ ∩ c⊥ )) = ((b ∪ c) ∩ (b ∪ c)⊥ )
37		dff 101	. . . . . . . 8 0 = ((b ∪ c) ∩ (b ∪ c)⊥ )
38	37	ax-r1 35	. . . . . . 7 ((b ∪ c) ∩ (b ∪ c)⊥ ) = 0
39	36, 38	ax-r2 36	. . . . . 6 ((b ∪ c) ∩ (b⊥ ∩ c⊥ )) = 0
40	39	lan 77	. . . . 5 (a ∩ ((b ∪ c) ∩ (b⊥ ∩ c⊥ ))) = (a ∩ 0)
41		an0 108	. . . . 5 (a ∩ 0) = 0
42	40, 41	ax-r2 36	. . . 4 (a ∩ ((b ∪ c) ∩ (b⊥ ∩ c⊥ ))) = 0
43	32, 42	ax-r2 36	. . 3 ((a ∩ (b ∪ c)) ∩ ((a ∩ b) ∪ (a ∩ c))⊥ ) = 0
44	2, 43	wwoml3 213	. 2 (((a ∩ b) ∪ (a ∩ c)) ≡ (a ∩ (b ∪ c))) = 1
45	1, 44	ax-r2 36	1 ((a ∩ (b ∪ c)) ≡ ((a ∩ b) ∪ (a ∩ c))) = 1

Syntax hints:   = wb 1   C wc 3  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9

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

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:  wwfh3  218