Theorem wwfh2 209

Description: Foulis-Holland Theorem (weak).

Hypotheses

Ref	Expression
wwfh2.1	a C b
wwfh2.2	c⊥ C a

Assertion

Ref	Expression
wwfh2	((b ∩ (a ∪ c)) ≡ ((b ∩ a) ∪ (b ∩ c))) = 1

Proof of Theorem wwfh2

Step	Hyp	Ref	Expression
1		bicom 88	. 2 ((b ∩ (a ∪ c)) ≡ ((b ∩ a) ∪ (b ∩ c))) = (((b ∩ a) ∪ (b ∩ c)) ≡ (b ∩ (a ∪ c)))
2		ledi 166	. . 3 ((b ∩ a) ∪ (b ∩ c)) ≤ (b ∩ (a ∪ c))
3		oran 79	. . . . . . . . . . 11 ((b ∩ a) ∪ (b ∩ c)) = ((b ∩ a)⊥ ∩ (b ∩ c)⊥ )⊥
4		df-a 39	. . . . . . . . . . . . . 14 (b ∩ a) = (b⊥ ∪ a⊥ )⊥
5	4	con2 64	. . . . . . . . . . . . 13 (b ∩ a)⊥ = (b⊥ ∪ a⊥ )
6	5	ran 71	. . . . . . . . . . . 12 ((b ∩ a)⊥ ∩ (b ∩ c)⊥ ) = ((b⊥ ∪ a⊥ ) ∩ (b ∩ c)⊥ )
7	6	ax-r4 36	. . . . . . . . . . 11 ((b ∩ a)⊥ ∩ (b ∩ c)⊥ )⊥ = ((b⊥ ∪ a⊥ ) ∩ (b ∩ c)⊥ )⊥
8	3, 7	ax-r2 35	. . . . . . . . . 10 ((b ∩ a) ∪ (b ∩ c)) = ((b⊥ ∪ a⊥ ) ∩ (b ∩ c)⊥ )⊥
9	8	con2 64	. . . . . . . . 9 ((b ∩ a) ∪ (b ∩ c))⊥ = ((b⊥ ∪ a⊥ ) ∩ (b ∩ c)⊥ )
10	9	lan 70	. . . . . . . 8 ((b ∩ (a ∪ c)) ∩ ((b ∩ a) ∪ (b ∩ c))⊥ ) = ((b ∩ (a ∪ c)) ∩ ((b⊥ ∪ a⊥ ) ∩ (b ∩ c)⊥ ))
11		an4 78	. . . . . . . . 9 ((b ∩ (a ∪ c)) ∩ ((b⊥ ∪ a⊥ ) ∩ (b ∩ c)⊥ )) = ((b ∩ (b⊥ ∪ a⊥ )) ∩ ((a ∪ c) ∩ (b ∩ c)⊥ ))
12		ax-a1 29	. . . . . . . . . . . . . 14 a = a⊥ ⊥
13	12	ax-r1 34	. . . . . . . . . . . . 13 a⊥ ⊥ = a
14		wwfh2.1	. . . . . . . . . . . . 13 a C b
15	13, 14	bctr 173	. . . . . . . . . . . 12 a⊥ ⊥ C b
16	15	wwcom3ii 207	. . . . . . . . . . 11 (b ∩ (b⊥ ∪ a⊥ )) = (b ∩ a⊥ )
17		ancom 68	. . . . . . . . . . 11 (b ∩ a⊥ ) = (a⊥ ∩ b)
18	16, 17	ax-r2 35	. . . . . . . . . 10 (b ∩ (b⊥ ∪ a⊥ )) = (a⊥ ∩ b)
19	18	ran 71	. . . . . . . . 9 ((b ∩ (b⊥ ∪ a⊥ )) ∩ ((a ∪ c) ∩ (b ∩ c)⊥ )) = ((a⊥ ∩ b) ∩ ((a ∪ c) ∩ (b ∩ c)⊥ ))
20	11, 19	ax-r2 35	. . . . . . . 8 ((b ∩ (a ∪ c)) ∩ ((b⊥ ∪ a⊥ ) ∩ (b ∩ c)⊥ )) = ((a⊥ ∩ b) ∩ ((a ∪ c) ∩ (b ∩ c)⊥ ))
21	10, 20	ax-r2 35	. . . . . . 7 ((b ∩ (a ∪ c)) ∩ ((b ∩ a) ∪ (b ∩ c))⊥ ) = ((a⊥ ∩ b) ∩ ((a ∪ c) ∩ (b ∩ c)⊥ ))
22		an4 78	. . . . . . 7 ((a⊥ ∩ b) ∩ ((a ∪ c) ∩ (b ∩ c)⊥ )) = ((a⊥ ∩ (a ∪ c)) ∩ (b ∩ (b ∩ c)⊥ ))
23	21, 22	ax-r2 35	. . . . . 6 ((b ∩ (a ∪ c)) ∩ ((b ∩ a) ∪ (b ∩ c))⊥ ) = ((a⊥ ∩ (a ∪ c)) ∩ (b ∩ (b ∩ c)⊥ ))
24	12	ax-r5 37	. . . . . . . . 9 (a ∪ c) = (a⊥ ⊥ ∪ c)
25	24	lan 70	. . . . . . . 8 (a⊥ ∩ (a ∪ c)) = (a⊥ ∩ (a⊥ ⊥ ∪ c))
26		wwfh2.2	. . . . . . . . . 10 c⊥ C a
27	26	comcom2 175	. . . . . . . . 9 c⊥ C a⊥
28	27	wwcom3ii 207	. . . . . . . 8 (a⊥ ∩ (a⊥ ⊥ ∪ c)) = (a⊥ ∩ c)
29	25, 28	ax-r2 35	. . . . . . 7 (a⊥ ∩ (a ∪ c)) = (a⊥ ∩ c)
30	29	ran 71	. . . . . 6 ((a⊥ ∩ (a ∪ c)) ∩ (b ∩ (b ∩ c)⊥ )) = ((a⊥ ∩ c) ∩ (b ∩ (b ∩ c)⊥ ))
31	23, 30	ax-r2 35	. . . . 5 ((b ∩ (a ∪ c)) ∩ ((b ∩ a) ∪ (b ∩ c))⊥ ) = ((a⊥ ∩ c) ∩ (b ∩ (b ∩ c)⊥ ))
32		anass 69	. . . . 5 ((a⊥ ∩ c) ∩ (b ∩ (b ∩ c)⊥ )) = (a⊥ ∩ (c ∩ (b ∩ (b ∩ c)⊥ )))
33	31, 32	ax-r2 35	. . . 4 ((b ∩ (a ∪ c)) ∩ ((b ∩ a) ∪ (b ∩ c))⊥ ) = (a⊥ ∩ (c ∩ (b ∩ (b ∩ c)⊥ )))
34		anass 69	. . . . . . . 8 ((b ∩ c) ∩ (b ∩ c)⊥ ) = (b ∩ (c ∩ (b ∩ c)⊥ ))
35	34	ax-r1 34	. . . . . . 7 (b ∩ (c ∩ (b ∩ c)⊥ )) = ((b ∩ c) ∩ (b ∩ c)⊥ )
36		an12 74	. . . . . . 7 (c ∩ (b ∩ (b ∩ c)⊥ )) = (b ∩ (c ∩ (b ∩ c)⊥ ))
37		dff 93	. . . . . . 7 0 = ((b ∩ c) ∩ (b ∩ c)⊥ )
38	35, 36, 37	3tr1 60	. . . . . 6 (c ∩ (b ∩ (b ∩ c)⊥ )) = 0
39	38	lan 70	. . . . 5 (a⊥ ∩ (c ∩ (b ∩ (b ∩ c)⊥ ))) = (a⊥ ∩ 0)
40		an0 100	. . . . 5 (a⊥ ∩ 0) = 0
41	39, 40	ax-r2 35	. . . 4 (a⊥ ∩ (c ∩ (b ∩ (b ∩ c)⊥ ))) = 0
42	33, 41	ax-r2 35	. . 3 ((b ∩ (a ∪ c)) ∩ ((b ∩ a) ∪ (b ∩ c))⊥ ) = 0
43	2, 42	wwoml3 205	. 2 (((b ∩ a) ∪ (b ∩ c)) ≡ (b ∩ (a ∪ c))) = 1
44	1, 43	ax-r2 35	1 ((b ∩ (a ∪ c)) ≡ ((b ∩ a) ∪ (b ∩ c))) = 1

Syntax hints:   = wb 1   C wc 3  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 9  0wf 10

This theorem is referenced by:  wwfh4 211

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

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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