Theorem wfh1 423

Description: Weak structural analog of Foulis-Holland Theorem. (Contributed by NM, 13-Oct-1997.)

Hypotheses

Ref	Expression
wfh.1	C (a, b) = 1
wfh.2	C (a, c) = 1

Assertion

Ref	Expression
wfh1	((a ∩ (b ∪ c)) ≡ ((a ∩ b) ∪ (a ∩ c))) = 1

Proof of Theorem wfh1

Step	Hyp	Ref	Expression
1		wledi 405	. . 3 (((a ∩ b) ∪ (a ∩ c)) ≤2 (a ∩ (b ∪ c))) = 1
2		ancom 74	. . . . . . 7 (a ∩ (b ∪ c)) = ((b ∪ c) ∩ a)
3	2	bi1 118	. . . . . 6 ((a ∩ (b ∪ c)) ≡ ((b ∪ c) ∩ a)) = 1
4		df-a 40	. . . . . . . . . 10 (a ∩ b) = (a⊥ ∪ b⊥ )⊥
5	4	bi1 118	. . . . . . . . 9 ((a ∩ b) ≡ (a⊥ ∪ b⊥ )⊥ ) = 1
6		df-a 40	. . . . . . . . . 10 (a ∩ c) = (a⊥ ∪ c⊥ )⊥
7	6	bi1 118	. . . . . . . . 9 ((a ∩ c) ≡ (a⊥ ∪ c⊥ )⊥ ) = 1
8	5, 7	w2or 372	. . . . . . . 8 (((a ∩ b) ∪ (a ∩ c)) ≡ ((a⊥ ∪ b⊥ )⊥ ∪ (a⊥ ∪ c⊥ )⊥ )) = 1
9		df-a 40	. . . . . . . . . . 11 ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ )) = ((a⊥ ∪ b⊥ )⊥ ∪ (a⊥ ∪ c⊥ )⊥ )⊥
10	9	bi1 118	. . . . . . . . . 10 (((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ )) ≡ ((a⊥ ∪ b⊥ )⊥ ∪ (a⊥ ∪ c⊥ )⊥ )⊥ ) = 1
11	10	wr1 197	. . . . . . . . 9 (((a⊥ ∪ b⊥ )⊥ ∪ (a⊥ ∪ c⊥ )⊥ )⊥ ≡ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))) = 1
12	11	wcon3 209	. . . . . . . 8 (((a⊥ ∪ b⊥ )⊥ ∪ (a⊥ ∪ c⊥ )⊥ ) ≡ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))⊥ ) = 1
13	8, 12	wr2 371	. . . . . . 7 (((a ∩ b) ∪ (a ∩ c)) ≡ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))⊥ ) = 1
14	13	wcon2 208	. . . . . 6 (((a ∩ b) ∪ (a ∩ c))⊥ ≡ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))) = 1
15	3, 14	w2an 373	. . . . 5 (((a ∩ (b ∪ c)) ∩ ((a ∩ b) ∪ (a ∩ c))⊥ ) ≡ (((b ∪ c) ∩ a) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ )))) = 1
16		anass 76	. . . . . . . 8 (((b ∪ c) ∩ a) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))) = ((b ∪ c) ∩ (a ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))))
17	16	bi1 118	. . . . . . 7 ((((b ∪ c) ∩ a) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))) ≡ ((b ∪ c) ∩ (a ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))))) = 1
18		wfh.1	. . . . . . . . . . . 12 C (a, b) = 1
19	18	wcomcom2 415	. . . . . . . . . . 11 C (a, b⊥ ) = 1
20	19	wcom3ii 419	. . . . . . . . . 10 ((a ∩ (a⊥ ∪ b⊥ )) ≡ (a ∩ b⊥ )) = 1
21		wfh.2	. . . . . . . . . . . 12 C (a, c) = 1
22	21	wcomcom2 415	. . . . . . . . . . 11 C (a, c⊥ ) = 1
23	22	wcom3ii 419	. . . . . . . . . 10 ((a ∩ (a⊥ ∪ c⊥ )) ≡ (a ∩ c⊥ )) = 1
24	20, 23	w2an 373	. . . . . . . . 9 (((a ∩ (a⊥ ∪ b⊥ )) ∩ (a ∩ (a⊥ ∪ c⊥ ))) ≡ ((a ∩ b⊥ ) ∩ (a ∩ c⊥ ))) = 1
25		anandi 114	. . . . . . . . . 10 (a ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))) = ((a ∩ (a⊥ ∪ b⊥ )) ∩ (a ∩ (a⊥ ∪ c⊥ )))
26	25	bi1 118	. . . . . . . . 9 ((a ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))) ≡ ((a ∩ (a⊥ ∪ b⊥ )) ∩ (a ∩ (a⊥ ∪ c⊥ )))) = 1
27		anandi 114	. . . . . . . . . 10 (a ∩ (b⊥ ∩ c⊥ )) = ((a ∩ b⊥ ) ∩ (a ∩ c⊥ ))
28	27	bi1 118	. . . . . . . . 9 ((a ∩ (b⊥ ∩ c⊥ )) ≡ ((a ∩ b⊥ ) ∩ (a ∩ c⊥ ))) = 1
29	24, 26, 28	w3tr1 374	. . . . . . . 8 ((a ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))) ≡ (a ∩ (b⊥ ∩ c⊥ ))) = 1
30	29	wlan 370	. . . . . . 7 (((b ∪ c) ∩ (a ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ )))) ≡ ((b ∪ c) ∩ (a ∩ (b⊥ ∩ c⊥ )))) = 1
31	17, 30	wr2 371	. . . . . 6 ((((b ∪ c) ∩ a) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))) ≡ ((b ∪ c) ∩ (a ∩ (b⊥ ∩ c⊥ )))) = 1
32		an12 81	. . . . . . 7 ((b ∪ c) ∩ (a ∩ (b⊥ ∩ c⊥ ))) = (a ∩ ((b ∪ c) ∩ (b⊥ ∩ c⊥ )))
33	32	bi1 118	. . . . . 6 (((b ∪ c) ∩ (a ∩ (b⊥ ∩ c⊥ ))) ≡ (a ∩ ((b ∪ c) ∩ (b⊥ ∩ c⊥ )))) = 1
34	31, 33	wr2 371	. . . . 5 ((((b ∪ c) ∩ a) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))) ≡ (a ∩ ((b ∪ c) ∩ (b⊥ ∩ c⊥ )))) = 1
35	15, 34	wr2 371	. . . 4 (((a ∩ (b ∪ c)) ∩ ((a ∩ b) ∪ (a ∩ c))⊥ ) ≡ (a ∩ ((b ∪ c) ∩ (b⊥ ∩ c⊥ )))) = 1
36		oran 87	. . . . . . . . . . 11 (b ∪ c) = (b⊥ ∩ c⊥ )⊥
37	36	bi1 118	. . . . . . . . . 10 ((b ∪ c) ≡ (b⊥ ∩ c⊥ )⊥ ) = 1
38	37	wr1 197	. . . . . . . . 9 ((b⊥ ∩ c⊥ )⊥ ≡ (b ∪ c)) = 1
39	38	wcon3 209	. . . . . . . 8 ((b⊥ ∩ c⊥ ) ≡ (b ∪ c)⊥ ) = 1
40	39	wlan 370	. . . . . . 7 (((b ∪ c) ∩ (b⊥ ∩ c⊥ )) ≡ ((b ∪ c) ∩ (b ∪ c)⊥ )) = 1
41		dff 101	. . . . . . . . 9 0 = ((b ∪ c) ∩ (b ∪ c)⊥ )
42	41	bi1 118	. . . . . . . 8 (0 ≡ ((b ∪ c) ∩ (b ∪ c)⊥ )) = 1
43	42	wr1 197	. . . . . . 7 (((b ∪ c) ∩ (b ∪ c)⊥ ) ≡ 0) = 1
44	40, 43	wr2 371	. . . . . 6 (((b ∪ c) ∩ (b⊥ ∩ c⊥ )) ≡ 0) = 1
45	44	wlan 370	. . . . 5 ((a ∩ ((b ∪ c) ∩ (b⊥ ∩ c⊥ ))) ≡ (a ∩ 0)) = 1
46		an0 108	. . . . . 6 (a ∩ 0) = 0
47	46	bi1 118	. . . . 5 ((a ∩ 0) ≡ 0) = 1
48	45, 47	wr2 371	. . . 4 ((a ∩ ((b ∪ c) ∩ (b⊥ ∩ c⊥ ))) ≡ 0) = 1
49	35, 48	wr2 371	. . 3 (((a ∩ (b ∪ c)) ∩ ((a ∩ b) ∪ (a ∩ c))⊥ ) ≡ 0) = 1
50	1, 49	wom5 381	. 2 (((a ∩ b) ∪ (a ∩ c)) ≡ (a ∩ (b ∪ c))) = 1
51	50	wr1 197	1 ((a ∩ (b ∪ c)) ≡ ((a ∩ b) ∪ (a ∩ c))) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   C wcmtr 29

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-wom 361

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134

This theorem is referenced by:  wfh3  425  wcom2or  427  wnbdi  429  wlem14  430  ska2  432  wddi1  1107