Theorem wfh1 405

Description: Weak structural analog of Foulis-Holland Theorem.

Hypotheses

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

Assertion

Ref	Expression
wfh1	((a ^ (b v c)) == ((a ^ b) v (a ^ c))) = 1

Proof of Theorem wfh1

Step	Hyp	Ref	Expression
1		wledi 387	. . 3 (((a ^ b) v (a ^ c)) =<2 (a ^ (b v c))) = 1
2		ancom 68	. . . . . . 7 (a ^ (b v c)) = ((b v c) ^ a)
3	2	bi1 110	. . . . . 6 ((a ^ (b v c)) == ((b v c) ^ a)) = 1
4		df-a 39	. . . . . . . . . 10 (a ^ b) = (a_|_ v b_|_)_|_
5	4	bi1 110	. . . . . . . . 9 ((a ^ b) == (a_|_ v b_|_)_|_) = 1
6		df-a 39	. . . . . . . . . 10 (a ^ c) = (a_|_ v c_|_)_|_
7	6	bi1 110	. . . . . . . . 9 ((a ^ c) == (a_|_ v c_|_)_|_) = 1
8	5, 7	w2or 354	. . . . . . . 8 (((a ^ b) v (a ^ c)) == ((a_|_ v b_|_)_|_ v (a_|_ v c_|_)_|_)) = 1
9		df-a 39	. . . . . . . . . . 11 ((a_|_ v b_|_) ^ (a_|_ v c_|_)) = ((a_|_ v b_|_)_|_ v (a_|_ v c_|_)_|_)_|_
10	9	bi1 110	. . . . . . . . . 10 (((a_|_ v b_|_) ^ (a_|_ v c_|_)) == ((a_|_ v b_|_)_|_ v (a_|_ v c_|_)_|_)_|_) = 1
11	10	wr1 189	. . . . . . . . 9 (((a_|_ v b_|_)_|_ v (a_|_ v c_|_)_|_)_|_ == ((a_|_ v b_|_) ^ (a_|_ v c_|_))) = 1
12	11	wcon3 201	. . . . . . . 8 (((a_|_ v b_|_)_|_ v (a_|_ v c_|_)_|_) == ((a_|_ v b_|_) ^ (a_|_ v c_|_))_|_) = 1
13	8, 12	wr2 353	. . . . . . 7 (((a ^ b) v (a ^ c)) == ((a_|_ v b_|_) ^ (a_|_ v c_|_))_|_) = 1
14	13	wcon2 200	. . . . . 6 (((a ^ b) v (a ^ c))_|_ == ((a_|_ v b_|_) ^ (a_|_ v c_|_))) = 1
15	3, 14	w2an 355	. . . . 5 (((a ^ (b v c)) ^ ((a ^ b) v (a ^ c))_|_) == (((b v c) ^ a) ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_)))) = 1
16		anass 69	. . . . . . . 8 (((b v c) ^ a) ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_))) = ((b v c) ^ (a ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_))))
17	16	bi1 110	. . . . . . 7 ((((b v c) ^ a) ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_))) == ((b v c) ^ (a ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_))))) = 1
18		wfh.1	. . . . . . . . . . . 12 C (a, b) = 1
19	18	wcomcom2 397	. . . . . . . . . . 11 C (a, b_|_) = 1
20	19	wcom3ii 401	. . . . . . . . . 10 ((a ^ (a_|_ v b_|_)) == (a ^ b_|_)) = 1
21		wfh.2	. . . . . . . . . . . 12 C (a, c) = 1
22	21	wcomcom2 397	. . . . . . . . . . 11 C (a, c_|_) = 1
23	22	wcom3ii 401	. . . . . . . . . 10 ((a ^ (a_|_ v c_|_)) == (a ^ c_|_)) = 1
24	20, 23	w2an 355	. . . . . . . . 9 (((a ^ (a_|_ v b_|_)) ^ (a ^ (a_|_ v c_|_))) == ((a ^ b_|_) ^ (a ^ c_|_))) = 1
25		anandi 106	. . . . . . . . . 10 (a ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_))) = ((a ^ (a_|_ v b_|_)) ^ (a ^ (a_|_ v c_|_)))
26	25	bi1 110	. . . . . . . . 9 ((a ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_))) == ((a ^ (a_|_ v b_|_)) ^ (a ^ (a_|_ v c_|_)))) = 1
27		anandi 106	. . . . . . . . . 10 (a ^ (b_|_ ^ c_|_)) = ((a ^ b_|_) ^ (a ^ c_|_))
28	27	bi1 110	. . . . . . . . 9 ((a ^ (b_|_ ^ c_|_)) == ((a ^ b_|_) ^ (a ^ c_|_))) = 1
29	24, 26, 28	w3tr1 356	. . . . . . . 8 ((a ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_))) == (a ^ (b_|_ ^ c_|_))) = 1
30	29	wlan 352	. . . . . . 7 (((b v c) ^ (a ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_)))) == ((b v c) ^ (a ^ (b_|_ ^ c_|_)))) = 1
31	17, 30	wr2 353	. . . . . 6 ((((b v c) ^ a) ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_))) == ((b v c) ^ (a ^ (b_|_ ^ c_|_)))) = 1
32		an12 74	. . . . . . 7 ((b v c) ^ (a ^ (b_|_ ^ c_|_))) = (a ^ ((b v c) ^ (b_|_ ^ c_|_)))
33	32	bi1 110	. . . . . 6 (((b v c) ^ (a ^ (b_|_ ^ c_|_))) == (a ^ ((b v c) ^ (b_|_ ^ c_|_)))) = 1
34	31, 33	wr2 353	. . . . 5 ((((b v c) ^ a) ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_))) == (a ^ ((b v c) ^ (b_|_ ^ c_|_)))) = 1
35	15, 34	wr2 353	. . . 4 (((a ^ (b v c)) ^ ((a ^ b) v (a ^ c))_|_) == (a ^ ((b v c) ^ (b_|_ ^ c_|_)))) = 1
36		oran 79	. . . . . . . . . . 11 (b v c) = (b_|_ ^ c_|_)_|_
37	36	bi1 110	. . . . . . . . . 10 ((b v c) == (b_|_ ^ c_|_)_|_) = 1
38	37	wr1 189	. . . . . . . . 9 ((b_|_ ^ c_|_)_|_ == (b v c)) = 1
39	38	wcon3 201	. . . . . . . 8 ((b_|_ ^ c_|_) == (b v c)_|_) = 1
40	39	wlan 352	. . . . . . 7 (((b v c) ^ (b_|_ ^ c_|_)) == ((b v c) ^ (b v c)_|_)) = 1
41		dff 93	. . . . . . . . 9 0 = ((b v c) ^ (b v c)_|_)
42	41	bi1 110	. . . . . . . 8 (0 == ((b v c) ^ (b v c)_|_)) = 1
43	42	wr1 189	. . . . . . 7 (((b v c) ^ (b v c)_|_) == 0) = 1
44	40, 43	wr2 353	. . . . . 6 (((b v c) ^ (b_|_ ^ c_|_)) == 0) = 1
45	44	wlan 352	. . . . 5 ((a ^ ((b v c) ^ (b_|_ ^ c_|_))) == (a ^ 0)) = 1
46		an0 100	. . . . . 6 (a ^ 0) = 0
47	46	bi1 110	. . . . 5 ((a ^ 0) == 0) = 1
48	45, 47	wr2 353	. . . 4 ((a ^ ((b v c) ^ (b_|_ ^ c_|_))) == 0) = 1
49	35, 48	wr2 353	. . 3 (((a ^ (b v c)) ^ ((a ^ b) v (a ^ c))_|_) == 0) = 1
50	1, 49	wom5 363	. 2 (((a ^ b) v (a ^ c)) == (a ^ (b v c))) = 1
51	50	wr1 189	1 ((a ^ (b v c)) == ((a ^ b) v (a ^ c))) = 1

Syntax hints:   = wb 1  _|_wn 4   == tb 5   v wo 6   ^ wa 7  1wt 9  0wf 10   C wcmtr 28

This theorem is referenced by:  wfh3 407  wcom2or 409  wnbdi 411  wlem14 412  ska2 414

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-wom 343

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le 121  df-le1 122  df-le2 123  df-cmtr 126

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