Theorem gsth 471

Description: Gudder-Schelp's Theorem. Beran, p. 262, Th. 4.1.

Hypotheses

Ref	Expression
gsth.1	a C b
gsth.2	b C c
gsth.3	a C (b ^ c)

Assertion

Ref	Expression
gsth	(a ^ b) C c

Proof of Theorem gsth

Step	Hyp	Ref	Expression
1		gsth.2	. . . . . 6 b C c
2		gsth.1	. . . . . . 7 a C b
3	2	comcom 435	. . . . . 6 b C a
4	1, 3	fh4rc 464	. . . . 5 ((a ^ b) v c) = ((a v c) ^ (b v c))
5	1	comcom2 175	. . . . . 6 b C c_|_
6	5, 3	fh4rc 464	. . . . 5 ((a ^ b) v c_|_) = ((a v c_|_) ^ (b v c_|_))
7	4, 6	2an 72	. . . 4 (((a ^ b) v c) ^ ((a ^ b) v c_|_)) = (((a v c) ^ (b v c)) ^ ((a v c_|_) ^ (b v c_|_)))
8		an4 78	. . . 4 (((a v c) ^ (b v c)) ^ ((a v c_|_) ^ (b v c_|_))) = (((a v c) ^ (a v c_|_)) ^ ((b v c) ^ (b v c_|_)))
9		an32 76	. . . . 5 (((a v c) ^ (a v c_|_)) ^ b) = (((a v c) ^ b) ^ (a v c_|_))
10	1	comd 438	. . . . . 6 b = ((b v c) ^ (b v c_|_))
11	10	lan 70	. . . . 5 (((a v c) ^ (a v c_|_)) ^ b) = (((a v c) ^ (a v c_|_)) ^ ((b v c) ^ (b v c_|_)))
12	3, 1	fh1r 455	. . . . . . 7 ((a v c) ^ b) = ((a ^ b) v (c ^ b))
13	12	ran 71	. . . . . 6 (((a v c) ^ b) ^ (a v c_|_)) = (((a ^ b) v (c ^ b)) ^ (a v c_|_))
14		lea 152	. . . . . . . . . 10 (a ^ b) =< a
15		leo 150	. . . . . . . . . 10 a =< (a v c_|_)
16	14, 15	letr 129	. . . . . . . . 9 (a ^ b) =< (a v c_|_)
17	16	lecom 172	. . . . . . . 8 (a ^ b) C (a v c_|_)
18	17	comcom 435	. . . . . . 7 (a v c_|_) C (a ^ b)
19		gsth.3	. . . . . . . . . . 11 a C (b ^ c)
20	19	comcom 435	. . . . . . . . . 10 (b ^ c) C a
21		coman2 178	. . . . . . . . . . 11 (b ^ c) C c
22	21	comcom2 175	. . . . . . . . . 10 (b ^ c) C c_|_
23	20, 22	com2or 465	. . . . . . . . 9 (b ^ c) C (a v c_|_)
24	23	comcom 435	. . . . . . . 8 (a v c_|_) C (b ^ c)
25		ancom 68	. . . . . . . 8 (b ^ c) = (c ^ b)
26	24, 25	cbtr 174	. . . . . . 7 (a v c_|_) C (c ^ b)
27	18, 26	fh1r 455	. . . . . 6 (((a ^ b) v (c ^ b)) ^ (a v c_|_)) = (((a ^ b) ^ (a v c_|_)) v ((c ^ b) ^ (a v c_|_)))
28	16	df2le2 128	. . . . . . . 8 ((a ^ b) ^ (a v c_|_)) = (a ^ b)
29		ancom 68	. . . . . . . . . 10 (c ^ b) = (b ^ c)
30	29	ran 71	. . . . . . . . 9 ((c ^ b) ^ (a v c_|_)) = ((b ^ c) ^ (a v c_|_))
31	20, 22	fh1 451	. . . . . . . . 9 ((b ^ c) ^ (a v c_|_)) = (((b ^ c) ^ a) v ((b ^ c) ^ c_|_))
32		anass 69	. . . . . . . . . . . 12 ((b ^ c) ^ c_|_) = (b ^ (c ^ c_|_))
33		dff 93	. . . . . . . . . . . . . 14 0 = (c ^ c_|_)
34	33	ax-r1 34	. . . . . . . . . . . . 13 (c ^ c_|_) = 0
35	34	lan 70	. . . . . . . . . . . 12 (b ^ (c ^ c_|_)) = (b ^ 0)
36		an0 100	. . . . . . . . . . . 12 (b ^ 0) = 0
37	32, 35, 36	3tr 62	. . . . . . . . . . 11 ((b ^ c) ^ c_|_) = 0
38	37	lor 66	. . . . . . . . . 10 (((b ^ c) ^ a) v ((b ^ c) ^ c_|_)) = (((b ^ c) ^ a) v 0)
39		or0 94	. . . . . . . . . 10 (((b ^ c) ^ a) v 0) = ((b ^ c) ^ a)
40	38, 39	ax-r2 35	. . . . . . . . 9 (((b ^ c) ^ a) v ((b ^ c) ^ c_|_)) = ((b ^ c) ^ a)
41	30, 31, 40	3tr 62	. . . . . . . 8 ((c ^ b) ^ (a v c_|_)) = ((b ^ c) ^ a)
42	28, 41	2or 67	. . . . . . 7 (((a ^ b) ^ (a v c_|_)) v ((c ^ b) ^ (a v c_|_))) = ((a ^ b) v ((b ^ c) ^ a))
43		ax-a2 30	. . . . . . 7 ((a ^ b) v ((b ^ c) ^ a)) = (((b ^ c) ^ a) v (a ^ b))
44		ancom 68	. . . . . . . . 9 ((b ^ c) ^ a) = (a ^ (b ^ c))
45		lea 152	. . . . . . . . . 10 (b ^ c) =< b
46	45	lelan 159	. . . . . . . . 9 (a ^ (b ^ c)) =< (a ^ b)
47	44, 46	bltr 130	. . . . . . . 8 ((b ^ c) ^ a) =< (a ^ b)
48	47	df-le2 123	. . . . . . 7 (((b ^ c) ^ a) v (a ^ b)) = (a ^ b)
49	42, 43, 48	3tr 62	. . . . . 6 (((a ^ b) ^ (a v c_|_)) v ((c ^ b) ^ (a v c_|_))) = (a ^ b)
50	13, 27, 49	3tr 62	. . . . 5 (((a v c) ^ b) ^ (a v c_|_)) = (a ^ b)
51	9, 11, 50	3tr2 61	. . . 4 (((a v c) ^ (a v c_|_)) ^ ((b v c) ^ (b v c_|_))) = (a ^ b)
52	7, 8, 51	3tr 62	. . 3 (((a ^ b) v c) ^ ((a ^ b) v c_|_)) = (a ^ b)
53	52	ax-r1 34	. 2 (a ^ b) = (((a ^ b) v c) ^ ((a ^ b) v c_|_))
54	53	df2c1 450	1 (a ^ b) C c

Syntax hints:   C wc 3  _|_wn 4   v wo 6   ^ wa 7  0wf 10

This theorem is referenced by:  gsth2 472

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-r3 421

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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