Theorem gsth 489

Description: Gudder-Schelp's Theorem. Beran, p. 262, Thm. 4.1. (Contributed by NM, 20-Sep-1998.)

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 453	. . . . . 6 b C a
4	1, 3	fh4rc 482	. . . . 5 ((a ∩ b) ∪ c) = ((a ∪ c) ∩ (b ∪ c))
5	1	comcom2 183	. . . . . 6 b C c⊥
6	5, 3	fh4rc 482	. . . . 5 ((a ∩ b) ∪ c⊥ ) = ((a ∪ c⊥ ) ∩ (b ∪ c⊥ ))
7	4, 6	2an 79	. . . 4 (((a ∩ b) ∪ c) ∩ ((a ∩ b) ∪ c⊥ )) = (((a ∪ c) ∩ (b ∪ c)) ∩ ((a ∪ c⊥ ) ∩ (b ∪ c⊥ )))
8		an4 86	. . . 4 (((a ∪ c) ∩ (b ∪ c)) ∩ ((a ∪ c⊥ ) ∩ (b ∪ c⊥ ))) = (((a ∪ c) ∩ (a ∪ c⊥ )) ∩ ((b ∪ c) ∩ (b ∪ c⊥ )))
9		an32 83	. . . . 5 (((a ∪ c) ∩ (a ∪ c⊥ )) ∩ b) = (((a ∪ c) ∩ b) ∩ (a ∪ c⊥ ))
10	1	comd 456	. . . . . 6 b = ((b ∪ c) ∩ (b ∪ c⊥ ))
11	10	lan 77	. . . . 5 (((a ∪ c) ∩ (a ∪ c⊥ )) ∩ b) = (((a ∪ c) ∩ (a ∪ c⊥ )) ∩ ((b ∪ c) ∩ (b ∪ c⊥ )))
12	3, 1	fh1r 473	. . . . . . 7 ((a ∪ c) ∩ b) = ((a ∩ b) ∪ (c ∩ b))
13	12	ran 78	. . . . . 6 (((a ∪ c) ∩ b) ∩ (a ∪ c⊥ )) = (((a ∩ b) ∪ (c ∩ b)) ∩ (a ∪ c⊥ ))
14		lea 160	. . . . . . . . . 10 (a ∩ b) ≤ a
15		leo 158	. . . . . . . . . 10 a ≤ (a ∪ c⊥ )
16	14, 15	letr 137	. . . . . . . . 9 (a ∩ b) ≤ (a ∪ c⊥ )
17	16	lecom 180	. . . . . . . 8 (a ∩ b) C (a ∪ c⊥ )
18	17	comcom 453	. . . . . . 7 (a ∪ c⊥ ) C (a ∩ b)
19		gsth.3	. . . . . . . . . . 11 a C (b ∩ c)
20	19	comcom 453	. . . . . . . . . 10 (b ∩ c) C a
21		coman2 186	. . . . . . . . . . 11 (b ∩ c) C c
22	21	comcom2 183	. . . . . . . . . 10 (b ∩ c) C c⊥
23	20, 22	com2or 483	. . . . . . . . 9 (b ∩ c) C (a ∪ c⊥ )
24	23	comcom 453	. . . . . . . 8 (a ∪ c⊥ ) C (b ∩ c)
25		ancom 74	. . . . . . . 8 (b ∩ c) = (c ∩ b)
26	24, 25	cbtr 182	. . . . . . 7 (a ∪ c⊥ ) C (c ∩ b)
27	18, 26	fh1r 473	. . . . . 6 (((a ∩ b) ∪ (c ∩ b)) ∩ (a ∪ c⊥ )) = (((a ∩ b) ∩ (a ∪ c⊥ )) ∪ ((c ∩ b) ∩ (a ∪ c⊥ )))
28	16	df2le2 136	. . . . . . . 8 ((a ∩ b) ∩ (a ∪ c⊥ )) = (a ∩ b)
29		ancom 74	. . . . . . . . . 10 (c ∩ b) = (b ∩ c)
30	29	ran 78	. . . . . . . . 9 ((c ∩ b) ∩ (a ∪ c⊥ )) = ((b ∩ c) ∩ (a ∪ c⊥ ))
31	20, 22	fh1 469	. . . . . . . . 9 ((b ∩ c) ∩ (a ∪ c⊥ )) = (((b ∩ c) ∩ a) ∪ ((b ∩ c) ∩ c⊥ ))
32		anass 76	. . . . . . . . . . . 12 ((b ∩ c) ∩ c⊥ ) = (b ∩ (c ∩ c⊥ ))
33		dff 101	. . . . . . . . . . . . . 14 0 = (c ∩ c⊥ )
34	33	ax-r1 35	. . . . . . . . . . . . 13 (c ∩ c⊥ ) = 0
35	34	lan 77	. . . . . . . . . . . 12 (b ∩ (c ∩ c⊥ )) = (b ∩ 0)
36		an0 108	. . . . . . . . . . . 12 (b ∩ 0) = 0
37	32, 35, 36	3tr 65	. . . . . . . . . . 11 ((b ∩ c) ∩ c⊥ ) = 0
38	37	lor 70	. . . . . . . . . 10 (((b ∩ c) ∩ a) ∪ ((b ∩ c) ∩ c⊥ )) = (((b ∩ c) ∩ a) ∪ 0)
39		or0 102	. . . . . . . . . 10 (((b ∩ c) ∩ a) ∪ 0) = ((b ∩ c) ∩ a)
40	38, 39	ax-r2 36	. . . . . . . . 9 (((b ∩ c) ∩ a) ∪ ((b ∩ c) ∩ c⊥ )) = ((b ∩ c) ∩ a)
41	30, 31, 40	3tr 65	. . . . . . . 8 ((c ∩ b) ∩ (a ∪ c⊥ )) = ((b ∩ c) ∩ a)
42	28, 41	2or 72	. . . . . . 7 (((a ∩ b) ∩ (a ∪ c⊥ )) ∪ ((c ∩ b) ∩ (a ∪ c⊥ ))) = ((a ∩ b) ∪ ((b ∩ c) ∩ a))
43		ax-a2 31	. . . . . . 7 ((a ∩ b) ∪ ((b ∩ c) ∩ a)) = (((b ∩ c) ∩ a) ∪ (a ∩ b))
44		ancom 74	. . . . . . . . 9 ((b ∩ c) ∩ a) = (a ∩ (b ∩ c))
45		lea 160	. . . . . . . . . 10 (b ∩ c) ≤ b
46	45	lelan 167	. . . . . . . . 9 (a ∩ (b ∩ c)) ≤ (a ∩ b)
47	44, 46	bltr 138	. . . . . . . 8 ((b ∩ c) ∩ a) ≤ (a ∩ b)
48	47	df-le2 131	. . . . . . 7 (((b ∩ c) ∩ a) ∪ (a ∩ b)) = (a ∩ b)
49	42, 43, 48	3tr 65	. . . . . 6 (((a ∩ b) ∩ (a ∪ c⊥ )) ∪ ((c ∩ b) ∩ (a ∪ c⊥ ))) = (a ∩ b)
50	13, 27, 49	3tr 65	. . . . 5 (((a ∪ c) ∩ b) ∩ (a ∪ c⊥ )) = (a ∩ b)
51	9, 11, 50	3tr2 64	. . . 4 (((a ∪ c) ∩ (a ∪ c⊥ )) ∩ ((b ∪ c) ∩ (b ∪ c⊥ ))) = (a ∩ b)
52	7, 8, 51	3tr 65	. . 3 (((a ∩ b) ∪ c) ∩ ((a ∩ b) ∪ c⊥ )) = (a ∩ b)
53	52	ax-r1 35	. 2 (a ∩ b) = (((a ∩ b) ∪ c) ∩ ((a ∩ b) ∪ c⊥ ))
54	53	df2c1 468	1 (a ∩ b) C c

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  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  ax-r3 439

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:  gsth2  490