Theorem gsth2 472

Description: Stronger version of Gudder-Schelp's Theorem. Beran, p. 263, Th. 4.2.

Hypotheses

Ref	Expression
gsth2.1	b C c
gsth2.2	a C (b ∩ c)

Assertion

Ref	Expression
gsth2	(a ∩ b) C c

Proof of Theorem gsth2

Step	Hyp	Ref	Expression
1		gsth2.1	. . . . 5 b C c
2	1	comcom 435	. . . 4 c C b
3		ancom 68	. . . . . . . . 9 (b ∩ (b⊥ ∪ a⊥ )) = ((b⊥ ∪ a⊥ ) ∩ b)
4		ax-a2 30	. . . . . . . . . 10 (b⊥ ∪ a⊥ ) = (a⊥ ∪ b⊥ )
5	4	ran 71	. . . . . . . . 9 ((b⊥ ∪ a⊥ ) ∩ b) = ((a⊥ ∪ b⊥ ) ∩ b)
6	3, 5	ax-r2 35	. . . . . . . 8 (b ∩ (b⊥ ∪ a⊥ )) = ((a⊥ ∪ b⊥ ) ∩ b)
7		comor2 444	. . . . . . . . . 10 (a⊥ ∪ b⊥ ) C b⊥
8	7	comcom7 442	. . . . . . . . 9 (a⊥ ∪ b⊥ ) C b
9		gsth2.2	. . . . . . . . . . . . 13 a C (b ∩ c)
10	9	comcom 435	. . . . . . . . . . . 12 (b ∩ c) C a
11	10	comcom2 175	. . . . . . . . . . 11 (b ∩ c) C a⊥
12		coman1 177	. . . . . . . . . . . 12 (b ∩ c) C b
13	12	comcom2 175	. . . . . . . . . . 11 (b ∩ c) C b⊥
14	11, 13	com2or 465	. . . . . . . . . 10 (b ∩ c) C (a⊥ ∪ b⊥ )
15	14	comcom 435	. . . . . . . . 9 (a⊥ ∪ b⊥ ) C (b ∩ c)
16	8, 1, 15	gsth 471	. . . . . . . 8 ((a⊥ ∪ b⊥ ) ∩ b) C c
17	6, 16	bctr 173	. . . . . . 7 (b ∩ (b⊥ ∪ a⊥ )) C c
18	17	comcom 435	. . . . . 6 c C (b ∩ (b⊥ ∪ a⊥ ))
19		df-a 39	. . . . . . 7 (b ∩ (b⊥ ∪ a⊥ )) = (b⊥ ∪ (b⊥ ∪ a⊥ )⊥ )⊥
20		df-a 39	. . . . . . . . . 10 (b ∩ a) = (b⊥ ∪ a⊥ )⊥
21	20	lor 66	. . . . . . . . 9 (b⊥ ∪ (b ∩ a)) = (b⊥ ∪ (b⊥ ∪ a⊥ )⊥ )
22	21	ax-r4 36	. . . . . . . 8 (b⊥ ∪ (b ∩ a))⊥ = (b⊥ ∪ (b⊥ ∪ a⊥ )⊥ )⊥
23	22	ax-r1 34	. . . . . . 7 (b⊥ ∪ (b⊥ ∪ a⊥ )⊥ )⊥ = (b⊥ ∪ (b ∩ a))⊥
24	19, 23	ax-r2 35	. . . . . 6 (b ∩ (b⊥ ∪ a⊥ )) = (b⊥ ∪ (b ∩ a))⊥
25	18, 24	cbtr 174	. . . . 5 c C (b⊥ ∪ (b ∩ a))⊥
26	25	comcom7 442	. . . 4 c C (b⊥ ∪ (b ∩ a))
27	2, 26	com2an 466	. . 3 c C (b ∩ (b⊥ ∪ (b ∩ a)))
28		omla 429	. . . 4 (b ∩ (b⊥ ∪ (b ∩ a))) = (b ∩ a)
29		ancom 68	. . . 4 (b ∩ a) = (a ∩ b)
30	28, 29	ax-r2 35	. . 3 (b ∩ (b⊥ ∪ (b ∩ a))) = (a ∩ b)
31	27, 30	cbtr 174	. 2 c C (a ∩ b)
32	31	comcom 435	1 (a ∩ b) C c

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

This theorem is referenced by:  gstho 473  oacom 991  oacom3 993

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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