Theorem mhlemlem1 856

Description: Lemma for Lemma 7.1 of Kalmbach, p. 91.

Hypothesis

Ref	Expression
mhlem.1	(a ∪ b) ≤ (c ∪ d)⊥

Assertion

Ref	Expression
mhlemlem1	(((a ∪ b) ∪ c) ∩ (a ∪ (c ∪ d))) = (a ∪ c)

Proof of Theorem mhlemlem1

Step	Hyp	Ref	Expression
1		leo 150	. . . . 5 a ≤ (a ∪ b)
2	1	ler 141	. . . 4 a ≤ ((a ∪ b) ∪ c)
3	2	lecom 172	. . 3 a C ((a ∪ b) ∪ c)
4		mhlem.1	. . . . . 6 (a ∪ b) ≤ (c ∪ d)⊥
5	1, 4	letr 129	. . . . 5 a ≤ (c ∪ d)⊥
6	5	lecom 172	. . . 4 a C (c ∪ d)⊥
7	6	comcom7 442	. . 3 a C (c ∪ d)
8	3, 7	fh2 452	. 2 (((a ∪ b) ∪ c) ∩ (a ∪ (c ∪ d))) = ((((a ∪ b) ∪ c) ∩ a) ∪ (((a ∪ b) ∪ c) ∩ (c ∪ d)))
9		ancom 68	. . . 4 (((a ∪ b) ∪ c) ∩ a) = (a ∩ ((a ∪ b) ∪ c))
10		ax-a3 31	. . . . 5 ((a ∪ b) ∪ c) = (a ∪ (b ∪ c))
11	10	lan 70	. . . 4 (a ∩ ((a ∪ b) ∪ c)) = (a ∩ (a ∪ (b ∪ c)))
12		a5c 113	. . . 4 (a ∩ (a ∪ (b ∪ c))) = a
13	9, 11, 12	3tr 62	. . 3 (((a ∪ b) ∪ c) ∩ a) = a
14		comor1 443	. . . . 5 (c ∪ d) C c
15	4	lecon3 149	. . . . . . 7 (c ∪ d) ≤ (a ∪ b)⊥
16	15	lecom 172	. . . . . 6 (c ∪ d) C (a ∪ b)⊥
17	16	comcom7 442	. . . . 5 (c ∪ d) C (a ∪ b)
18	14, 17	fh1rc 461	. . . 4 (((a ∪ b) ∪ c) ∩ (c ∪ d)) = (((a ∪ b) ∩ (c ∪ d)) ∪ (c ∩ (c ∪ d)))
19	4	ortha 420	. . . . 5 ((a ∪ b) ∩ (c ∪ d)) = 0
20		a5c 113	. . . . 5 (c ∩ (c ∪ d)) = c
21	19, 20	2or 67	. . . 4 (((a ∪ b) ∩ (c ∪ d)) ∪ (c ∩ (c ∪ d))) = (0 ∪ c)
22		or0r 95	. . . 4 (0 ∪ c) = c
23	18, 21, 22	3tr 62	. . 3 (((a ∪ b) ∪ c) ∩ (c ∪ d)) = c
24	13, 23	2or 67	. 2 ((((a ∪ b) ∪ c) ∩ a) ∪ (((a ∪ b) ∪ c) ∩ (c ∪ d))) = (a ∪ c)
25	8, 24	ax-r2 35	1 (((a ∪ b) ∪ c) ∩ (a ∪ (c ∪ d))) = (a ∪ c)

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 10

This theorem is referenced by:  mhlemlem2 857  mhlem 858

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

metamath.org