Theorem testmod 1213

Description: A modular law experiment. (Contributed by NM, 21-Apr-2012.)

Assertion

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

Proof of Theorem testmod

Step	Hyp	Ref	Expression
1		leao1 162	. . . . . . . 8 ((a ∪ c) ∩ (b ∪ d)) ≤ ((a ∪ c) ∪ ((b ∪ c) ∩ (d ∪ a)))
2	1	mli 1126	. . . . . . 7 ((((a ∪ c) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ d) ∪ ((a ∪ c) ∩ (b ∪ d))) = (((a ∪ c) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))
3		orass 75	. . . . . . . 8 ((a ∪ c) ∪ ((b ∪ c) ∩ (d ∪ a))) = (a ∪ (c ∪ ((b ∪ c) ∩ (d ∪ a))))
4	3	ran 78	. . . . . . 7 (((a ∪ c) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))) = ((a ∪ (c ∪ ((b ∪ c) ∩ (d ∪ a)))) ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))
5	2, 4	tr 62	. . . . . 6 ((((a ∪ c) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ d) ∪ ((a ∪ c) ∩ (b ∪ d))) = ((a ∪ (c ∪ ((b ∪ c) ∩ (d ∪ a)))) ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))
6	5	lan 77	. . . . 5 (b ∩ ((((a ∪ c) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ d) ∪ ((a ∪ c) ∩ (b ∪ d)))) = (b ∩ ((a ∪ (c ∪ ((b ∪ c) ∩ (d ∪ a)))) ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))))
7	6	ror 71	. . . 4 ((b ∩ ((((a ∪ c) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ d) ∪ ((a ∪ c) ∩ (b ∪ d)))) ∪ a) = ((b ∩ ((a ∪ (c ∪ ((b ∪ c) ∩ (d ∪ a)))) ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))) ∪ a)
8		an12 81	. . . . 5 (b ∩ ((a ∪ (c ∪ ((b ∪ c) ∩ (d ∪ a)))) ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))) = ((a ∪ (c ∪ ((b ∪ c) ∩ (d ∪ a)))) ∩ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))))
9	8	ror 71	. . . 4 ((b ∩ ((a ∪ (c ∪ ((b ∪ c) ∩ (d ∪ a)))) ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))) ∪ a) = (((a ∪ (c ∪ ((b ∪ c) ∩ (d ∪ a)))) ∩ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))) ∪ a)
10	7, 9	tr 62	. . 3 ((b ∩ ((((a ∪ c) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ d) ∪ ((a ∪ c) ∩ (b ∪ d)))) ∪ a) = (((a ∪ (c ∪ ((b ∪ c) ∩ (d ∪ a)))) ∩ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))) ∪ a)
11		leo 158	. . . 4 a ≤ (a ∪ (c ∪ ((b ∪ c) ∩ (d ∪ a))))
12	11	mli 1126	. . 3 (((a ∪ (c ∪ ((b ∪ c) ∩ (d ∪ a)))) ∩ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))) ∪ a) = ((a ∪ (c ∪ ((b ∪ c) ∩ (d ∪ a)))) ∩ ((b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))) ∪ a))
13		orcom 73	. . . . 5 (a ∪ (c ∪ ((b ∪ c) ∩ (d ∪ a)))) = ((c ∪ ((b ∪ c) ∩ (d ∪ a))) ∪ a)
14		or32 82	. . . . 5 ((c ∪ ((b ∪ c) ∩ (d ∪ a))) ∪ a) = ((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a)))
15	13, 14	tr 62	. . . 4 (a ∪ (c ∪ ((b ∪ c) ∩ (d ∪ a)))) = ((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a)))
16		orcom 73	. . . 4 ((b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))) ∪ a) = (a ∪ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))))
17	15, 16	2an 79	. . 3 ((a ∪ (c ∪ ((b ∪ c) ∩ (d ∪ a)))) ∩ ((b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))) ∪ a)) = (((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ (a ∪ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))))
18	10, 12, 17	3tr 65	. 2 ((b ∩ ((((a ∪ c) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ d) ∪ ((a ∪ c) ∩ (b ∪ d)))) ∪ a) = (((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ (a ∪ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))))
19	18	cm 61	1 (((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ (a ∪ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))))) = ((b ∩ ((((a ∪ c) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ d) ∪ ((a ∪ c) ∩ (b ∪ d)))) ∪ a)

This proof depends on syntax axioms:   = wb 1   ∪ wo 6   ∩ wa 7

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-ml 1122

This proof depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131

This theorem is used by:  testmod1  1214