Theorem testmod3 1217

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

Assertion

Ref	Expression
testmod3	(((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ (a ∪ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))))) = (a ∪ (((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))))

Proof of Theorem testmod3

Step	Hyp	Ref	Expression
1		orcom 73	. . . 4 (a ∪ (((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))))) = ((((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))) ∪ a)
2		leor 159	. . . . . 6 a ≤ (c ∪ a)
3	2	ler 149	. . . . 5 a ≤ ((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a)))
4	3	mli 1126	. . . 4 ((((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))) ∪ a) = (((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ ((b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))) ∪ a))
5	1, 4	tr 62	. . 3 (a ∪ (((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))))) = (((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ ((b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))) ∪ a))
6		orcom 73	. . . 4 ((b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))) ∪ a) = (a ∪ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))))
7	6	lan 77	. . 3 (((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ ((b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))) ∪ a)) = (((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ (a ∪ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))))
8	5, 7	tr 62	. 2 (a ∪ (((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))))) = (((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ (a ∪ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))))
9	8	cm 61	1 (((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ (a ∪ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))))) = (a ∪ (((c ∪ a) ∪ ((b ∪ c) ∩ (d ∪ a))) ∩ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))))

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: (None)