Theorem test2 785

Description: Part of an attempt to crack a potential Kalmbach axiom.

Assertion

Ref	Expression
test2	(a ∪ b) ≤ ((a ≡ b)⊥ ∪ ((c ∪ (a ∩ b)) ∩ (c⊥ ∪ (a ∩ b))))

Proof of Theorem test2

Step	Hyp	Ref	Expression
1		dfnb 87	. . . . 5 (a ≡ b)⊥ = ((a ∪ b) ∩ (a⊥ ∪ b⊥ ))
2		anidm 103	. . . . 5 ((a ∩ b) ∩ (a ∩ b)) = (a ∩ b)
3	1, 2	2or 67	. . . 4 ((a ≡ b)⊥ ∪ ((a ∩ b) ∩ (a ∩ b))) = (((a ∪ b) ∩ (a⊥ ∪ b⊥ )) ∪ (a ∩ b))
4		comor1 443	. . . . . . 7 (a ∪ b) C a
5		comor2 444	. . . . . . 7 (a ∪ b) C b
6	4, 5	com2an 466	. . . . . 6 (a ∪ b) C (a ∩ b)
7	4	comcom2 175	. . . . . . 7 (a ∪ b) C a⊥
8	5	comcom2 175	. . . . . . 7 (a ∪ b) C b⊥
9	7, 8	com2or 465	. . . . . 6 (a ∪ b) C (a⊥ ∪ b⊥ )
10	6, 9	fh4r 458	. . . . 5 (((a ∪ b) ∩ (a⊥ ∪ b⊥ )) ∪ (a ∩ b)) = (((a ∪ b) ∪ (a ∩ b)) ∩ ((a⊥ ∪ b⊥ ) ∪ (a ∩ b)))
11		ax-a2 30	. . . . . . . 8 ((a ∪ b) ∪ (a ∩ b)) = ((a ∩ b) ∪ (a ∪ b))
12		lea 152	. . . . . . . . . 10 (a ∩ b) ≤ a
13		leo 150	. . . . . . . . . 10 a ≤ (a ∪ b)
14	12, 13	letr 129	. . . . . . . . 9 (a ∩ b) ≤ (a ∪ b)
15	14	df-le2 123	. . . . . . . 8 ((a ∩ b) ∪ (a ∪ b)) = (a ∪ b)
16	11, 15	ax-r2 35	. . . . . . 7 ((a ∪ b) ∪ (a ∩ b)) = (a ∪ b)
17		df-a 39	. . . . . . . . 9 (a ∩ b) = (a⊥ ∪ b⊥ )⊥
18	17	lor 66	. . . . . . . 8 ((a⊥ ∪ b⊥ ) ∪ (a ∩ b)) = ((a⊥ ∪ b⊥ ) ∪ (a⊥ ∪ b⊥ )⊥ )
19		df-t 40	. . . . . . . . 9 1 = ((a⊥ ∪ b⊥ ) ∪ (a⊥ ∪ b⊥ )⊥ )
20	19	ax-r1 34	. . . . . . . 8 ((a⊥ ∪ b⊥ ) ∪ (a⊥ ∪ b⊥ )⊥ ) = 1
21	18, 20	ax-r2 35	. . . . . . 7 ((a⊥ ∪ b⊥ ) ∪ (a ∩ b)) = 1
22	16, 21	2an 72	. . . . . 6 (((a ∪ b) ∪ (a ∩ b)) ∩ ((a⊥ ∪ b⊥ ) ∪ (a ∩ b))) = ((a ∪ b) ∩ 1)
23		an1 98	. . . . . 6 ((a ∪ b) ∩ 1) = (a ∪ b)
24	22, 23	ax-r2 35	. . . . 5 (((a ∪ b) ∪ (a ∩ b)) ∩ ((a⊥ ∪ b⊥ ) ∪ (a ∩ b))) = (a ∪ b)
25	10, 24	ax-r2 35	. . . 4 (((a ∪ b) ∩ (a⊥ ∪ b⊥ )) ∪ (a ∩ b)) = (a ∪ b)
26	3, 25	ax-r2 35	. . 3 ((a ≡ b)⊥ ∪ ((a ∩ b) ∩ (a ∩ b))) = (a ∪ b)
27	26	ax-r1 34	. 2 (a ∪ b) = ((a ≡ b)⊥ ∪ ((a ∩ b) ∩ (a ∩ b)))
28		leor 151	. . . 4 (a ∩ b) ≤ (c ∪ (a ∩ b))
29		leor 151	. . . 4 (a ∩ b) ≤ (c⊥ ∪ (a ∩ b))
30	28, 29	le2an 161	. . 3 ((a ∩ b) ∩ (a ∩ b)) ≤ ((c ∪ (a ∩ b)) ∩ (c⊥ ∪ (a ∩ b)))
31	30	lelor 158	. 2 ((a ≡ b)⊥ ∪ ((a ∩ b) ∩ (a ∩ b))) ≤ ((a ≡ b)⊥ ∪ ((c ∪ (a ∩ b)) ∩ (c⊥ ∪ (a ∩ b))))
32	27, 31	bltr 130	1 (a ∪ b) ≤ ((a ≡ b)⊥ ∪ ((c ∪ (a ∩ b)) ∩ (c⊥ ∪ (a ∩ b))))

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 9

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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