Theorem mlaconjolem 885

Description: Lemma for OML proof of Mladen's conjecture. (Contributed by NM, 10-Mar-2002.)

Assertion

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

Proof of Theorem mlaconjolem

Step	Hyp	Ref	Expression
1		orbile 843	. 2 ((a ≡ c) ∪ (b ≡ c)) ≤ (((a ∩ b) →2 c) ∩ (c →1 (a ∪ b)))
2		df-i2 45	. . . . 5 ((a ∩ b) →2 c) = (c ∪ ((a ∩ b)⊥ ∩ c⊥ ))
3		oran3 93	. . . . . . . 8 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
4	3	ran 78	. . . . . . 7 ((a⊥ ∪ b⊥ ) ∩ c⊥ ) = ((a ∩ b)⊥ ∩ c⊥ )
5	4	lor 70	. . . . . 6 (c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) = (c ∪ ((a ∩ b)⊥ ∩ c⊥ ))
6	5	ax-r1 35	. . . . 5 (c ∪ ((a ∩ b)⊥ ∩ c⊥ )) = (c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ ))
7	2, 6	ax-r2 36	. . . 4 ((a ∩ b) →2 c) = (c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ ))
8		df-i1 44	. . . 4 (c →1 (a ∪ b)) = (c⊥ ∪ (c ∩ (a ∪ b)))
9	7, 8	2an 79	. . 3 (((a ∩ b) →2 c) ∩ (c →1 (a ∪ b))) = ((c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) ∩ (c⊥ ∪ (c ∩ (a ∪ b))))
10		comor1 461	. . . . 5 (c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) C c
11	10	comcom2 183	. . . 4 (c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) C c⊥
12		leao1 162	. . . . . 6 (c ∩ (a ∪ b)) ≤ (c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ ))
13	12	lecom 180	. . . . 5 (c ∩ (a ∪ b)) C (c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ ))
14	13	comcom 453	. . . 4 (c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) C (c ∩ (a ∪ b))
15	11, 14	fh1 469	. . 3 ((c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) ∩ (c⊥ ∪ (c ∩ (a ∪ b)))) = (((c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) ∩ c⊥ ) ∪ ((c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) ∩ (c ∩ (a ∪ b))))
16		ancom 74	. . . . . . . 8 ((a⊥ ∪ b⊥ ) ∩ c⊥ ) = (c⊥ ∩ (a⊥ ∪ b⊥ ))
17	16	lor 70	. . . . . . 7 (c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) = (c ∪ (c⊥ ∩ (a⊥ ∪ b⊥ )))
18	17	ran 78	. . . . . 6 ((c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) ∩ c⊥ ) = ((c ∪ (c⊥ ∩ (a⊥ ∪ b⊥ ))) ∩ c⊥ )
19		ancom 74	. . . . . 6 ((c ∪ (c⊥ ∩ (a⊥ ∪ b⊥ ))) ∩ c⊥ ) = (c⊥ ∩ (c ∪ (c⊥ ∩ (a⊥ ∪ b⊥ ))))
20		omlan 448	. . . . . 6 (c⊥ ∩ (c ∪ (c⊥ ∩ (a⊥ ∪ b⊥ )))) = (c⊥ ∩ (a⊥ ∪ b⊥ ))
21	18, 19, 20	3tr 65	. . . . 5 ((c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) ∩ c⊥ ) = (c⊥ ∩ (a⊥ ∪ b⊥ ))
22		ancom 74	. . . . . 6 ((c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) ∩ (c ∩ (a ∪ b))) = ((c ∩ (a ∪ b)) ∩ (c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )))
23	12	df2le2 136	. . . . . 6 ((c ∩ (a ∪ b)) ∩ (c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ ))) = (c ∩ (a ∪ b))
24	22, 23	ax-r2 36	. . . . 5 ((c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) ∩ (c ∩ (a ∪ b))) = (c ∩ (a ∪ b))
25	21, 24	2or 72	. . . 4 (((c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) ∩ c⊥ ) ∪ ((c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) ∩ (c ∩ (a ∪ b)))) = ((c⊥ ∩ (a⊥ ∪ b⊥ )) ∪ (c ∩ (a ∪ b)))
26		ax-a2 31	. . . 4 ((c⊥ ∩ (a⊥ ∪ b⊥ )) ∪ (c ∩ (a ∪ b))) = ((c ∩ (a ∪ b)) ∪ (c⊥ ∩ (a⊥ ∪ b⊥ )))
27	25, 26	ax-r2 36	. . 3 (((c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) ∩ c⊥ ) ∪ ((c ∪ ((a⊥ ∪ b⊥ ) ∩ c⊥ )) ∩ (c ∩ (a ∪ b)))) = ((c ∩ (a ∪ b)) ∪ (c⊥ ∩ (a⊥ ∪ b⊥ )))
28	9, 15, 27	3tr 65	. 2 (((a ∩ b) →2 c) ∩ (c →1 (a ∪ b))) = ((c ∩ (a ∪ b)) ∪ (c⊥ ∩ (a⊥ ∪ b⊥ )))
29	1, 28	lbtr 139	1 ((a ≡ c) ∪ (b ≡ c)) ≤ ((c ∩ (a ∪ b)) ∪ (c⊥ ∩ (a⊥ ∪ b⊥ )))

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  mlaconjo  886