Theorem mloa 1018

Description: Mladen's OA. (Contributed by NM, 20-Nov-1998.)

Assertion

Ref	Expression
mloa	((a ≡ b) ∩ ((b ≡ c) ∪ ((b ∪ (a ≡ b)) ∩ (c ∪ (a ≡ c))))) ≤ (c ∪ (a ≡ c))

Proof of Theorem mloa

Step	Hyp	Ref	Expression
1		lea 160	. . . 4 ((a →2 b) ∩ (b →2 a)) ≤ (a →2 b)
2		ax-a3 32	. . . . . 6 (((b ∩ c) ∪ (b⊥ ∩ c⊥ )) ∪ ((a →2 b) ∩ (a →2 c))) = ((b ∩ c) ∪ ((b⊥ ∩ c⊥ ) ∪ ((a →2 b) ∩ (a →2 c))))
3		or12 80	. . . . . . 7 ((b ∩ c) ∪ ((b⊥ ∩ c⊥ ) ∪ ((a →2 b) ∩ (a →2 c)))) = ((b⊥ ∩ c⊥ ) ∪ ((b ∩ c) ∪ ((a →2 b) ∩ (a →2 c))))
4		anor3 90	. . . . . . . 8 (b⊥ ∩ c⊥ ) = (b ∪ c)⊥
5	4	ax-r5 38	. . . . . . 7 ((b⊥ ∩ c⊥ ) ∪ ((b ∩ c) ∪ ((a →2 b) ∩ (a →2 c)))) = ((b ∪ c)⊥ ∪ ((b ∩ c) ∪ ((a →2 b) ∩ (a →2 c))))
6	3, 5	ax-r2 36	. . . . . 6 ((b ∩ c) ∪ ((b⊥ ∩ c⊥ ) ∪ ((a →2 b) ∩ (a →2 c)))) = ((b ∪ c)⊥ ∪ ((b ∩ c) ∪ ((a →2 b) ∩ (a →2 c))))
7	2, 6	ax-r2 36	. . . . 5 (((b ∩ c) ∪ (b⊥ ∩ c⊥ )) ∪ ((a →2 b) ∩ (a →2 c))) = ((b ∪ c)⊥ ∪ ((b ∩ c) ∪ ((a →2 b) ∩ (a →2 c))))
8		leo 158	. . . . . . . . 9 b ≤ (b ∪ (a⊥ ∩ b⊥ ))
9		df-i2 45	. . . . . . . . . 10 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
10	9	ax-r1 35	. . . . . . . . 9 (b ∪ (a⊥ ∩ b⊥ )) = (a →2 b)
11	8, 10	lbtr 139	. . . . . . . 8 b ≤ (a →2 b)
12		leo 158	. . . . . . . . 9 c ≤ (c ∪ (a⊥ ∩ c⊥ ))
13		df-i2 45	. . . . . . . . . 10 (a →2 c) = (c ∪ (a⊥ ∩ c⊥ ))
14	13	ax-r1 35	. . . . . . . . 9 (c ∪ (a⊥ ∩ c⊥ )) = (a →2 c)
15	12, 14	lbtr 139	. . . . . . . 8 c ≤ (a →2 c)
16	11, 15	le2an 169	. . . . . . 7 (b ∩ c) ≤ ((a →2 b) ∩ (a →2 c))
17		id 59	. . . . . . . 8 ((a →2 b) ∩ (a →2 c)) = ((a →2 b) ∩ (a →2 c))
18	17	bile 142	. . . . . . 7 ((a →2 b) ∩ (a →2 c)) ≤ ((a →2 b) ∩ (a →2 c))
19	16, 18	lel2or 170	. . . . . 6 ((b ∩ c) ∪ ((a →2 b) ∩ (a →2 c))) ≤ ((a →2 b) ∩ (a →2 c))
20	19	lelor 166	. . . . 5 ((b ∪ c)⊥ ∪ ((b ∩ c) ∪ ((a →2 b) ∩ (a →2 c)))) ≤ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
21	7, 20	bltr 138	. . . 4 (((b ∩ c) ∪ (b⊥ ∩ c⊥ )) ∪ ((a →2 b) ∩ (a →2 c))) ≤ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
22	1, 21	le2an 169	. . 3 (((a →2 b) ∩ (b →2 a)) ∩ (((b ∩ c) ∪ (b⊥ ∩ c⊥ )) ∪ ((a →2 b) ∩ (a →2 c)))) ≤ ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))))
23		oal2 999	. . 3 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) ≤ (a →2 c)
24	22, 23	letr 137	. 2 (((a →2 b) ∩ (b →2 a)) ∩ (((b ∩ c) ∪ (b⊥ ∩ c⊥ )) ∪ ((a →2 b) ∩ (a →2 c)))) ≤ (a →2 c)
25		u2lembi 721	. . 3 ((a →2 b) ∩ (b →2 a)) = (a ≡ b)
26		dfb 94	. . . . 5 (b ≡ c) = ((b ∩ c) ∪ (b⊥ ∩ c⊥ ))
27	26	ax-r1 35	. . . 4 ((b ∩ c) ∪ (b⊥ ∩ c⊥ )) = (b ≡ c)
28		i2bi 722	. . . . 5 (a →2 b) = (b ∪ (a ≡ b))
29		i2bi 722	. . . . 5 (a →2 c) = (c ∪ (a ≡ c))
30	28, 29	2an 79	. . . 4 ((a →2 b) ∩ (a →2 c)) = ((b ∪ (a ≡ b)) ∩ (c ∪ (a ≡ c)))
31	27, 30	2or 72	. . 3 (((b ∩ c) ∪ (b⊥ ∩ c⊥ )) ∪ ((a →2 b) ∩ (a →2 c))) = ((b ≡ c) ∪ ((b ∪ (a ≡ b)) ∩ (c ∪ (a ≡ c))))
32	25, 31	2an 79	. 2 (((a →2 b) ∩ (b →2 a)) ∩ (((b ∩ c) ∪ (b⊥ ∩ c⊥ )) ∪ ((a →2 b) ∩ (a →2 c)))) = ((a ≡ b) ∩ ((b ≡ c) ∪ ((b ∪ (a ≡ b)) ∩ (c ∪ (a ≡ c)))))
33	24, 32, 29	le3tr2 141	1 ((a ≡ b) ∩ ((b ≡ c) ∪ ((b ∪ (a ≡ b)) ∩ (c ∪ (a ≡ c))))) ≤ (c ∪ (a ≡ c))

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7   →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  ax-3oa 998

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: (None)