Theorem i3orlem8 541

Description: Lemma for Kalmbach implication OR builder.

Assertion

Ref	Expression
i3orlem8	(((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ ) ≤ ((a →3 b)⊥ ∪ ((a ∪ c) →3 (b ∪ c)))

Proof of Theorem i3orlem8

Step	Hyp	Ref	Expression
1		anass 69	. . . . . 6 (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ ) = ((a ∪ b) ∩ ((a ∪ b⊥ ) ∩ a⊥ ))
2		ancom 68	. . . . . . 7 ((a ∪ b⊥ ) ∩ a⊥ ) = (a⊥ ∩ (a ∪ b⊥ ))
3	2	lan 70	. . . . . 6 ((a ∪ b) ∩ ((a ∪ b⊥ ) ∩ a⊥ )) = ((a ∪ b) ∩ (a⊥ ∩ (a ∪ b⊥ )))
4	1, 3	ax-r2 35	. . . . 5 (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ ) = ((a ∪ b) ∩ (a⊥ ∩ (a ∪ b⊥ )))
5		leor 151	. . . . 5 ((a ∪ b) ∩ (a⊥ ∩ (a ∪ b⊥ ))) ≤ (((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a ∩ b))) ∪ ((a ∪ b) ∩ (a⊥ ∩ (a ∪ b⊥ ))))
6	4, 5	bltr 130	. . . 4 (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ ) ≤ (((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a ∩ b))) ∪ ((a ∪ b) ∩ (a⊥ ∩ (a ∪ b⊥ ))))
7		i3n1 241	. . . . . . 7 (a⊥ →3 b⊥ ) = (((a ∩ b⊥ ) ∪ (a ∩ b)) ∪ (a⊥ ∩ (a ∪ b⊥ )))
8	7	lan 70	. . . . . 6 ((a ∪ b) ∩ (a⊥ →3 b⊥ )) = ((a ∪ b) ∩ (((a ∩ b⊥ ) ∪ (a ∩ b)) ∪ (a⊥ ∩ (a ∪ b⊥ ))))
9		comor1 443	. . . . . . . . 9 (a ∪ b) C a
10		comor2 444	. . . . . . . . . 10 (a ∪ b) C b
11	10	comcom2 175	. . . . . . . . 9 (a ∪ b) C b⊥
12	9, 11	com2an 466	. . . . . . . 8 (a ∪ b) C (a ∩ b⊥ )
13	9, 10	com2an 466	. . . . . . . 8 (a ∪ b) C (a ∩ b)
14	12, 13	com2or 465	. . . . . . 7 (a ∪ b) C ((a ∩ b⊥ ) ∪ (a ∩ b))
15	9	comcom2 175	. . . . . . . 8 (a ∪ b) C a⊥
16	9, 11	com2or 465	. . . . . . . 8 (a ∪ b) C (a ∪ b⊥ )
17	15, 16	com2an 466	. . . . . . 7 (a ∪ b) C (a⊥ ∩ (a ∪ b⊥ ))
18	14, 17	fh1 451	. . . . . 6 ((a ∪ b) ∩ (((a ∩ b⊥ ) ∪ (a ∩ b)) ∪ (a⊥ ∩ (a ∪ b⊥ )))) = (((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a ∩ b))) ∪ ((a ∪ b) ∩ (a⊥ ∩ (a ∪ b⊥ ))))
19	8, 18	ax-r2 35	. . . . 5 ((a ∪ b) ∩ (a⊥ →3 b⊥ )) = (((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a ∩ b))) ∪ ((a ∪ b) ∩ (a⊥ ∩ (a ∪ b⊥ ))))
20	19	ax-r1 34	. . . 4 (((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a ∩ b))) ∪ ((a ∪ b) ∩ (a⊥ ∩ (a ∪ b⊥ )))) = ((a ∪ b) ∩ (a⊥ →3 b⊥ ))
21	6, 20	lbtr 131	. . 3 (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ ) ≤ ((a ∪ b) ∩ (a⊥ →3 b⊥ ))
22	21	ler 141	. 2 (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ ) ≤ (((a ∪ b) ∩ (a⊥ →3 b⊥ )) ∪ ((a ∪ c) →3 (b ∪ c)))
23		i3orlem6 539	. . 3 ((a →3 b)⊥ ∪ ((a ∪ c) →3 (b ∪ c))) = (((a ∪ b) ∩ (a⊥ →3 b⊥ )) ∪ ((a ∪ c) →3 (b ∪ c)))
24	23	ax-r1 34	. 2 (((a ∪ b) ∩ (a⊥ →3 b⊥ )) ∪ ((a ∪ c) →3 (b ∪ c))) = ((a →3 b)⊥ ∪ ((a ∪ c) →3 (b ∪ c)))
25	22, 24	lbtr 131	1 (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ ) ≤ ((a →3 b)⊥ ∪ ((a ∪ c) →3 (b ∪ c)))

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 15

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-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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