Theorem u1lem8 758

Description: Lemma used in study of orthoarguesian law.

Assertion

Ref	Expression
u1lem8	((a →1 b) ∩ (a⊥ →1 b)) = ((a ∩ b) ∪ (a⊥ ∩ b))

Proof of Theorem u1lem8

Step	Hyp	Ref	Expression
1		df-i1 43	. . 3 (a →1 b) = (a⊥ ∪ (a ∩ b))
2		df-i1 43	. . . 4 (a⊥ →1 b) = (a⊥ ⊥ ∪ (a⊥ ∩ b))
3		ax-a1 29	. . . . . 6 a = a⊥ ⊥
4	3	ax-r5 37	. . . . 5 (a ∪ (a⊥ ∩ b)) = (a⊥ ⊥ ∪ (a⊥ ∩ b))
5	4	ax-r1 34	. . . 4 (a⊥ ⊥ ∪ (a⊥ ∩ b)) = (a ∪ (a⊥ ∩ b))
6	2, 5	ax-r2 35	. . 3 (a⊥ →1 b) = (a ∪ (a⊥ ∩ b))
7	1, 6	2an 72	. 2 ((a →1 b) ∩ (a⊥ →1 b)) = ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a⊥ ∩ b)))
8		comor1 443	. . . 4 (a ∪ (a⊥ ∩ b)) C a
9	8	comcom2 175	. . 3 (a ∪ (a⊥ ∩ b)) C a⊥
10		coman1 177	. . . . 5 (a ∩ b) C a
11	10	comcom2 175	. . . . . 6 (a ∩ b) C a⊥
12		coman2 178	. . . . . 6 (a ∩ b) C b
13	11, 12	com2an 466	. . . . 5 (a ∩ b) C (a⊥ ∩ b)
14	10, 13	com2or 465	. . . 4 (a ∩ b) C (a ∪ (a⊥ ∩ b))
15	14	comcom 435	. . 3 (a ∪ (a⊥ ∩ b)) C (a ∩ b)
16	9, 15	fh1r 455	. 2 ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a⊥ ∩ b))) = ((a⊥ ∩ (a ∪ (a⊥ ∩ b))) ∪ ((a ∩ b) ∩ (a ∪ (a⊥ ∩ b))))
17		omlan 430	. . . 4 (a⊥ ∩ (a ∪ (a⊥ ∩ b))) = (a⊥ ∩ b)
18		lea 152	. . . . . 6 (a ∩ b) ≤ a
19		leo 150	. . . . . 6 a ≤ (a ∪ (a⊥ ∩ b))
20	18, 19	letr 129	. . . . 5 (a ∩ b) ≤ (a ∪ (a⊥ ∩ b))
21	20	df2le2 128	. . . 4 ((a ∩ b) ∩ (a ∪ (a⊥ ∩ b))) = (a ∩ b)
22	17, 21	2or 67	. . 3 ((a⊥ ∩ (a ∪ (a⊥ ∩ b))) ∪ ((a ∩ b) ∩ (a ∪ (a⊥ ∩ b)))) = ((a⊥ ∩ b) ∪ (a ∩ b))
23		ax-a2 30	. . 3 ((a⊥ ∩ b) ∪ (a ∩ b)) = ((a ∩ b) ∪ (a⊥ ∩ b))
24	22, 23	ax-r2 35	. 2 ((a⊥ ∩ (a ∪ (a⊥ ∩ b))) ∪ ((a ∩ b) ∩ (a ∪ (a⊥ ∩ b)))) = ((a ∩ b) ∪ (a⊥ ∩ b))
25	7, 16, 24	3tr 62	1 ((a →1 b) ∩ (a⊥ →1 b)) = ((a ∩ b) ∪ (a⊥ ∩ b))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13

This theorem is referenced by:  oa4to4u 953  oa4to4u2 954  oa3-u1 971  oa3-u2 972

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-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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