Theorem u1lem8 776

Description: Lemma used in study of orthoarguesian law. (Contributed by NM, 27-Dec-1998.)

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 44	. . 3 (a →1 b) = (a⊥ ∪ (a ∩ b))
2		df-i1 44	. . . 4 (a⊥ →1 b) = (a⊥ ⊥ ∪ (a⊥ ∩ b))
3		ax-a1 30	. . . . . 6 a = a⊥ ⊥
4	3	ax-r5 38	. . . . 5 (a ∪ (a⊥ ∩ b)) = (a⊥ ⊥ ∪ (a⊥ ∩ b))
5	4	ax-r1 35	. . . 4 (a⊥ ⊥ ∪ (a⊥ ∩ b)) = (a ∪ (a⊥ ∩ b))
6	2, 5	ax-r2 36	. . 3 (a⊥ →1 b) = (a ∪ (a⊥ ∩ b))
7	1, 6	2an 79	. 2 ((a →1 b) ∩ (a⊥ →1 b)) = ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a⊥ ∩ b)))
8		comor1 461	. . . 4 (a ∪ (a⊥ ∩ b)) C a
9	8	comcom2 183	. . 3 (a ∪ (a⊥ ∩ b)) C a⊥
10		coman1 185	. . . . 5 (a ∩ b) C a
11	10	comcom2 183	. . . . . 6 (a ∩ b) C a⊥
12		coman2 186	. . . . . 6 (a ∩ b) C b
13	11, 12	com2an 484	. . . . 5 (a ∩ b) C (a⊥ ∩ b)
14	10, 13	com2or 483	. . . 4 (a ∩ b) C (a ∪ (a⊥ ∩ b))
15	14	comcom 453	. . 3 (a ∪ (a⊥ ∩ b)) C (a ∩ b)
16	9, 15	fh1r 473	. 2 ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a⊥ ∩ b))) = ((a⊥ ∩ (a ∪ (a⊥ ∩ b))) ∪ ((a ∩ b) ∩ (a ∪ (a⊥ ∩ b))))
17		omlan 448	. . . 4 (a⊥ ∩ (a ∪ (a⊥ ∩ b))) = (a⊥ ∩ b)
18		lea 160	. . . . . 6 (a ∩ b) ≤ a
19		leo 158	. . . . . 6 a ≤ (a ∪ (a⊥ ∩ b))
20	18, 19	letr 137	. . . . 5 (a ∩ b) ≤ (a ∪ (a⊥ ∩ b))
21	20	df2le2 136	. . . 4 ((a ∩ b) ∩ (a ∪ (a⊥ ∩ b))) = (a ∩ b)
22	17, 21	2or 72	. . 3 ((a⊥ ∩ (a ∪ (a⊥ ∩ b))) ∪ ((a ∩ b) ∩ (a ∪ (a⊥ ∩ b)))) = ((a⊥ ∩ b) ∪ (a ∩ b))
23		ax-a2 31	. . 3 ((a⊥ ∩ b) ∪ (a ∩ b)) = ((a ∩ b) ∪ (a⊥ ∩ b))
24	22, 23	ax-r2 36	. 2 ((a⊥ ∩ (a ∪ (a⊥ ∩ b))) ∪ ((a ∩ b) ∩ (a ∪ (a⊥ ∩ b)))) = ((a ∩ b) ∪ (a⊥ ∩ b))
25	7, 16, 24	3tr 65	1 ((a →1 b) ∩ (a⊥ →1 b)) = ((a ∩ b) ∪ (a⊥ ∩ b))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12

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-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  oa4to4u  973  oa4to4u2  974  oa3-u1  991  oa3-u2  992