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Theorem oau 929

Description: Transformation lemma for studying the orthoarguesian law. (Contributed by NM, 28-Dec-1998.)

**Hypothesis**
Ref	Expression
oau.1	(a ∩ ((a →₁c) ∪ b)) ≤ c

**Assertion**
Ref	Expression
oau	b ≤ (a →₁c)

**Proof of Theorem oau**
Step	Hyp	Ref	Expression
1		ax-a2 31	. . 3 (b ∪ (a →₁c)) = ((a →₁c) ∪ b)
2		lea 160	. . . . . . . 8 (a ∩ ((a →₁c) ∪ b)) ≤ a
3		oau.1	. . . . . . . 8 (a ∩ ((a →₁c) ∪ b)) ≤ c
4	2, 3	ler2an 173	. . . . . . 7 (a ∩ ((a →₁c) ∪ b)) ≤ (a ∩ c)
5		u1lemaa 600	. . . . . . . 8 ((a →₁c) ∩ a) = (a ∩ c)
6	5	ax-r1 35	. . . . . . 7 (a ∩ c) = ((a →₁c) ∩ a)
7	4, 6	lbtr 139	. . . . . 6 (a ∩ ((a →₁c) ∪ b)) ≤ ((a →₁c) ∩ a)
8	7	lelor 166	. . . . 5 ((a →₁c) ∪ (a ∩ ((a →₁c) ∪ b))) ≤ ((a →₁c) ∪ ((a →₁c) ∩ a))
9		u1lemc1 680	. . . . . . . 8 a C (a →₁c)
10	9	comcom 453	. . . . . . 7 (a →₁c) C a
11		comorr 184	. . . . . . 7 (a →₁c) C ((a →₁c) ∪ b)
12	10, 11	fh3 471	. . . . . 6 ((a →₁c) ∪ (a ∩ ((a →₁c) ∪ b))) = (((a →₁c) ∪ a) ∩ ((a →₁c) ∪ ((a →₁c) ∪ b)))
13		u1lemoa 620	. . . . . . 7 ((a →₁c) ∪ a) = 1
14		ax-a3 32	. . . . . . . . 9 (((a →₁c) ∪ (a →₁c)) ∪ b) = ((a →₁c) ∪ ((a →₁c) ∪ b))
15	14	ax-r1 35	. . . . . . . 8 ((a →₁c) ∪ ((a →₁c) ∪ b)) = (((a →₁c) ∪ (a →₁c)) ∪ b)
16		oridm 110	. . . . . . . . 9 ((a →₁c) ∪ (a →₁c)) = (a →₁c)
17	16	ax-r5 38	. . . . . . . 8 (((a →₁c) ∪ (a →₁c)) ∪ b) = ((a →₁c) ∪ b)
18	15, 17	ax-r2 36	. . . . . . 7 ((a →₁c) ∪ ((a →₁c) ∪ b)) = ((a →₁c) ∪ b)
19	13, 18	2an 79	. . . . . 6 (((a →₁c) ∪ a) ∩ ((a →₁c) ∪ ((a →₁c) ∪ b))) = (1 ∩ ((a →₁c) ∪ b))
20		ancom 74	. . . . . . 7 (1 ∩ ((a →₁c) ∪ b)) = (((a →₁c) ∪ b) ∩ 1)
21		an1 106	. . . . . . 7 (((a →₁c) ∪ b) ∩ 1) = ((a →₁c) ∪ b)
22	20, 21	ax-r2 36	. . . . . 6 (1 ∩ ((a →₁c) ∪ b)) = ((a →₁c) ∪ b)
23	12, 19, 22	3tr 65	. . . . 5 ((a →₁c) ∪ (a ∩ ((a →₁c) ∪ b))) = ((a →₁c) ∪ b)
24		orabs 120	. . . . 5 ((a →₁c) ∪ ((a →₁c) ∩ a)) = (a →₁c)
25	8, 23, 24	le3tr2 141	. . . 4 ((a →₁c) ∪ b) ≤ (a →₁c)
26		leo 158	. . . 4 (a →₁c) ≤ ((a →₁c) ∪ b)
27	25, 26	lebi 145	. . 3 ((a →₁c) ∪ b) = (a →₁c)
28	1, 27	ax-r2 36	. 2 (b ∪ (a →₁c)) = (a →₁c)
29	28	df-le1 130	1 b ≤ (a →₁c)

Colors of variables: term

Syntax hints: ≤ wle 2 ∪ wo 6 ∩ wa 7 1wt 8 →₁wi1 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: oa4uto4g 975

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