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Theorem oau 909

Description: Transformation lemma for studying the orthoarguesian law.

**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 30	. . 3 (b ∪ (a →₁c)) = ((a →₁c) ∪ b)
2		lea 152	. . . . . . . 8 (a ∩ ((a →₁c) ∪ b)) ≤ a
3		oau.1	. . . . . . . 8 (a ∩ ((a →₁c) ∪ b)) ≤ c
4	2, 3	ler2an 165	. . . . . . 7 (a ∩ ((a →₁c) ∪ b)) ≤ (a ∩ c)
5		u1lemaa 582	. . . . . . . 8 ((a →₁c) ∩ a) = (a ∩ c)
6	5	ax-r1 34	. . . . . . 7 (a ∩ c) = ((a →₁c) ∩ a)
7	4, 6	lbtr 131	. . . . . 6 (a ∩ ((a →₁c) ∪ b)) ≤ ((a →₁c) ∩ a)
8	7	lelor 158	. . . . 5 ((a →₁c) ∪ (a ∩ ((a →₁c) ∪ b))) ≤ ((a →₁c) ∪ ((a →₁c) ∩ a))
9		u1lemc1 662	. . . . . . . 8 a C (a →₁c)
10	9	comcom 435	. . . . . . 7 (a →₁c) C a
11		comorr 176	. . . . . . 7 (a →₁c) C ((a →₁c) ∪ b)
12	10, 11	fh3 453	. . . . . 6 ((a →₁c) ∪ (a ∩ ((a →₁c) ∪ b))) = (((a →₁c) ∪ a) ∩ ((a →₁c) ∪ ((a →₁c) ∪ b)))
13		u1lemoa 602	. . . . . . 7 ((a →₁c) ∪ a) = 1
14		ax-a3 31	. . . . . . . . 9 (((a →₁c) ∪ (a →₁c)) ∪ b) = ((a →₁c) ∪ ((a →₁c) ∪ b))
15	14	ax-r1 34	. . . . . . . 8 ((a →₁c) ∪ ((a →₁c) ∪ b)) = (((a →₁c) ∪ (a →₁c)) ∪ b)
16		oridm 102	. . . . . . . . 9 ((a →₁c) ∪ (a →₁c)) = (a →₁c)
17	16	ax-r5 37	. . . . . . . 8 (((a →₁c) ∪ (a →₁c)) ∪ b) = ((a →₁c) ∪ b)
18	15, 17	ax-r2 35	. . . . . . 7 ((a →₁c) ∪ ((a →₁c) ∪ b)) = ((a →₁c) ∪ b)
19	13, 18	2an 72	. . . . . 6 (((a →₁c) ∪ a) ∩ ((a →₁c) ∪ ((a →₁c) ∪ b))) = (1 ∩ ((a →₁c) ∪ b))
20		ancom 68	. . . . . . 7 (1 ∩ ((a →₁c) ∪ b)) = (((a →₁c) ∪ b) ∩ 1)
21		an1 98	. . . . . . 7 (((a →₁c) ∪ b) ∩ 1) = ((a →₁c) ∪ b)
22	20, 21	ax-r2 35	. . . . . 6 (1 ∩ ((a →₁c) ∪ b)) = ((a →₁c) ∪ b)
23	12, 19, 22	3tr 62	. . . . 5 ((a →₁c) ∪ (a ∩ ((a →₁c) ∪ b))) = ((a →₁c) ∪ b)
24		a5b 112	. . . . 5 ((a →₁c) ∪ ((a →₁c) ∩ a)) = (a →₁c)
25	8, 23, 24	le3tr2 133	. . . 4 ((a →₁c) ∪ b) ≤ (a →₁c)
26		leo 150	. . . 4 (a →₁c) ≤ ((a →₁c) ∪ b)
27	25, 26	lebi 137	. . 3 ((a →₁c) ∪ b) = (a →₁c)
28	1, 27	ax-r2 35	. 2 (b ∪ (a →₁c)) = (a →₁c)
29	28	df-le1 122	1 b ≤ (a →₁c)

Colors of variables: term

Syntax hints: ≤ wle 2 ∪ wo 6 ∩ wa 7 1wt 9 →₁wi1 13

This theorem is referenced by: oa4uto4g 955

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

metamath.org