Theorem oacom2 1012

Description: Commutation law requiring OA. (Contributed by NM, 19-Nov-1998.)

Hypothesis

Ref	Expression
oacom2.1	d ≤ ((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))))

Assertion

Ref	Expression
oacom2	d C ((a →2 b) ∩ (a →2 c))

Proof of Theorem oacom2

Step	Hyp	Ref	Expression
1		oacom2.1	. . . 4 d ≤ ((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))))
2		lear 161	. . . 4 ((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
3	1, 2	letr 137	. . 3 d ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
4	3	lecom 180	. 2 d C ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
5		lea 160	. . . 4 (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) ≤ d
6		lea 160	. . . . 5 ((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) ≤ (a →2 b)
7	1, 6	letr 137	. . . 4 d ≤ (a →2 b)
8	5, 7	letr 137	. . 3 (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) ≤ (a →2 b)
9	8	lecom 180	. 2 (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) C (a →2 b)
10	4, 9	oacom 1011	1 d C ((a →2 b) ∩ (a →2 c))

Syntax hints:   ≤ wle 2   C wc 3   ∪ wo 6   ∩ wa 7   →₀ wi0 11   →₂ wi2 13

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  ax-3oa 998

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)