Theorem 3vcom 813

Description: 3-variable commutation theorem. (Contributed by NM, 19-Mar-1999.)

Assertion

Ref	Expression
3vcom	((a →1 c) ∪ (b →1 c)) C ((a⊥ →1 c) ∩ (b⊥ →1 c))

Proof of Theorem 3vcom

Step	Hyp	Ref	Expression
1		oran3 93	. . . . 5 ((a⊥ →1 c)⊥ ∪ (b⊥ →1 c)⊥ ) = ((a⊥ →1 c) ∩ (b⊥ →1 c))⊥
2	1	ax-r1 35	. . . 4 ((a⊥ →1 c) ∩ (b⊥ →1 c))⊥ = ((a⊥ →1 c)⊥ ∪ (b⊥ →1 c)⊥ )
3		u1lem9ab 779	. . . . . 6 (a⊥ →1 c)⊥ ≤ (a →1 c)
4		u1lem9ab 779	. . . . . 6 (b⊥ →1 c)⊥ ≤ (b →1 c)
5	3, 4	le2or 168	. . . . 5 ((a⊥ →1 c)⊥ ∪ (b⊥ →1 c)⊥ ) ≤ ((a →1 c) ∪ (b →1 c))
6	5	lecom 180	. . . 4 ((a⊥ →1 c)⊥ ∪ (b⊥ →1 c)⊥ ) C ((a →1 c) ∪ (b →1 c))
7	2, 6	bctr 181	. . 3 ((a⊥ →1 c) ∩ (b⊥ →1 c))⊥ C ((a →1 c) ∪ (b →1 c))
8	7	comcom6 459	. 2 ((a⊥ →1 c) ∩ (b⊥ →1 c)) C ((a →1 c) ∪ (b →1 c))
9	8	comcom 453	1 ((a →1 c) ∪ (b →1 c)) C ((a⊥ →1 c) ∩ (b⊥ →1 c))

Syntax hints:   C wc 3  ^⊥ 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-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: (None)