Theorem mulcompq 10088

Description: Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
mulcompq	⊢ (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴)

Proof of Theorem mulcompq

Step	Hyp	Ref	Expression
1		mulcompi 10032	. . . 4 ⊢ ((1st ‘𝐴) ·N (1st ‘𝐵)) = ((1st ‘𝐵) ·N (1st ‘𝐴))
2		mulcompi 10032	. . . 4 ⊢ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) = ((2nd ‘𝐵) ·N (2nd ‘𝐴))
3	1, 2	opeq12i 4627	. . 3 ⊢ ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ = ⟨((1st ‘𝐵) ·N (1st ‘𝐴)), ((2nd ‘𝐵) ·N (2nd ‘𝐴))⟩
4		mulpipq2 10075	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
5		mulpipq2 10075	. . . 4 ⊢ ((𝐵 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐵 ·pQ 𝐴) = ⟨((1st ‘𝐵) ·N (1st ‘𝐴)), ((2nd ‘𝐵) ·N (2nd ‘𝐴))⟩)
6	5	ancoms 452	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 ·pQ 𝐴) = ⟨((1st ‘𝐵) ·N (1st ‘𝐴)), ((2nd ‘𝐵) ·N (2nd ‘𝐴))⟩)
7	3, 4, 6	3eqtr4a 2886	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴))
8		mulpqf 10082	. . . 4 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N)
9	8	fdmi 6287	. . 3 ⊢ dom ·pQ = ((N × N) × (N × N))
10	9	ndmovcom 7080	. 2 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴))
11	7, 10	pm2.61i 177	1 ⊢ (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴)

Syntax hints:   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ⟨cop 4402   × cxp 5339  ‘cfv 6122  (class class class)co 6904  1^st c1st 7425  2^nd c2nd 7426  Ncnpi 9980   ·_N cmi 9982   ·_pQ cmpq 9985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-reu 3123  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-pss 3813  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-tp 4401  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-tr 4975  df-id 5249  df-eprel 5254  df-po 5262  df-so 5263  df-fr 5300  df-we 5302  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-pred 5919  df-ord 5965  df-on 5966  df-lim 5967  df-suc 5968  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-om 7326  df-1st 7427  df-2nd 7428  df-wrecs 7671  df-recs 7733  df-rdg 7771  df-oadd 7829  df-omul 7830  df-ni 10008  df-mi 10010  df-mpq 10045

This theorem is referenced by:  mulcomnq  10089  mulerpq  10093