Theorem mulcompq 10867

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 10811	. . . 4 ⊢ ((1st ‘𝐴) ·N (1st ‘𝐵)) = ((1st ‘𝐵) ·N (1st ‘𝐴))
2		mulcompi 10811	. . . 4 ⊢ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) = ((2nd ‘𝐵) ·N (2nd ‘𝐴))
3	1, 2	opeq12i 4835	. . 3 ⊢ ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ = ⟨((1st ‘𝐵) ·N (1st ‘𝐴)), ((2nd ‘𝐵) ·N (2nd ‘𝐴))⟩
4		mulpipq2 10854	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
5		mulpipq2 10854	. . . 4 ⊢ ((𝐵 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐵 ·pQ 𝐴) = ⟨((1st ‘𝐵) ·N (1st ‘𝐴)), ((2nd ‘𝐵) ·N (2nd ‘𝐴))⟩)
6	5	ancoms 458	. . 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 2798	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴))
8		mulpqf 10861	. . . 4 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N)
9	8	fdmi 6674	. . 3 ⊢ dom ·pQ = ((N × N) × (N × N))
10	9	ndmovcom 7547	. 2 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴))
11	7, 10	pm2.61i 182	1 ⊢ (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴)

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4587   × cxp 5623  ‘cfv 6493  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  Ncnpi 10759   ·_N cmi 10761   ·_pQ cmpq 10764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-oadd 8403  df-omul 8404  df-ni 10787  df-mi 10789  df-mpq 10824

This theorem is referenced by:  mulcomnq  10868  mulerpq  10872