Theorem mulcompq 10919

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 10863	. . . 4 ⊢ ((1st ‘𝐴) ·N (1st ‘𝐵)) = ((1st ‘𝐵) ·N (1st ‘𝐴))
2		mulcompi 10863	. . . 4 ⊢ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) = ((2nd ‘𝐵) ·N (2nd ‘𝐴))
3	1, 2	opeq12i 4862	. . 3 ⊢ ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ = ⟨((1st ‘𝐵) ·N (1st ‘𝐴)), ((2nd ‘𝐵) ·N (2nd ‘𝐴))⟩
4		mulpipq2 10906	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
5		mulpipq2 10906	. . . 4 ⊢ ((𝐵 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐵 ·pQ 𝐴) = ⟨((1st ‘𝐵) ·N (1st ‘𝐴)), ((2nd ‘𝐵) ·N (2nd ‘𝐴))⟩)
6	5	ancoms 459	. . 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 2797	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴))
8		mulpqf 10913	. . . 4 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N)
9	8	fdmi 6707	. . 3 ⊢ dom ·pQ = ((N × N) × (N × N))
10	9	ndmovcom 7568	. 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 396   = wceq 1541   ∈ wcel 2106  ⟨cop 4619   × cxp 5658  ‘cfv 6523  (class class class)co 7384  1^st c1st 7946  2^nd c2nd 7947  Ncnpi 10811   ·_N cmi 10813   ·_pQ cmpq 10816

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5283  ax-nul 5290  ax-pr 5411  ax-un 7699

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3372  df-rab 3426  df-v 3468  df-sbc 3765  df-csb 3881  df-dif 3938  df-un 3940  df-in 3942  df-ss 3952  df-pss 3954  df-nul 4310  df-if 4514  df-pw 4589  df-sn 4614  df-pr 4616  df-op 4620  df-uni 4893  df-iun 4983  df-br 5133  df-opab 5195  df-mpt 5216  df-tr 5250  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5615  df-we 5617  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6280  df-ord 6347  df-on 6348  df-lim 6349  df-suc 6350  df-iota 6475  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7387  df-oprab 7388  df-mpo 7389  df-om 7830  df-1st 7948  df-2nd 7949  df-frecs 8239  df-wrecs 8270  df-recs 8344  df-rdg 8383  df-oadd 8443  df-omul 8444  df-ni 10839  df-mi 10841  df-mpq 10876

This theorem is referenced by:  mulcomnq  10920  mulerpq  10924