MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulcompq Structured version   Visualization version   GIF version

Theorem mulcompq 10977
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
StepHypRef Expression
1 mulcompi 10921 . . . 4 ((1st𝐴) ·N (1st𝐵)) = ((1st𝐵) ·N (1st𝐴))
2 mulcompi 10921 . . . 4 ((2nd𝐴) ·N (2nd𝐵)) = ((2nd𝐵) ·N (2nd𝐴))
31, 2opeq12i 4880 . . 3 ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ = ⟨((1st𝐵) ·N (1st𝐴)), ((2nd𝐵) ·N (2nd𝐴))⟩
4 mulpipq2 10964 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
5 mulpipq2 10964 . . . 4 ((𝐵 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐵 ·pQ 𝐴) = ⟨((1st𝐵) ·N (1st𝐴)), ((2nd𝐵) ·N (2nd𝐴))⟩)
65ancoms 457 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 ·pQ 𝐴) = ⟨((1st𝐵) ·N (1st𝐴)), ((2nd𝐵) ·N (2nd𝐴))⟩)
73, 4, 63eqtr4a 2791 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴))
8 mulpqf 10971 . . . 4 ·pQ :((N × N) × (N × N))⟶(N × N)
98fdmi 6734 . . 3 dom ·pQ = ((N × N) × (N × N))
109ndmovcom 7608 . 2 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴))
117, 10pm2.61i 182 1 (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wcel 2098  cop 4636   × cxp 5676  cfv 6549  (class class class)co 7419  1st c1st 7992  2nd c2nd 7993  Ncnpi 10869   ·N cmi 10871   ·pQ cmpq 10874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-oadd 8491  df-omul 8492  df-ni 10897  df-mi 10899  df-mpq 10934
This theorem is referenced by:  mulcomnq  10978  mulerpq  10982
  Copyright terms: Public domain W3C validator