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Theorem mpomulf 11241
Description: Multiplication is an operation on complex numbers. Version of ax-mulf 11226 using maps-to notation, proved from the axioms of set theory and ax-mulcl 11208. (Contributed by GG, 16-Mar-2025.)
Assertion
Ref Expression
mpomulf (π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦)):(β„‚ Γ— β„‚)βŸΆβ„‚
Distinct variable group:   π‘₯,𝑦

Proof of Theorem mpomulf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqid 2728 . . 3 (π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦)) = (π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦))
2 ovex 7459 . . 3 (π‘₯ Β· 𝑦) ∈ V
31, 2fnmpoi 8080 . 2 (π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦)) Fn (β„‚ Γ— β„‚)
4 simpll 765 . . . . 5 (((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) ∧ 𝑧 = (π‘₯ Β· 𝑦)) β†’ π‘₯ ∈ β„‚)
5 simplr 767 . . . . 5 (((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) ∧ 𝑧 = (π‘₯ Β· 𝑦)) β†’ 𝑦 ∈ β„‚)
6 mulcl 11230 . . . . . . 7 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) β†’ (π‘₯ Β· 𝑦) ∈ β„‚)
7 eleq1a 2824 . . . . . . 7 ((π‘₯ Β· 𝑦) ∈ β„‚ β†’ (𝑧 = (π‘₯ Β· 𝑦) β†’ 𝑧 ∈ β„‚))
86, 7syl 17 . . . . . 6 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) β†’ (𝑧 = (π‘₯ Β· 𝑦) β†’ 𝑧 ∈ β„‚))
98imp 405 . . . . 5 (((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) ∧ 𝑧 = (π‘₯ Β· 𝑦)) β†’ 𝑧 ∈ β„‚)
104, 5, 93jca 1125 . . . 4 (((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) ∧ 𝑧 = (π‘₯ Β· 𝑦)) β†’ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚ ∧ 𝑧 ∈ β„‚))
1110ssoprab2i 7537 . . 3 {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) ∧ 𝑧 = (π‘₯ Β· 𝑦))} βŠ† {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚ ∧ 𝑧 ∈ β„‚)}
12 df-mpo 7431 . . 3 (π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦)) = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) ∧ 𝑧 = (π‘₯ Β· 𝑦))}
13 dfxp3 8071 . . 3 ((β„‚ Γ— β„‚) Γ— β„‚) = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚ ∧ 𝑧 ∈ β„‚)}
1411, 12, 133sstr4i 4025 . 2 (π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦)) βŠ† ((β„‚ Γ— β„‚) Γ— β„‚)
15 dff2 7114 . 2 ((π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦)):(β„‚ Γ— β„‚)βŸΆβ„‚ ↔ ((π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦)) Fn (β„‚ Γ— β„‚) ∧ (π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦)) βŠ† ((β„‚ Γ— β„‚) Γ— β„‚)))
163, 14, 15mpbir2an 709 1 (π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦)):(β„‚ Γ— β„‚)βŸΆβ„‚
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3949   Γ— cxp 5680   Fn wfn 6548  βŸΆwf 6549  (class class class)co 7426  {coprab 7427   ∈ cmpo 7428  β„‚cc 11144   Β· cmul 11151
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746  ax-mulcl 11208
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000
This theorem is referenced by:  mpomulex  13012  cncrng  21323  mpomulcn  24805  mpodvdsmulf1o  27146  fsumdvdsmul  27147
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