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Theorem mpomulf 11207
Description: Multiplication is an operation on complex numbers. Version of ax-mulf 11192 using maps-to notation, proved from the axioms of set theory and ax-mulcl 11174. (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 2726 . . 3 (π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦)) = (π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦))
2 ovex 7438 . . 3 (π‘₯ Β· 𝑦) ∈ V
31, 2fnmpoi 8055 . 2 (π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦)) Fn (β„‚ Γ— β„‚)
4 simpll 764 . . . . 5 (((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) ∧ 𝑧 = (π‘₯ Β· 𝑦)) β†’ π‘₯ ∈ β„‚)
5 simplr 766 . . . . 5 (((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) ∧ 𝑧 = (π‘₯ Β· 𝑦)) β†’ 𝑦 ∈ β„‚)
6 mulcl 11196 . . . . . . 7 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) β†’ (π‘₯ Β· 𝑦) ∈ β„‚)
7 eleq1a 2822 . . . . . . 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 406 . . . . 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 7515 . . 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 7410 . . 3 (π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦)) = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) ∧ 𝑧 = (π‘₯ Β· 𝑦))}
13 dfxp3 8046 . . 3 ((β„‚ Γ— β„‚) Γ— β„‚) = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚ ∧ 𝑧 ∈ β„‚)}
1411, 12, 133sstr4i 4020 . 2 (π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦)) βŠ† ((β„‚ Γ— β„‚) Γ— β„‚)
15 dff2 7094 . 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 708 1 (π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦)):(β„‚ Γ— β„‚)βŸΆβ„‚
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943   Γ— cxp 5667   Fn wfn 6532  βŸΆwf 6533  (class class class)co 7405  {coprab 7406   ∈ cmpo 7407  β„‚cc 11110   Β· cmul 11117
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722  ax-mulcl 11174
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975
This theorem is referenced by:  mpomulex  12978  cncrng  21277  mpomulcn  24740  mpodvdsmulf1o  27081  fsumdvdsmul  27082
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