Theorem mpomulf 11196

Description: Multiplication is an operation on complex numbers. Version of ax-mulf 11181 using maps-to notation, proved from the axioms of set theory and ax-mulcl 11163. (Contributed by GG, 16-Mar-2025.)

Assertion

Ref	Expression
mpomulf	⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ

Distinct variable group:   𝑥,𝑦

Proof of Theorem mpomulf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))
2		ovex 7445	. . 3 ⊢ (𝑥 · 𝑦) ∈ V
3	1, 2	fnmpoi 8068	. 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) Fn (ℂ × ℂ)
4		simpll 778	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦)) → 𝑥 ∈ ℂ)
5		simplr 780	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦)) → 𝑦 ∈ ℂ)
6		mulcl 11185	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
7		eleq1a 2858	. . . . . . 7 ⊢ ((𝑥 · 𝑦) ∈ ℂ → (𝑧 = (𝑥 · 𝑦) → 𝑧 ∈ ℂ))
8	6, 7	syl 18	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑧 = (𝑥 · 𝑦) → 𝑧 ∈ ℂ))
9	8	imp 411	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦)) → 𝑧 ∈ ℂ)
10	4, 5, 9	3jca 1146	. . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦)) → (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ))
11	10	ssoprab2i 7523	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦))} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)}
12		df-mpo 7417	. . 3 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦))}
13		dfxp3 8059	. . 3 ⊢ ((ℂ × ℂ) × ℂ) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)}
14	11, 12, 13	3sstr4i 3989	. 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ⊆ ((ℂ × ℂ) × ℂ)
15		dff2 7096	. 2 ⊢ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ ↔ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) Fn (ℂ × ℂ) ∧ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ⊆ ((ℂ × ℂ) × ℂ)))
16	3, 14, 15	mpbir2an 723	1 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ⊆ wss 3906   × cxp 5661   Fn wfn 6533  ⟶wf 6534  (class class class)co 7412  {coprab 7413   ∈ cmpo 7414  ℂcc 11099   · cmul 11106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734  ax-mulcl 11163

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988

This theorem is referenced by:  mpomulex  13015  cncrng  21524  mpomulcn  25007  mpodvdsmulf1o  27336  fsumdvdsmul  27337