Theorem mpomulf 11229

Description: Multiplication is an operation on complex numbers. Version of ax-mulf 11214 using maps-to notation, proved from the axioms of set theory and ax-mulcl 11196. (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 2736	. . 3 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))
2		ovex 7443	. . 3 ⊢ (𝑥 · 𝑦) ∈ V
3	1, 2	fnmpoi 8074	. 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) Fn (ℂ × ℂ)
4		simpll 766	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦)) → 𝑥 ∈ ℂ)
5		simplr 768	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦)) → 𝑦 ∈ ℂ)
6		mulcl 11218	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
7		eleq1a 2830	. . . . . . 7 ⊢ ((𝑥 · 𝑦) ∈ ℂ → (𝑧 = (𝑥 · 𝑦) → 𝑧 ∈ ℂ))
8	6, 7	syl 17	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑧 = (𝑥 · 𝑦) → 𝑧 ∈ ℂ))
9	8	imp 406	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦)) → 𝑧 ∈ ℂ)
10	4, 5, 9	3jca 1128	. . . 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 7415	. . 3 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦))}
13		dfxp3 8065	. . 3 ⊢ ((ℂ × ℂ) × ℂ) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)}
14	11, 12, 13	3sstr4i 4015	. 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ⊆ ((ℂ × ℂ) × ℂ)
15		dff2 7094	. 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 711	1 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3931   × cxp 5657   Fn wfn 6531  ⟶wf 6532  (class class class)co 7410  {coprab 7411   ∈ cmpo 7412  ℂcc 11132   · cmul 11139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734  ax-mulcl 11196

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994

This theorem is referenced by:  mpomulex  13011  cncrng  21356  mpomulcn  24814  mpodvdsmulf1o  27161  fsumdvdsmul  27162