Theorem mpomulf 8062

Description: Multiplication is an operation on complex numbers. Version of ax-mulf 8048 using maps-to notation, proved from the axioms of set theory and ax-mulcl 8023. (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		mulcl 8052	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
2	1	rgen2 2592	. . 3 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 · 𝑦) ∈ ℂ
3		eqid 2205	. . . 4 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))
4	3	fnmpo 6288	. . 3 ⊢ (∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 · 𝑦) ∈ ℂ → (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) Fn (ℂ × ℂ))
5	2, 4	ax-mp 5	. 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) Fn (ℂ × ℂ)
6		simpll 527	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦)) → 𝑥 ∈ ℂ)
7		simplr 528	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦)) → 𝑦 ∈ ℂ)
8		eleq1a 2277	. . . . . . 7 ⊢ ((𝑥 · 𝑦) ∈ ℂ → (𝑧 = (𝑥 · 𝑦) → 𝑧 ∈ ℂ))
9	1, 8	syl 14	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑧 = (𝑥 · 𝑦) → 𝑧 ∈ ℂ))
10	9	imp 124	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦)) → 𝑧 ∈ ℂ)
11	6, 7, 10	3jca 1180	. . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦)) → (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ))
12	11	ssoprab2i 6034	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦))} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)}
13		df-mpo 5949	. . 3 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦))}
14		dfxp3 6280	. . 3 ⊢ ((ℂ × ℂ) × ℂ) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)}
15	12, 13, 14	3sstr4i 3234	. 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ⊆ ((ℂ × ℂ) × ℂ)
16		dff2 5724	. 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)))
17	5, 15, 16	mpbir2an 945	1 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 981   = wceq 1373   ∈ wcel 2176  ∀wral 2484   ⊆ wss 3166   × cxp 4673   Fn wfn 5266  ⟶wf 5267  (class class class)co 5944  {coprab 5945   ∈ cmpo 5946  ℂcc 7923   · cmul 7930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-mulcl 8023

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227

This theorem is referenced by:  mpomulcn  15038  mpodvdsmulf1o  15462  fsumdvdsmul  15463