Theorem mpomulf 11170

Description: Multiplication is an operation on complex numbers. Version of ax-mulf 11155 using maps-to notation, proved from the axioms of set theory and ax-mulcl 11137. (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 2730	. . 3 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))
2		ovex 7423	. . 3 ⊢ (𝑥 · 𝑦) ∈ V
3	1, 2	fnmpoi 8052	. 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 11159	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
7		eleq1a 2824	. . . . . . 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 7503	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦))} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)}
12		df-mpo 7395	. . 3 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 · 𝑦))}
13		dfxp3 8043	. . 3 ⊢ ((ℂ × ℂ) × ℂ) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)}
14	11, 12, 13	3sstr4i 4001	. 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ⊆ ((ℂ × ℂ) × ℂ)
15		dff2 7074	. 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 3917   × cxp 5639   Fn wfn 6509  ⟶wf 6510  (class class class)co 7390  {coprab 7391   ∈ cmpo 7392  ℂcc 11073   · cmul 11080

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714  ax-mulcl 11137

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972

This theorem is referenced by:  mpomulex  12956  cncrng  21307  mpomulcn  24765  mpodvdsmulf1o  27111  fsumdvdsmul  27112