Theorem mpoaddf 11294

Description: Addition is an operation on complex numbers. Version of ax-addf 11279 using maps-to notation, proved from the axioms of set theory and ax-addcl 11260. (Contributed by GG, 31-Mar-2025.)

Assertion

Ref	Expression
mpoaddf	⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)):(ℂ × ℂ)⟶ℂ

Distinct variable group:   𝑥,𝑦

Proof of Theorem mpoaddf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2760	. . 3 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))
2		ovex 7453	. . 3 ⊢ (𝑥 + 𝑦) ∈ V
3	1, 2	fnmpoi 8081	. 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) Fn (ℂ × ℂ)
4		simpll 779	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 + 𝑦)) → 𝑥 ∈ ℂ)
5		simplr 781	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 + 𝑦)) → 𝑦 ∈ ℂ)
6		addcl 11282	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
7		eleq1a 2855	. . . . . . 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 412	. . . . 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 7531	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 + 𝑦))} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)}
12		df-mpo 7425	. . 3 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑧 = (𝑥 + 𝑦))}
13		dfxp3 8072	. . 3 ⊢ ((ℂ × ℂ) × ℂ) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)}
14	11, 12, 13	3sstr4i 3959	. 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) ⊆ ((ℂ × ℂ) × ℂ)
15		dff2 7099	. 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 724	1 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)):(ℂ × ℂ)⟶ℂ

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   × cxp 5648   Fn wfn 6533  ⟶wf 6534  (class class class)co 7420  {coprab 7421   ∈ cmpo 7422  ℂcc 11198   + caddc 11203

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751  ax-addcl 11260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002

This theorem is used by:  mpoaddex  13116