Theorem ovmul 35155

Description: Multiplication of complex numbers produces the same value as multiplication expressed in maps-to notation of the same complex numbers. (Contributed by GG, 16-Mar-2025.)

Assertion

Ref	Expression
ovmul	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝐵) = (𝐴 · 𝐵))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem ovmul

Step	Hyp	Ref	Expression
1		mulcl 11193	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
2		oveq12 7417	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 · 𝑦) = (𝐴 · 𝐵))
3		eqidd 2733	. . 3 ⊢ (𝑥 = 𝐴 → ℂ = ℂ)
4		eqid 2732	. . 3 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))
5	2, 3, 4	ovmpox 7560	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 · 𝐵) ∈ ℂ) → (𝐴(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝐵) = (𝐴 · 𝐵))
6	1, 5	mpd3an3 1462	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝐵) = (𝐴 · 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  (class class class)co 7408   ∈ cmpo 7410  ℂcc 11107   · cmul 11114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-mulcl 11171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413

This theorem is referenced by:  gg-expcn  35159  gg-negcncf  35161  gg-dvcnp2  35169  gg-dvmulbr  35170  gg-dvcobr  35171  gg-cmvth  35176  gg-dvfsumle  35177  gg-dvfsumlem2  35178