Theorem fdivval 45885

Description: The quotient of two functions into the complex numbers. (Contributed by AV, 15-May-2020.)

Assertion

Ref	Expression
fdivval	⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 /f 𝐺) = ((𝐹 ∘f / 𝐺) ↾ (𝐺 supp 0)))

Proof of Theorem fdivval

Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fdiv 45884	. . 3 ⊢ /f = (𝑓 ∈ V, 𝑔 ∈ V ↦ ((𝑓 ∘f / 𝑔) ↾ (𝑔 supp 0)))
2	1	a1i 11	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → /f = (𝑓 ∈ V, 𝑔 ∈ V ↦ ((𝑓 ∘f / 𝑔) ↾ (𝑔 supp 0))))
3		oveq12 7284	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 ∘f / 𝑔) = (𝐹 ∘f / 𝐺))
4		oveq1 7282	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 supp 0) = (𝐺 supp 0))
5	4	adantl 482	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑔 supp 0) = (𝐺 supp 0))
6	3, 5	reseq12d 5892	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓 ∘f / 𝑔) ↾ (𝑔 supp 0)) = ((𝐹 ∘f / 𝐺) ↾ (𝐺 supp 0)))
7	6	adantl 482	. 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → ((𝑓 ∘f / 𝑔) ↾ (𝑔 supp 0)) = ((𝐹 ∘f / 𝐺) ↾ (𝐺 supp 0)))
8		elex 3450	. . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
9	8	adantr 481	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐹 ∈ V)
10		elex 3450	. . 3 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
11	10	adantl 482	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ V)
12		funmpt 6472	. . . 4 ⊢ Fun (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥)))
13		offval3 7825	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘f / 𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥))))
14	13	funeqd 6456	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (Fun (𝐹 ∘f / 𝐺) ↔ Fun (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥)))))
15	12, 14	mpbiri 257	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → Fun (𝐹 ∘f / 𝐺))
16		ovex 7308	. . 3 ⊢ (𝐺 supp 0) ∈ V
17		resfunexg 7091	. . 3 ⊢ ((Fun (𝐹 ∘f / 𝐺) ∧ (𝐺 supp 0) ∈ V) → ((𝐹 ∘f / 𝐺) ↾ (𝐺 supp 0)) ∈ V)
18	15, 16, 17	sylancl 586	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘f / 𝐺) ↾ (𝐺 supp 0)) ∈ V)
19	2, 7, 9, 11, 18	ovmpod 7425	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 /f 𝐺) = ((𝐹 ∘f / 𝐺) ↾ (𝐺 supp 0)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∩ cin 3886   ↦ cmpt 5157  dom cdm 5589   ↾ cres 5591  Fun wfun 6427  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277   ∘_f cof 7531   supp csupp 7977  0cc0 10871   / cdiv 11632   /_f cfdiv 45883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-fdiv 45884

This theorem is referenced by:  fdivmpt  45886