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Theorem dvcoapbr 13311

Description: The chain rule for derivatives at a point. The 𝑢 # 𝐶 → (𝐺‘𝑢) # (𝐺‘𝐶) hypothesis constrains what functions work for 𝐺. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 21-Dec-2023.)

**Hypotheses**
Ref	Expression
dvco.f	⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
dvco.x	⊢ (𝜑 → 𝑋 ⊆ 𝑆)
dvco.g	⊢ (𝜑 → 𝐺:𝑌⟶𝑋)
dvco.y	⊢ (𝜑 → 𝑌 ⊆ 𝑇)
dvcoap.gap	⊢ (𝜑 → ∀𝑢 ∈ 𝑌 (𝑢 # 𝐶 → (𝐺‘𝑢) # (𝐺‘𝐶)))
dvcobr.s	⊢ (𝜑 → 𝑆 ⊆ ℂ)
dvcobr.t	⊢ (𝜑 → 𝑇 ⊆ ℂ)
dvco.bf	⊢ (𝜑 → (𝐺‘𝐶)(𝑆 D 𝐹)𝐾)
dvco.bg	⊢ (𝜑 → 𝐶(𝑇 D 𝐺)𝐿)
dvcoap.j	⊢ 𝐽 = (MetOpen‘(abs ∘ − ))

**Assertion**
Ref	Expression
dvcoapbr	⊢ (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿))

Distinct variable groups: 𝑢,𝐶 𝑢,𝐺 𝑢,𝑌 Allowed substitution hints: 𝜑(𝑢) 𝑆(𝑢) 𝑇(𝑢) 𝐹(𝑢) 𝐽(𝑢) 𝐾(𝑢) 𝐿(𝑢) 𝑋(𝑢)

Proof of Theorem dvcoapbr Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvco.bg	. . . 4 ⊢ (𝜑 → 𝐶(𝑇 D 𝐺)𝐿)
2		eqid 2165	. . . . 5 ⊢ (𝐽 ↾_t 𝑇) = (𝐽 ↾_t 𝑇)
3		dvcoap.j	. . . . 5 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
4		eqid 2165	. . . . 5 ⊢ (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))
5		dvcobr.t	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ ℂ)
6		dvco.g	. . . . . 6 ⊢ (𝜑 → 𝐺:𝑌⟶𝑋)
7		dvco.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
8		dvcobr.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
9	7, 8	sstrd 3152	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
10	6, 9	fssd 5350	. . . . 5 ⊢ (𝜑 → 𝐺:𝑌⟶ℂ)
11		dvco.y	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑇)
12	2, 3, 4, 5, 10, 11	eldvap 13291	. . . 4 ⊢ (𝜑 → (𝐶(𝑇 D 𝐺)𝐿 ↔ (𝐶 ∈ ((int‘(𝐽 ↾_t 𝑇))‘𝑌) ∧ 𝐿 ∈ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))))
13	1, 12	mpbid 146	. . 3 ⊢ (𝜑 → (𝐶 ∈ ((int‘(𝐽 ↾_t 𝑇))‘𝑌) ∧ 𝐿 ∈ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶)))
14	13	simpld 111	. 2 ⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾_t 𝑇))‘𝑌))
15		dvco.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
16	15	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝐹:𝑋⟶ℂ)
17	6	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝐺:𝑌⟶𝑋)
18		elrabi 2879	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} → 𝑧 ∈ 𝑌)
19	18	adantl 275	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝑧 ∈ 𝑌)
20	17, 19	ffvelrnd 5621	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝑧) ∈ 𝑋)
21	16, 20	ffvelrnd 5621	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐹‘(𝐺‘𝑧)) ∈ ℂ)
22	5, 10, 11	dvbss 13294	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑇 D 𝐺) ⊆ 𝑌)
23		cnex 7877	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
24	23	a1i 9	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℂ ∈ V)
25	24, 5	ssexd 4122	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ V)
26		elpm2r 6632	. . . . . . . . . . . . 13 ⊢ (((ℂ ∈ V ∧ 𝑇 ∈ V) ∧ (𝐺:𝑌⟶ℂ ∧ 𝑌 ⊆ 𝑇)) → 𝐺 ∈ (ℂ ↑_pm 𝑇))
27	24, 25, 10, 11, 26	syl22anc 1229	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (ℂ ↑_pm 𝑇))
28		reldvg 13288	. . . . . . . . . . . 12 ⊢ ((𝑇 ⊆ ℂ ∧ 𝐺 ∈ (ℂ ↑_pm 𝑇)) → Rel (𝑇 D 𝐺))
29	5, 27, 28	syl2anc 409	. . . . . . . . . . 11 ⊢ (𝜑 → Rel (𝑇 D 𝐺))
30		releldm 4839	. . . . . . . . . . 11 ⊢ ((Rel (𝑇 D 𝐺) ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐶 ∈ dom (𝑇 D 𝐺))
31	29, 1, 30	syl2anc 409	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ dom (𝑇 D 𝐺))
32	22, 31	sseldd 3143	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑌)
33	32	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝐶 ∈ 𝑌)
34	17, 33	ffvelrnd 5621	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝐶) ∈ 𝑋)
35	16, 34	ffvelrnd 5621	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ)
36	21, 35	subcld 8209	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) ∈ ℂ)
37	10	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝐺:𝑌⟶ℂ)
38	37, 19	ffvelrnd 5621	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝑧) ∈ ℂ)
39	37, 33	ffvelrnd 5621	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝐶) ∈ ℂ)
40	38, 39	subcld 8209	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((𝐺‘𝑧) − (𝐺‘𝐶)) ∈ ℂ)
41	9	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝑋 ⊆ ℂ)
42	41, 20	sseldd 3143	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝑧) ∈ ℂ)
43	41, 34	sseldd 3143	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝐶) ∈ ℂ)
44		breq1 3985	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (𝑤 # 𝐶 ↔ 𝑧 # 𝐶))
45	44	elrab 2882	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↔ (𝑧 ∈ 𝑌 ∧ 𝑧 # 𝐶))
46	45	simprbi 273	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} → 𝑧 # 𝐶)
47	46	adantl 275	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝑧 # 𝐶)
48		breq1 3985	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → (𝑢 # 𝐶 ↔ 𝑧 # 𝐶))
49		fveq2 5486	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → (𝐺‘𝑢) = (𝐺‘𝑧))
50	49	breq1d 3992	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → ((𝐺‘𝑢) # (𝐺‘𝐶) ↔ (𝐺‘𝑧) # (𝐺‘𝐶)))
51	48, 50	imbi12d 233	. . . . . . . 8 ⊢ (𝑢 = 𝑧 → ((𝑢 # 𝐶 → (𝐺‘𝑢) # (𝐺‘𝐶)) ↔ (𝑧 # 𝐶 → (𝐺‘𝑧) # (𝐺‘𝐶))))
52		dvcoap.gap	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑢 ∈ 𝑌 (𝑢 # 𝐶 → (𝐺‘𝑢) # (𝐺‘𝐶)))
53	52	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ∀𝑢 ∈ 𝑌 (𝑢 # 𝐶 → (𝐺‘𝑢) # (𝐺‘𝐶)))
54	51, 53, 19	rspcdva 2835	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝑧 # 𝐶 → (𝐺‘𝑧) # (𝐺‘𝐶)))
55	47, 54	mpd 13	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝑧) # (𝐺‘𝐶))
56	42, 43, 55	subap0d 8542	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((𝐺‘𝑧) − (𝐺‘𝐶)) # 0)
57	36, 40, 56	divclapd 8686	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) ∈ ℂ)
58	11, 5	sstrd 3152	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
59	10, 58, 32	dvlemap 13289	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ)
60		ssidd 3163	. . . 4 ⊢ (𝜑 → ℂ ⊆ ℂ)
61	3	cntoptopon 13172	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ)
62		txtopon 12902	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐽 ∈ (TopOn‘ℂ)) → (𝐽 ×_t 𝐽) ∈ (TopOn‘(ℂ × ℂ)))
63	61, 61, 62	mp2an 423	. . . . 5 ⊢ (𝐽 ×_t 𝐽) ∈ (TopOn‘(ℂ × ℂ))
64	63	toponrestid 12659	. . . 4 ⊢ (𝐽 ×_t 𝐽) = ((𝐽 ×_t 𝐽) ↾_t (ℂ × ℂ))
65		breq1 3985	. . . . . 6 ⊢ (𝑤 = (𝐺‘𝑧) → (𝑤 # (𝐺‘𝐶) ↔ (𝐺‘𝑧) # (𝐺‘𝐶)))
66	65, 20, 55	elrabd 2884	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝑧) ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)})
67	15	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → 𝐹:𝑋⟶ℂ)
68		elrabi 2879	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)} → 𝑦 ∈ 𝑋)
69	68	adantl 275	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → 𝑦 ∈ 𝑋)
70	67, 69	ffvelrnd 5621	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝐹‘𝑦) ∈ ℂ)
71	6	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → 𝐺:𝑌⟶𝑋)
72	32	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → 𝐶 ∈ 𝑌)
73	71, 72	ffvelrnd 5621	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝐺‘𝐶) ∈ 𝑋)
74	67, 73	ffvelrnd 5621	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ)
75	70, 74	subcld 8209	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → ((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) ∈ ℂ)
76	9	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → 𝑋 ⊆ ℂ)
77	76, 69	sseldd 3143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → 𝑦 ∈ ℂ)
78	76, 73	sseldd 3143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝐺‘𝐶) ∈ ℂ)
79	77, 78	subcld 8209	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝑦 − (𝐺‘𝐶)) ∈ ℂ)
80		breq1 3985	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤 # (𝐺‘𝐶) ↔ 𝑦 # (𝐺‘𝐶)))
81	80	elrab 2882	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)} ↔ (𝑦 ∈ 𝑋 ∧ 𝑦 # (𝐺‘𝐶)))
82	81	simprbi 273	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)} → 𝑦 # (𝐺‘𝐶))
83	82	adantl 275	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → 𝑦 # (𝐺‘𝐶))
84	77, 78, 83	subap0d 8542	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝑦 − (𝐺‘𝐶)) # 0)
85	75, 79, 84	divclapd 8686	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))) ∈ ℂ)
86		limcresi 13275	. . . . . . 7 ⊢ (𝐺 lim_ℂ 𝐶) ⊆ ((𝐺 ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) lim_ℂ 𝐶)
87	6	feqmptd 5539	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)))
88	87	reseq1d 4883	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) = ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}))
89		ssrab2 3227	. . . . . . . . . 10 ⊢ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ⊆ 𝑌
90		resmpt 4932	. . . . . . . . . 10 ⊢ ({𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ⊆ 𝑌 → ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) = (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (𝐺‘𝑧)))
91	89, 90	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) = (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (𝐺‘𝑧))
92	88, 91	eqtrdi 2215	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) = (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (𝐺‘𝑧)))
93	92	oveq1d 5857	. . . . . . 7 ⊢ (𝜑 → ((𝐺 ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) lim_ℂ 𝐶) = ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
94	86, 93	sseqtrid 3192	. . . . . 6 ⊢ (𝜑 → (𝐺 lim_ℂ 𝐶) ⊆ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
95		eqid 2165	. . . . . . . . . 10 ⊢ (𝐽 ↾_t 𝑌) = (𝐽 ↾_t 𝑌)
96	95, 3	dvcnp2cntop 13303	. . . . . . . . 9 ⊢ (((𝑇 ⊆ ℂ ∧ 𝐺:𝑌⟶ℂ ∧ 𝑌 ⊆ 𝑇) ∧ 𝐶 ∈ dom (𝑇 D 𝐺)) → 𝐺 ∈ (((𝐽 ↾_t 𝑌) CnP 𝐽)‘𝐶))
97	5, 10, 11, 31, 96	syl31anc 1231	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (((𝐽 ↾_t 𝑌) CnP 𝐽)‘𝐶))
98	3, 95	cnplimccntop 13279	. . . . . . . . 9 ⊢ ((𝑌 ⊆ ℂ ∧ 𝐶 ∈ 𝑌) → (𝐺 ∈ (((𝐽 ↾_t 𝑌) CnP 𝐽)‘𝐶) ↔ (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶))))
99	58, 32, 98	syl2anc 409	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∈ (((𝐽 ↾_t 𝑌) CnP 𝐽)‘𝐶) ↔ (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶))))
100	97, 99	mpbid 146	. . . . . . 7 ⊢ (𝜑 → (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶)))
101	100	simprd 113	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶))
102	94, 101	sseldd 3143	. . . . 5 ⊢ (𝜑 → (𝐺‘𝐶) ∈ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
103		dvco.bf	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝐶)(𝑆 D 𝐹)𝐾)
104		eqid 2165	. . . . . . . 8 ⊢ (𝐽 ↾_t 𝑆) = (𝐽 ↾_t 𝑆)
105		eqid 2165	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)} ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) = (𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)} ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))))
106	104, 3, 105, 8, 15, 7	eldvap 13291	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝐶)(𝑆 D 𝐹)𝐾 ↔ ((𝐺‘𝐶) ∈ ((int‘(𝐽 ↾_t 𝑆))‘𝑋) ∧ 𝐾 ∈ ((𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)} ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) lim_ℂ (𝐺‘𝐶)))))
107	103, 106	mpbid 146	. . . . . 6 ⊢ (𝜑 → ((𝐺‘𝐶) ∈ ((int‘(𝐽 ↾_t 𝑆))‘𝑋) ∧ 𝐾 ∈ ((𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)} ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) lim_ℂ (𝐺‘𝐶))))
108	107	simprd 113	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ((𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)} ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) lim_ℂ (𝐺‘𝐶)))
109		fveq2 5486	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑧) → (𝐹‘𝑦) = (𝐹‘(𝐺‘𝑧)))
110	109	oveq1d 5857	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑧) → ((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))))
111		oveq1 5849	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 − (𝐺‘𝐶)) = ((𝐺‘𝑧) − (𝐺‘𝐶)))
112	110, 111	oveq12d 5860	. . . . 5 ⊢ (𝑦 = (𝐺‘𝑧) → (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))
113	66, 85, 102, 108, 112	limccoap 13287	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) lim_ℂ 𝐶))
114	13	simprd 113	. . . 4 ⊢ (𝜑 → 𝐿 ∈ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
115	3	mulcncntop 13194	. . . . 5 ⊢ · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
116	8, 15, 7	dvcl 13292	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐺‘𝐶)(𝑆 D 𝐹)𝐾) → 𝐾 ∈ ℂ)
117	103, 116	mpdan 418	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℂ)
118	5, 10, 11	dvcl 13292	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐿 ∈ ℂ)
119	1, 118	mpdan 418	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℂ)
120	117, 119	opelxpd 4637	. . . . 5 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (ℂ × ℂ))
121	63	toponunii 12655	. . . . . 6 ⊢ (ℂ × ℂ) = ∪ (𝐽 ×_t 𝐽)
122	121	cncnpi 12868	. . . . 5 ⊢ (( · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽) ∧ ⟨𝐾, 𝐿⟩ ∈ (ℂ × ℂ)) → · ∈ (((𝐽 ×_t 𝐽) CnP 𝐽)‘⟨𝐾, 𝐿⟩))
123	115, 120, 122	sylancr 411	. . . 4 ⊢ (𝜑 → · ∈ (((𝐽 ×_t 𝐽) CnP 𝐽)‘⟨𝐾, 𝐿⟩))
124	57, 59, 60, 60, 3, 64, 113, 114, 123	limccnp2cntop 13286	. . 3 ⊢ (𝜑 → (𝐾 · 𝐿) ∈ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ ((((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) lim_ℂ 𝐶))
125	42, 43	subcld 8209	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((𝐺‘𝑧) − (𝐺‘𝐶)) ∈ ℂ)
126	58	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝑌 ⊆ ℂ)
127	126, 19	sseldd 3143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝑧 ∈ ℂ)
128	126, 33	sseldd 3143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝐶 ∈ ℂ)
129	127, 128	subcld 8209	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝑧 − 𝐶) ∈ ℂ)
130	127, 128, 47	subap0d 8542	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝑧 − 𝐶) # 0)
131	36, 125, 129, 56, 130	dmdcanap2d 8717	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)))
132		fvco3 5557	. . . . . . . . 9 ⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
133	17, 19, 132	syl2anc 409	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
134		fvco3 5557	. . . . . . . . 9 ⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝐶 ∈ 𝑌) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
135	17, 33, 134	syl2anc 409	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
136	133, 135	oveq12d 5860	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))))
137	136	oveq1d 5857	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶)) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)))
138	131, 137	eqtr4d 2201	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶)))
139	138	mpteq2dva 4072	. . . 4 ⊢ (𝜑 → (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ ((((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) = (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))))
140	139	oveq1d 5857	. . 3 ⊢ (𝜑 → ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ ((((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) lim_ℂ 𝐶) = ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
141	124, 140	eleqtrd 2245	. 2 ⊢ (𝜑 → (𝐾 · 𝐿) ∈ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
142		eqid 2165	. . 3 ⊢ (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶)))
143		fco 5353	. . . 4 ⊢ ((𝐹:𝑋⟶ℂ ∧ 𝐺:𝑌⟶𝑋) → (𝐹 ∘ 𝐺):𝑌⟶ℂ)
144	15, 6, 143	syl2anc 409	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝑌⟶ℂ)
145	2, 3, 142, 5, 144, 11	eldvap 13291	. 2 ⊢ (𝜑 → (𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿) ↔ (𝐶 ∈ ((int‘(𝐽 ↾_t 𝑇))‘𝑌) ∧ (𝐾 · 𝐿) ∈ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))))
146	14, 141, 145	mpbir2and 934	1 ⊢ (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1343 ∈ wcel 2136 ∀wral 2444 {crab 2448 Vcvv 2726 ⊆ wss 3116 ⟨cop 3579 class class class wbr 3982 ↦ cmpt 4043 × cxp 4602 dom cdm 4604 ↾ cres 4606 ∘ ccom 4608 Rel wrel 4609 ⟶wf 5184 ‘cfv 5188 (class class class)co 5842 ↑_pm cpm 6615 ℂcc 7751 · cmul 7758 − cmin 8069 # cap 8479 / cdiv 8568 abscabs 10939 ↾_t crest 12556 MetOpencmopn 12625 TopOnctopon 12648 intcnt 12733 Cn ccn 12825 CnP ccnp 12826 ×_t ctx 12892 lim_ℂ climc 13263 D cdv 13264

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873 ax-addf 7875 ax-mulf 7876

This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-isom 5197 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-map 6616 df-pm 6617 df-sup 6949 df-inf 6950 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-xneg 9708 df-xadd 9709 df-seqfrec 10381 df-exp 10455 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-rest 12558 df-topgen 12577 df-psmet 12627 df-xmet 12628 df-met 12629 df-bl 12630 df-mopn 12631 df-top 12636 df-topon 12649 df-bases 12681 df-ntr 12736 df-cn 12828 df-cnp 12829 df-tx 12893 df-cncf 13198 df-limced 13265 df-dvap 13266

This theorem is referenced by: dvef 13328

W3C validator