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Theorem dvcoapbr 15294

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 2207	. . . . 5 ⊢ (𝐽 ↾_t 𝑇) = (𝐽 ↾_t 𝑇)
3		dvcoap.j	. . . . 5 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
4		eqid 2207	. . . . 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 3211	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
10	6, 9	fssd 5458	. . . . 5 ⊢ (𝜑 → 𝐺:𝑌⟶ℂ)
11		dvco.y	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑇)
12	2, 3, 4, 5, 10, 11	eldvap 15269	. . . 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 147	. . 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 112	. 2 ⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾_t 𝑇))‘𝑌))
15		dvco.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
16	15	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝐹:𝑋⟶ℂ)
17	6	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝐺:𝑌⟶𝑋)
18		elrabi 2933	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} → 𝑧 ∈ 𝑌)
19	18	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝑧 ∈ 𝑌)
20	17, 19	ffvelcdmd 5739	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝑧) ∈ 𝑋)
21	16, 20	ffvelcdmd 5739	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐹‘(𝐺‘𝑧)) ∈ ℂ)
22	5, 10, 11	dvbss 15272	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑇 D 𝐺) ⊆ 𝑌)
23		cnex 8084	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
24	23	a1i 9	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℂ ∈ V)
25	24, 5	ssexd 4200	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ V)
26		elpm2r 6776	. . . . . . . . . . . . 13 ⊢ (((ℂ ∈ V ∧ 𝑇 ∈ V) ∧ (𝐺:𝑌⟶ℂ ∧ 𝑌 ⊆ 𝑇)) → 𝐺 ∈ (ℂ ↑_pm 𝑇))
27	24, 25, 10, 11, 26	syl22anc 1251	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (ℂ ↑_pm 𝑇))
28		reldvg 15266	. . . . . . . . . . . 12 ⊢ ((𝑇 ⊆ ℂ ∧ 𝐺 ∈ (ℂ ↑_pm 𝑇)) → Rel (𝑇 D 𝐺))
29	5, 27, 28	syl2anc 411	. . . . . . . . . . 11 ⊢ (𝜑 → Rel (𝑇 D 𝐺))
30		releldm 4932	. . . . . . . . . . 11 ⊢ ((Rel (𝑇 D 𝐺) ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐶 ∈ dom (𝑇 D 𝐺))
31	29, 1, 30	syl2anc 411	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ dom (𝑇 D 𝐺))
32	22, 31	sseldd 3202	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑌)
33	32	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝐶 ∈ 𝑌)
34	17, 33	ffvelcdmd 5739	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝐶) ∈ 𝑋)
35	16, 34	ffvelcdmd 5739	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ)
36	21, 35	subcld 8418	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) ∈ ℂ)
37	10	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝐺:𝑌⟶ℂ)
38	37, 19	ffvelcdmd 5739	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝑧) ∈ ℂ)
39	37, 33	ffvelcdmd 5739	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝐶) ∈ ℂ)
40	38, 39	subcld 8418	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((𝐺‘𝑧) − (𝐺‘𝐶)) ∈ ℂ)
41	9	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝑋 ⊆ ℂ)
42	41, 20	sseldd 3202	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝑧) ∈ ℂ)
43	41, 34	sseldd 3202	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝐶) ∈ ℂ)
44		breq1 4062	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (𝑤 # 𝐶 ↔ 𝑧 # 𝐶))
45	44	elrab 2936	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↔ (𝑧 ∈ 𝑌 ∧ 𝑧 # 𝐶))
46	45	simprbi 275	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} → 𝑧 # 𝐶)
47	46	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝑧 # 𝐶)
48		breq1 4062	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → (𝑢 # 𝐶 ↔ 𝑧 # 𝐶))
49		fveq2 5599	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → (𝐺‘𝑢) = (𝐺‘𝑧))
50	49	breq1d 4069	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → ((𝐺‘𝑢) # (𝐺‘𝐶) ↔ (𝐺‘𝑧) # (𝐺‘𝐶)))
51	48, 50	imbi12d 234	. . . . . . . 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 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ∀𝑢 ∈ 𝑌 (𝑢 # 𝐶 → (𝐺‘𝑢) # (𝐺‘𝐶)))
54	51, 53, 19	rspcdva 2889	. . . . . . 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 8752	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((𝐺‘𝑧) − (𝐺‘𝐶)) # 0)
57	36, 40, 56	divclapd 8898	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) ∈ ℂ)
58	11, 5	sstrd 3211	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
59	10, 58, 32	dvlemap 15267	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ)
60		ssidd 3222	. . . 4 ⊢ (𝜑 → ℂ ⊆ ℂ)
61	3	cntoptopon 15119	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ)
62		txtopon 14849	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐽 ∈ (TopOn‘ℂ)) → (𝐽 ×_t 𝐽) ∈ (TopOn‘(ℂ × ℂ)))
63	61, 61, 62	mp2an 426	. . . . 5 ⊢ (𝐽 ×_t 𝐽) ∈ (TopOn‘(ℂ × ℂ))
64	63	toponrestid 14608	. . . 4 ⊢ (𝐽 ×_t 𝐽) = ((𝐽 ×_t 𝐽) ↾_t (ℂ × ℂ))
65		breq1 4062	. . . . . 6 ⊢ (𝑤 = (𝐺‘𝑧) → (𝑤 # (𝐺‘𝐶) ↔ (𝐺‘𝑧) # (𝐺‘𝐶)))
66	65, 20, 55	elrabd 2938	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝑧) ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)})
67	15	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → 𝐹:𝑋⟶ℂ)
68		elrabi 2933	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)} → 𝑦 ∈ 𝑋)
69	68	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → 𝑦 ∈ 𝑋)
70	67, 69	ffvelcdmd 5739	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝐹‘𝑦) ∈ ℂ)
71	6	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → 𝐺:𝑌⟶𝑋)
72	32	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → 𝐶 ∈ 𝑌)
73	71, 72	ffvelcdmd 5739	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝐺‘𝐶) ∈ 𝑋)
74	67, 73	ffvelcdmd 5739	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ)
75	70, 74	subcld 8418	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → ((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) ∈ ℂ)
76	9	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → 𝑋 ⊆ ℂ)
77	76, 69	sseldd 3202	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → 𝑦 ∈ ℂ)
78	76, 73	sseldd 3202	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝐺‘𝐶) ∈ ℂ)
79	77, 78	subcld 8418	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝑦 − (𝐺‘𝐶)) ∈ ℂ)
80		breq1 4062	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤 # (𝐺‘𝐶) ↔ 𝑦 # (𝐺‘𝐶)))
81	80	elrab 2936	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)} ↔ (𝑦 ∈ 𝑋 ∧ 𝑦 # (𝐺‘𝐶)))
82	81	simprbi 275	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)} → 𝑦 # (𝐺‘𝐶))
83	82	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → 𝑦 # (𝐺‘𝐶))
84	77, 78, 83	subap0d 8752	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝑦 − (𝐺‘𝐶)) # 0)
85	75, 79, 84	divclapd 8898	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))) ∈ ℂ)
86		limcresi 15253	. . . . . . 7 ⊢ (𝐺 lim_ℂ 𝐶) ⊆ ((𝐺 ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) lim_ℂ 𝐶)
87	6	feqmptd 5655	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)))
88	87	reseq1d 4977	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) = ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}))
89		ssrab2 3286	. . . . . . . . . 10 ⊢ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ⊆ 𝑌
90		resmpt 5026	. . . . . . . . . 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 2256	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) = (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (𝐺‘𝑧)))
93	92	oveq1d 5982	. . . . . . 7 ⊢ (𝜑 → ((𝐺 ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) lim_ℂ 𝐶) = ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
94	86, 93	sseqtrid 3251	. . . . . 6 ⊢ (𝜑 → (𝐺 lim_ℂ 𝐶) ⊆ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
95		eqid 2207	. . . . . . . . . 10 ⊢ (𝐽 ↾_t 𝑌) = (𝐽 ↾_t 𝑌)
96	95, 3	dvcnp2cntop 15286	. . . . . . . . 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 1253	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (((𝐽 ↾_t 𝑌) CnP 𝐽)‘𝐶))
98	3, 95	cnplimccntop 15257	. . . . . . . . 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 411	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∈ (((𝐽 ↾_t 𝑌) CnP 𝐽)‘𝐶) ↔ (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶))))
100	97, 99	mpbid 147	. . . . . . 7 ⊢ (𝜑 → (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶)))
101	100	simprd 114	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶))
102	94, 101	sseldd 3202	. . . . 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 2207	. . . . . . . 8 ⊢ (𝐽 ↾_t 𝑆) = (𝐽 ↾_t 𝑆)
105		eqid 2207	. . . . . . . 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 15269	. . . . . . 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 147	. . . . . 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 114	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ((𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)} ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) lim_ℂ (𝐺‘𝐶)))
109		fveq2 5599	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑧) → (𝐹‘𝑦) = (𝐹‘(𝐺‘𝑧)))
110	109	oveq1d 5982	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑧) → ((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))))
111		oveq1 5974	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 − (𝐺‘𝐶)) = ((𝐺‘𝑧) − (𝐺‘𝐶)))
112	110, 111	oveq12d 5985	. . . . 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 15265	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) lim_ℂ 𝐶))
114	13	simprd 114	. . . 4 ⊢ (𝜑 → 𝐿 ∈ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
115	3	mulcncntop 15151	. . . . 5 ⊢ · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
116	8, 15, 7	dvcl 15270	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐺‘𝐶)(𝑆 D 𝐹)𝐾) → 𝐾 ∈ ℂ)
117	103, 116	mpdan 421	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℂ)
118	5, 10, 11	dvcl 15270	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐿 ∈ ℂ)
119	1, 118	mpdan 421	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℂ)
120	117, 119	opelxpd 4726	. . . . 5 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (ℂ × ℂ))
121	63	toponunii 14604	. . . . . 6 ⊢ (ℂ × ℂ) = ∪ (𝐽 ×_t 𝐽)
122	121	cncnpi 14815	. . . . 5 ⊢ (( · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽) ∧ ⟨𝐾, 𝐿⟩ ∈ (ℂ × ℂ)) → · ∈ (((𝐽 ×_t 𝐽) CnP 𝐽)‘⟨𝐾, 𝐿⟩))
123	115, 120, 122	sylancr 414	. . . 4 ⊢ (𝜑 → · ∈ (((𝐽 ×_t 𝐽) CnP 𝐽)‘⟨𝐾, 𝐿⟩))
124	57, 59, 60, 60, 3, 64, 113, 114, 123	limccnp2cntop 15264	. . 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 8418	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((𝐺‘𝑧) − (𝐺‘𝐶)) ∈ ℂ)
126	58	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝑌 ⊆ ℂ)
127	126, 19	sseldd 3202	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝑧 ∈ ℂ)
128	126, 33	sseldd 3202	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝐶 ∈ ℂ)
129	127, 128	subcld 8418	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝑧 − 𝐶) ∈ ℂ)
130	127, 128, 47	subap0d 8752	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝑧 − 𝐶) # 0)
131	36, 125, 129, 56, 130	dmdcanap2d 8929	. . . . . 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 5673	. . . . . . . . 9 ⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
133	17, 19, 132	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
134		fvco3 5673	. . . . . . . . 9 ⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝐶 ∈ 𝑌) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
135	17, 33, 134	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
136	133, 135	oveq12d 5985	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))))
137	136	oveq1d 5982	. . . . . 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 2243	. . . . 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 4150	. . . 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 5982	. . 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 2286	. 2 ⊢ (𝜑 → (𝐾 · 𝐿) ∈ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
142		eqid 2207	. . 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 5461	. . . 4 ⊢ ((𝐹:𝑋⟶ℂ ∧ 𝐺:𝑌⟶𝑋) → (𝐹 ∘ 𝐺):𝑌⟶ℂ)
144	15, 6, 143	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝑌⟶ℂ)
145	2, 3, 142, 5, 144, 11	eldvap 15269	. 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 947	1 ⊢ (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1373 ∈ wcel 2178 ∀wral 2486 {crab 2490 Vcvv 2776 ⊆ wss 3174 ⟨cop 3646 class class class wbr 4059 ↦ cmpt 4121 × cxp 4691 dom cdm 4693 ↾ cres 4695 ∘ ccom 4697 Rel wrel 4698 ⟶wf 5286 ‘cfv 5290 (class class class)co 5967 ↑_pm cpm 6759 ℂcc 7958 · cmul 7965 − cmin 8278 # cap 8689 / cdiv 8780 abscabs 11423 ↾_t crest 13186 MetOpencmopn 14418 TopOnctopon 14597 intcnt 14680 Cn ccn 14772 CnP ccnp 14773 ×_t ctx 14839 lim_ℂ climc 15241 D cdv 15242

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4175 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-iinf 4654 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-mulrcl 8059 ax-addcom 8060 ax-mulcom 8061 ax-addass 8062 ax-mulass 8063 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-1rid 8067 ax-0id 8068 ax-rnegex 8069 ax-precex 8070 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-ltwlin 8073 ax-pre-lttrn 8074 ax-pre-apti 8075 ax-pre-ltadd 8076 ax-pre-mulgt0 8077 ax-pre-mulext 8078 ax-arch 8079 ax-caucvg 8080 ax-addf 8082 ax-mulf 8083

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rmo 2494 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-if 3580 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-iun 3943 df-br 4060 df-opab 4122 df-mpt 4123 df-tr 4159 df-id 4358 df-po 4361 df-iso 4362 df-iord 4431 df-on 4433 df-ilim 4434 df-suc 4436 df-iom 4657 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-isom 5299 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-1st 6249 df-2nd 6250 df-recs 6414 df-frec 6500 df-map 6760 df-pm 6761 df-sup 7112 df-inf 7113 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147 df-le 8148 df-sub 8280 df-neg 8281 df-reap 8683 df-ap 8690 df-div 8781 df-inn 9072 df-2 9130 df-3 9131 df-4 9132 df-n0 9331 df-z 9408 df-uz 9684 df-q 9776 df-rp 9811 df-xneg 9929 df-xadd 9930 df-seqfrec 10630 df-exp 10721 df-cj 11268 df-re 11269 df-im 11270 df-rsqrt 11424 df-abs 11425 df-rest 13188 df-topgen 13207 df-psmet 14420 df-xmet 14421 df-met 14422 df-bl 14423 df-mopn 14424 df-top 14585 df-topon 14598 df-bases 14630 df-ntr 14683 df-cn 14775 df-cnp 14776 df-tx 14840 df-cncf 15158 df-limced 15243 df-dvap 15244

This theorem is referenced by: dvef 15314

W3C validator