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Theorem dvcoapbr 13838

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 2177	. . . . 5 ⊢ (𝐽 ↾_t 𝑇) = (𝐽 ↾_t 𝑇)
3		dvcoap.j	. . . . 5 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
4		eqid 2177	. . . . 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 3165	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
10	6, 9	fssd 5374	. . . . 5 ⊢ (𝜑 → 𝐺:𝑌⟶ℂ)
11		dvco.y	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑇)
12	2, 3, 4, 5, 10, 11	eldvap 13818	. . . 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 2890	. . . . . . . . 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 5648	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝑧) ∈ 𝑋)
21	16, 20	ffvelcdmd 5648	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐹‘(𝐺‘𝑧)) ∈ ℂ)
22	5, 10, 11	dvbss 13821	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑇 D 𝐺) ⊆ 𝑌)
23		cnex 7926	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
24	23	a1i 9	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℂ ∈ V)
25	24, 5	ssexd 4140	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ V)
26		elpm2r 6660	. . . . . . . . . . . . 13 ⊢ (((ℂ ∈ V ∧ 𝑇 ∈ V) ∧ (𝐺:𝑌⟶ℂ ∧ 𝑌 ⊆ 𝑇)) → 𝐺 ∈ (ℂ ↑_pm 𝑇))
27	24, 25, 10, 11, 26	syl22anc 1239	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (ℂ ↑_pm 𝑇))
28		reldvg 13815	. . . . . . . . . . . 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 4858	. . . . . . . . . . 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 3156	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑌)
33	32	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝐶 ∈ 𝑌)
34	17, 33	ffvelcdmd 5648	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝐶) ∈ 𝑋)
35	16, 34	ffvelcdmd 5648	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ)
36	21, 35	subcld 8258	. . . . 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 5648	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝑧) ∈ ℂ)
39	37, 33	ffvelcdmd 5648	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝐶) ∈ ℂ)
40	38, 39	subcld 8258	. . . . 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 3156	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝑧) ∈ ℂ)
43	41, 34	sseldd 3156	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝐶) ∈ ℂ)
44		breq1 4003	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (𝑤 # 𝐶 ↔ 𝑧 # 𝐶))
45	44	elrab 2893	. . . . . . . . 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 4003	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → (𝑢 # 𝐶 ↔ 𝑧 # 𝐶))
49		fveq2 5511	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → (𝐺‘𝑢) = (𝐺‘𝑧))
50	49	breq1d 4010	. . . . . . . . 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 2846	. . . . . . 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 8591	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((𝐺‘𝑧) − (𝐺‘𝐶)) # 0)
57	36, 40, 56	divclapd 8736	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) ∈ ℂ)
58	11, 5	sstrd 3165	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
59	10, 58, 32	dvlemap 13816	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ)
60		ssidd 3176	. . . 4 ⊢ (𝜑 → ℂ ⊆ ℂ)
61	3	cntoptopon 13699	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ)
62		txtopon 13429	. . . . . 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 13186	. . . 4 ⊢ (𝐽 ×_t 𝐽) = ((𝐽 ×_t 𝐽) ↾_t (ℂ × ℂ))
65		breq1 4003	. . . . . 6 ⊢ (𝑤 = (𝐺‘𝑧) → (𝑤 # (𝐺‘𝐶) ↔ (𝐺‘𝑧) # (𝐺‘𝐶)))
66	65, 20, 55	elrabd 2895	. . . . 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 2890	. . . . . . . . 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 5648	. . . . . . 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 5648	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝐺‘𝐶) ∈ 𝑋)
74	67, 73	ffvelcdmd 5648	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ)
75	70, 74	subcld 8258	. . . . . 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 3156	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → 𝑦 ∈ ℂ)
78	76, 73	sseldd 3156	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝐺‘𝐶) ∈ ℂ)
79	77, 78	subcld 8258	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝑦 − (𝐺‘𝐶)) ∈ ℂ)
80		breq1 4003	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤 # (𝐺‘𝐶) ↔ 𝑦 # (𝐺‘𝐶)))
81	80	elrab 2893	. . . . . . . . 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 8591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝑦 − (𝐺‘𝐶)) # 0)
85	75, 79, 84	divclapd 8736	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))) ∈ ℂ)
86		limcresi 13802	. . . . . . 7 ⊢ (𝐺 lim_ℂ 𝐶) ⊆ ((𝐺 ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) lim_ℂ 𝐶)
87	6	feqmptd 5565	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)))
88	87	reseq1d 4902	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) = ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}))
89		ssrab2 3240	. . . . . . . . . 10 ⊢ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ⊆ 𝑌
90		resmpt 4951	. . . . . . . . . 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 2226	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) = (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (𝐺‘𝑧)))
93	92	oveq1d 5884	. . . . . . 7 ⊢ (𝜑 → ((𝐺 ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) lim_ℂ 𝐶) = ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
94	86, 93	sseqtrid 3205	. . . . . 6 ⊢ (𝜑 → (𝐺 lim_ℂ 𝐶) ⊆ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
95		eqid 2177	. . . . . . . . . 10 ⊢ (𝐽 ↾_t 𝑌) = (𝐽 ↾_t 𝑌)
96	95, 3	dvcnp2cntop 13830	. . . . . . . . 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 1241	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (((𝐽 ↾_t 𝑌) CnP 𝐽)‘𝐶))
98	3, 95	cnplimccntop 13806	. . . . . . . . 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 3156	. . . . 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 2177	. . . . . . . 8 ⊢ (𝐽 ↾_t 𝑆) = (𝐽 ↾_t 𝑆)
105		eqid 2177	. . . . . . . 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 13818	. . . . . . 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 5511	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑧) → (𝐹‘𝑦) = (𝐹‘(𝐺‘𝑧)))
110	109	oveq1d 5884	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑧) → ((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))))
111		oveq1 5876	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 − (𝐺‘𝐶)) = ((𝐺‘𝑧) − (𝐺‘𝐶)))
112	110, 111	oveq12d 5887	. . . . 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 13814	. . . 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 13721	. . . . 5 ⊢ · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
116	8, 15, 7	dvcl 13819	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐺‘𝐶)(𝑆 D 𝐹)𝐾) → 𝐾 ∈ ℂ)
117	103, 116	mpdan 421	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℂ)
118	5, 10, 11	dvcl 13819	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐿 ∈ ℂ)
119	1, 118	mpdan 421	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℂ)
120	117, 119	opelxpd 4656	. . . . 5 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (ℂ × ℂ))
121	63	toponunii 13182	. . . . . 6 ⊢ (ℂ × ℂ) = ∪ (𝐽 ×_t 𝐽)
122	121	cncnpi 13395	. . . . 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 13813	. . 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 8258	. . . . . . 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 3156	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝑧 ∈ ℂ)
128	126, 33	sseldd 3156	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝐶 ∈ ℂ)
129	127, 128	subcld 8258	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝑧 − 𝐶) ∈ ℂ)
130	127, 128, 47	subap0d 8591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝑧 − 𝐶) # 0)
131	36, 125, 129, 56, 130	dmdcanap2d 8767	. . . . . 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 5583	. . . . . . . . 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 5583	. . . . . . . . 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 5887	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))))
137	136	oveq1d 5884	. . . . . 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 2213	. . . . 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 4090	. . . 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 5884	. . 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 2256	. 2 ⊢ (𝜑 → (𝐾 · 𝐿) ∈ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
142		eqid 2177	. . 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 5377	. . . 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 13818	. 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 944	1 ⊢ (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∀wral 2455 {crab 2459 Vcvv 2737 ⊆ wss 3129 ⟨cop 3594 class class class wbr 4000 ↦ cmpt 4061 × cxp 4621 dom cdm 4623 ↾ cres 4625 ∘ ccom 4627 Rel wrel 4628 ⟶wf 5208 ‘cfv 5212 (class class class)co 5869 ↑_pm cpm 6643 ℂcc 7800 · cmul 7807 − cmin 8118 # cap 8528 / cdiv 8618 abscabs 10990 ↾_t crest 12636 MetOpencmopn 13152 TopOnctopon 13175 intcnt 13260 Cn ccn 13352 CnP ccnp 13353 ×_t ctx 13419 lim_ℂ climc 13790 D cdv 13791

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922 ax-addf 7924 ax-mulf 7925

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-map 6644 df-pm 6645 df-sup 6977 df-inf 6978 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-rp 9641 df-xneg 9759 df-xadd 9760 df-seqfrec 10432 df-exp 10506 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-rest 12638 df-topgen 12657 df-psmet 13154 df-xmet 13155 df-met 13156 df-bl 13157 df-mopn 13158 df-top 13163 df-topon 13176 df-bases 13208 df-ntr 13263 df-cn 13355 df-cnp 13356 df-tx 13420 df-cncf 13725 df-limced 13792 df-dvap 13793

This theorem is referenced by: dvef 13855

W3C validator