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Theorem dvcoapbr 15933

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 2238	. . . . 5 ⊢ (𝐽 ↾_t 𝑇) = (𝐽 ↾_t 𝑇)
3		dvcoap.j	. . . . 5 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
4		eqid 2238	. . . . 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 3258	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
10	6, 9	fssd 5550	. . . . 5 ⊢ (𝜑 → 𝐺:𝑌⟶ℂ)
11		dvco.y	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑇)
12	2, 3, 4, 5, 10, 11	eldvap 15908	. . . 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 2979	. . . . . . . . 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 5847	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝑧) ∈ 𝑋)
21	16, 20	ffvelcdmd 5847	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐹‘(𝐺‘𝑧)) ∈ ℂ)
22	5, 10, 11	dvbss 15911	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑇 D 𝐺) ⊆ 𝑌)
23		cnex 8309	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
24	23	a1i 9	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℂ ∈ V)
25	24, 5	ssexd 4275	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ V)
26		elpm2r 6944	. . . . . . . . . . . . 13 ⊢ (((ℂ ∈ V ∧ 𝑇 ∈ V) ∧ (𝐺:𝑌⟶ℂ ∧ 𝑌 ⊆ 𝑇)) → 𝐺 ∈ (ℂ ↑_pm 𝑇))
27	24, 25, 10, 11, 26	syl22anc 1279	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (ℂ ↑_pm 𝑇))
28		reldvg 15905	. . . . . . . . . . . 12 ⊢ ((𝑇 ⊆ ℂ ∧ 𝐺 ∈ (ℂ ↑_pm 𝑇)) → Rel (𝑇 D 𝐺))
29	5, 27, 28	syl2anc 415	. . . . . . . . . . 11 ⊢ (𝜑 → Rel (𝑇 D 𝐺))
30		releldm 5019	. . . . . . . . . . 11 ⊢ ((Rel (𝑇 D 𝐺) ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐶 ∈ dom (𝑇 D 𝐺))
31	29, 1, 30	syl2anc 415	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ dom (𝑇 D 𝐺))
32	22, 31	sseldd 3249	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑌)
33	32	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝐶 ∈ 𝑌)
34	17, 33	ffvelcdmd 5847	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝐶) ∈ 𝑋)
35	16, 34	ffvelcdmd 5847	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ)
36	21, 35	subcld 8644	. . . . 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 5847	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝑧) ∈ ℂ)
39	37, 33	ffvelcdmd 5847	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝐶) ∈ ℂ)
40	38, 39	subcld 8644	. . . . 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 3249	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝑧) ∈ ℂ)
43	41, 34	sseldd 3249	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝐺‘𝐶) ∈ ℂ)
44		breq1 4134	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (𝑤 # 𝐶 ↔ 𝑧 # 𝐶))
45	44	elrab 2982	. . . . . . . . 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 4134	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → (𝑢 # 𝐶 ↔ 𝑧 # 𝐶))
49		fveq2 5698	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → (𝐺‘𝑢) = (𝐺‘𝑧))
50	49	breq1d 4141	. . . . . . . . 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 2934	. . . . . . 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 8980	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((𝐺‘𝑧) − (𝐺‘𝐶)) # 0)
57	36, 40, 56	divclapd 9128	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) ∈ ℂ)
58	11, 5	sstrd 3258	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
59	10, 58, 32	dvlemap 15906	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ)
60		ssidd 3269	. . . 4 ⊢ (𝜑 → ℂ ⊆ ℂ)
61	3	cntoptopon 15758	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ)
62		txtopon 15488	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐽 ∈ (TopOn‘ℂ)) → (𝐽 ×_t 𝐽) ∈ (TopOn‘(ℂ × ℂ)))
63	61, 61, 62	mp2an 430	. . . . 5 ⊢ (𝐽 ×_t 𝐽) ∈ (TopOn‘(ℂ × ℂ))
64	63	toponrestid 15247	. . . 4 ⊢ (𝐽 ×_t 𝐽) = ((𝐽 ×_t 𝐽) ↾_t (ℂ × ℂ))
65		breq1 4134	. . . . . 6 ⊢ (𝑤 = (𝐺‘𝑧) → (𝑤 # (𝐺‘𝐶) ↔ (𝐺‘𝑧) # (𝐺‘𝐶)))
66	65, 20, 55	elrabd 2984	. . . . 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 2979	. . . . . . . . 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 5847	. . . . . . 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 5847	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝐺‘𝐶) ∈ 𝑋)
74	67, 73	ffvelcdmd 5847	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ)
75	70, 74	subcld 8644	. . . . . 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 3249	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → 𝑦 ∈ ℂ)
78	76, 73	sseldd 3249	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝐺‘𝐶) ∈ ℂ)
79	77, 78	subcld 8644	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝑦 − (𝐺‘𝐶)) ∈ ℂ)
80		breq1 4134	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤 # (𝐺‘𝐶) ↔ 𝑦 # (𝐺‘𝐶)))
81	80	elrab 2982	. . . . . . . . 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 8980	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (𝑦 − (𝐺‘𝐶)) # 0)
85	75, 79, 84	divclapd 9128	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝑋 ∣ 𝑤 # (𝐺‘𝐶)}) → (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))) ∈ ℂ)
86		limcresi 15892	. . . . . . 7 ⊢ (𝐺 lim_ℂ 𝐶) ⊆ ((𝐺 ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) lim_ℂ 𝐶)
87	6	feqmptd 5759	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)))
88	87	reseq1d 5064	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) = ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}))
89		ssrab2 3333	. . . . . . . . . 10 ⊢ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ⊆ 𝑌
90		resmpt 5114	. . . . . . . . . 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 2287	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) = (𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (𝐺‘𝑧)))
93	92	oveq1d 6103	. . . . . . 7 ⊢ (𝜑 → ((𝐺 ↾ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) lim_ℂ 𝐶) = ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
94	86, 93	sseqtrid 3298	. . . . . 6 ⊢ (𝜑 → (𝐺 lim_ℂ 𝐶) ⊆ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
95		eqid 2238	. . . . . . . . . 10 ⊢ (𝐽 ↾_t 𝑌) = (𝐽 ↾_t 𝑌)
96	95, 3	dvcnp2cntop 15925	. . . . . . . . 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 1281	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (((𝐽 ↾_t 𝑌) CnP 𝐽)‘𝐶))
98	3, 95	cnplimccntop 15896	. . . . . . . . 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 415	. . . . . . . 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 3249	. . . . 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 2238	. . . . . . . 8 ⊢ (𝐽 ↾_t 𝑆) = (𝐽 ↾_t 𝑆)
105		eqid 2238	. . . . . . . 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 15908	. . . . . . 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 5698	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑧) → (𝐹‘𝑦) = (𝐹‘(𝐺‘𝑧)))
110	109	oveq1d 6103	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑧) → ((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))))
111		oveq1 6095	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 − (𝐺‘𝐶)) = ((𝐺‘𝑧) − (𝐺‘𝐶)))
112	110, 111	oveq12d 6106	. . . . 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 15904	. . . 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 15790	. . . . 5 ⊢ · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
116	8, 15, 7	dvcl 15909	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐺‘𝐶)(𝑆 D 𝐹)𝐾) → 𝐾 ∈ ℂ)
117	103, 116	mpdan 425	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℂ)
118	5, 10, 11	dvcl 15909	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐿 ∈ ℂ)
119	1, 118	mpdan 425	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℂ)
120	117, 119	opelxpd 4809	. . . . 5 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (ℂ × ℂ))
121	63	toponunii 15243	. . . . . 6 ⊢ (ℂ × ℂ) = ∪ (𝐽 ×_t 𝐽)
122	121	cncnpi 15454	. . . . 5 ⊢ (( · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽) ∧ ⟨𝐾, 𝐿⟩ ∈ (ℂ × ℂ)) → · ∈ (((𝐽 ×_t 𝐽) CnP 𝐽)‘⟨𝐾, 𝐿⟩))
123	115, 120, 122	sylancr 418	. . . 4 ⊢ (𝜑 → · ∈ (((𝐽 ×_t 𝐽) CnP 𝐽)‘⟨𝐾, 𝐿⟩))
124	57, 59, 60, 60, 3, 64, 113, 114, 123	limccnp2cntop 15903	. . 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 8644	. . . . . . 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 3249	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝑧 ∈ ℂ)
128	126, 33	sseldd 3249	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → 𝐶 ∈ ℂ)
129	127, 128	subcld 8644	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝑧 − 𝐶) ∈ ℂ)
130	127, 128, 47	subap0d 8980	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (𝑧 − 𝐶) # 0)
131	36, 125, 129, 56, 130	dmdcanap2d 9159	. . . . . 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 5779	. . . . . . . . 9 ⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
133	17, 19, 132	syl2anc 415	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
134		fvco3 5779	. . . . . . . . 9 ⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝐶 ∈ 𝑌) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
135	17, 33, 134	syl2anc 415	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
136	133, 135	oveq12d 6106	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶}) → (((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))))
137	136	oveq1d 6103	. . . . . 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 2274	. . . . 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 4223	. . . 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 6103	. . 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 2317	. 2 ⊢ (𝜑 → (𝐾 · 𝐿) ∈ ((𝑧 ∈ {𝑤 ∈ 𝑌 ∣ 𝑤 # 𝐶} ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
142		eqid 2238	. . 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 5555	. . . 4 ⊢ ((𝐹:𝑋⟶ℂ ∧ 𝐺:𝑌⟶𝑋) → (𝐹 ∘ 𝐺):𝑌⟶ℂ)
144	15, 6, 143	syl2anc 415	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝑌⟶ℂ)
145	2, 3, 142, 5, 144, 11	eldvap 15908	. 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 957	1 ⊢ (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1402 ∈ wcel 2209 ∀wral 2528 {crab 2532 Vcvv 2821 ⊆ wss 3220 ⟨cop 3712 class class class wbr 4131 ↦ cmpt 4193 × cxp 4774 dom cdm 4776 ↾ cres 4778 ∘ ccom 4780 Rel wrel 4781 ⟶wf 5376 ‘cfv 5380 (class class class)co 6088 ↑_pm cpm 6927 ℂcc 8183 · cmul 8190 − cmin 8504 # cap 8917 / cdiv 9010 abscabs 11785 ↾_t crest 13652 MetOpencmopn 14987 TopOnctopon 15236 intcnt 15319 Cn ccn 15411 CnP ccnp 15412 ×_t ctx 15478 lim_ℂ climc 15880 D cdv 15881

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305 ax-addf 8307 ax-mulf 8308

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-map 6928 df-pm 6929 df-sup 7330 df-inf 7331 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-n0 9575 df-z 9656 df-uz 9938 df-q 10036 df-rp 10072 df-xneg 10191 df-xadd 10192 df-seqfrec 10906 df-exp 10997 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787 df-rest 13654 df-topgen 13673 df-psmet 14989 df-xmet 14990 df-met 14991 df-bl 14992 df-mopn 14993 df-top 15224 df-topon 15237 df-bases 15269 df-ntr 15322 df-cn 15414 df-cnp 15415 df-tx 15479 df-cncf 15797 df-limced 15882 df-dvap 15883

This theorem is used by: dvef 15953

W3C validator