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Theorem dvcobr 26105

Description: The chain rule for derivatives at a point. For the (simpler but more limited) function version, see dvco 26106. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) Avoid ax-mulf 11175 and remove unnecessary hypotheses. (Revised by GG, 16-Mar-2025.)

**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)
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)
dvco.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)

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

Proof of Theorem dvcobr Dummy variables 𝑦 𝑧 𝑢 𝑣 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 2763	. . . . 5 ⊢ (𝐽 ↾_t 𝑇) = (𝐽 ↾_t 𝑇)
3		dvco.j	. . . . 5 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
4		eqid 2763	. . . . 5 ⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ (𝑌 ∖ {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 3947	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
10	6, 9	fssd 6723	. . . . 5 ⊢ (𝜑 → 𝐺:𝑌⟶ℂ)
11		dvco.y	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑇)
12	2, 3, 4, 5, 10, 11	eldv 26057	. . . 4 ⊢ (𝜑 → (𝐶(𝑇 D 𝐺)𝐿 ↔ (𝐶 ∈ ((int‘(𝐽 ↾_t 𝑇))‘𝑌) ∧ 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))))
13	1, 12	mpbid 235	. . 3 ⊢ (𝜑 → (𝐶 ∈ ((int‘(𝐽 ↾_t 𝑇))‘𝑌) ∧ 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶)))
14	13	simpld 499	. 2 ⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾_t 𝑇))‘𝑌))
15		dvco.bf	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝐶)(𝑆 D 𝐹)𝐾)
16		dvco.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
17	8, 16, 7	dvcl 26058	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺‘𝐶)(𝑆 D 𝐹)𝐾) → 𝐾 ∈ ℂ)
18	15, 17	mpdan 699	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℂ)
19	18	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝐾 ∈ ℂ)
20	16	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐹:𝑋⟶ℂ)
21		eldifi 4085	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) → 𝑧 ∈ 𝑌)
22		ffvelcdm 7076	. . . . . . . . . . . 12 ⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → (𝐺‘𝑧) ∈ 𝑋)
23	6, 21, 22	syl2an 607	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝑧) ∈ 𝑋)
24	20, 23	ffvelcdmd 7080	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐹‘(𝐺‘𝑧)) ∈ ℂ)
25	24	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐹‘(𝐺‘𝑧)) ∈ ℂ)
26	6	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐺:𝑌⟶𝑋)
27	5, 10, 11	dvbss 26060	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝑇 D 𝐺) ⊆ 𝑌)
28		reldv 26029	. . . . . . . . . . . . . . 15 ⊢ Rel (𝑇 D 𝐺)
29		releldm 5934	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝑇 D 𝐺) ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐶 ∈ dom (𝑇 D 𝐺))
30	28, 1, 29	sylancr 598	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ dom (𝑇 D 𝐺))
31	27, 30	sseldd 3938	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ 𝑌)
32	31	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐶 ∈ 𝑌)
33	26, 32	ffvelcdmd 7080	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝐶) ∈ 𝑋)
34	20, 33	ffvelcdmd 7080	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ)
35	34	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ)
36	25, 35	subcld 11564	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) ∈ ℂ)
37	10	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝐺:𝑌⟶ℂ)
38	21	ad2antlr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝑧 ∈ 𝑌)
39	37, 38	ffvelcdmd 7080	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐺‘𝑧) ∈ ℂ)
40	31	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝐶 ∈ 𝑌)
41	37, 40	ffvelcdmd 7080	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐺‘𝐶) ∈ ℂ)
42	39, 41	subcld 11564	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐺‘𝑧) − (𝐺‘𝐶)) ∈ ℂ)
43		simpr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ¬ (𝐺‘𝑧) = (𝐺‘𝐶))
44	39, 41	subeq0ad 11574	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) = 0 ↔ (𝐺‘𝑧) = (𝐺‘𝐶)))
45	44	necon3abid 2994	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) ≠ 0 ↔ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)))
46	43, 45	mpbird 260	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐺‘𝑧) − (𝐺‘𝐶)) ≠ 0)
47	36, 42, 46	divcld 11986	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) ∈ ℂ)
48	19, 47	ifclda 4523	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) ∈ ℂ)
49	11, 5	sstrd 3947	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
50	10, 49, 31	dvlem 26055	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ)
51		ssidd 3960	. . . . . 6 ⊢ (𝜑 → ℂ ⊆ ℂ)
52	3	cnfldtopon 24939	. . . . . . . 8 ⊢ 𝐽 ∈ (TopOn‘ℂ)
53		txtopon 23748	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐽 ∈ (TopOn‘ℂ)) → (𝐽 ×_t 𝐽) ∈ (TopOn‘(ℂ × ℂ)))
54	52, 52, 53	mp2an 704	. . . . . . 7 ⊢ (𝐽 ×_t 𝐽) ∈ (TopOn‘(ℂ × ℂ))
55	54	toponrestid 23078	. . . . . 6 ⊢ (𝐽 ×_t 𝐽) = ((𝐽 ×_t 𝐽) ↾_t (ℂ × ℂ))
56	23	anim1i 626	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶)) → ((𝐺‘𝑧) ∈ 𝑋 ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶)))
57		eldifsn 4753	. . . . . . . . 9 ⊢ ((𝐺‘𝑧) ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↔ ((𝐺‘𝑧) ∈ 𝑋 ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶)))
58	56, 57	sylibr 237	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶)) → (𝐺‘𝑧) ∈ (𝑋 ∖ {(𝐺‘𝐶)}))
59	58	anasss 471	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶))) → (𝐺‘𝑧) ∈ (𝑋 ∖ {(𝐺‘𝐶)}))
60		eldifsni 4758	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) → 𝑦 ≠ (𝐺‘𝐶))
61		ifnefalse 4499	. . . . . . . . . 10 ⊢ (𝑦 ≠ (𝐺‘𝐶) → if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) = (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))))
62	60, 61	syl 18	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) → if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) = (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))))
63	62	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)})) → if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) = (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))))
64	6, 31	ffvelcdmd 7080	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝐶) ∈ 𝑋)
65	16, 9, 64	dvlem 26055	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)})) → (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))) ∈ ℂ)
66	63, 65	eqeltrd 2863	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)})) → if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) ∈ ℂ)
67		limcresi 26044	. . . . . . . . 9 ⊢ (𝐺 lim_ℂ 𝐶) ⊆ ((𝐺 ↾ (𝑌 ∖ {𝐶})) lim_ℂ 𝐶)
68	6	feqmptd 6949	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)))
69	68	reseq1d 5977	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ↾ (𝑌 ∖ {𝐶})) = ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ (𝑌 ∖ {𝐶})))
70		difss 4090	. . . . . . . . . . . 12 ⊢ (𝑌 ∖ {𝐶}) ⊆ 𝑌
71		resmpt 6039	. . . . . . . . . . . 12 ⊢ ((𝑌 ∖ {𝐶}) ⊆ 𝑌 → ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ (𝑌 ∖ {𝐶})) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)))
72	70, 71	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ (𝑌 ∖ {𝐶})) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧))
73	69, 72	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ↾ (𝑌 ∖ {𝐶})) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)))
74	73	oveq1d 7425	. . . . . . . . 9 ⊢ (𝜑 → ((𝐺 ↾ (𝑌 ∖ {𝐶})) lim_ℂ 𝐶) = ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
75	67, 74	sseqtrid 3979	. . . . . . . 8 ⊢ (𝜑 → (𝐺 lim_ℂ 𝐶) ⊆ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
76		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝐽 ↾_t 𝑌) = (𝐽 ↾_t 𝑌)
77	76, 3	dvcnp2 26079	. . . . . . . . . . 11 ⊢ (((𝑇 ⊆ ℂ ∧ 𝐺:𝑌⟶ℂ ∧ 𝑌 ⊆ 𝑇) ∧ 𝐶 ∈ dom (𝑇 D 𝐺)) → 𝐺 ∈ (((𝐽 ↾_t 𝑌) CnP 𝐽)‘𝐶))
78	5, 10, 11, 30, 77	syl31anc 1400	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (((𝐽 ↾_t 𝑌) CnP 𝐽)‘𝐶))
79	3, 76	cnplimc 26046	. . . . . . . . . . 11 ⊢ ((𝑌 ⊆ ℂ ∧ 𝐶 ∈ 𝑌) → (𝐺 ∈ (((𝐽 ↾_t 𝑌) CnP 𝐽)‘𝐶) ↔ (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶))))
80	49, 31, 79	syl2anc 595	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ∈ (((𝐽 ↾_t 𝑌) CnP 𝐽)‘𝐶) ↔ (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶))))
81	78, 80	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶)))
82	81	simprd 500	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶))
83	75, 82	sseldd 3938	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝐶) ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
84		eqid 2763	. . . . . . . . . . 11 ⊢ (𝐽 ↾_t 𝑆) = (𝐽 ↾_t 𝑆)
85		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) = (𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))))
86	84, 3, 85, 8, 16, 7	eldv 26057	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐺‘𝐶)(𝑆 D 𝐹)𝐾 ↔ ((𝐺‘𝐶) ∈ ((int‘(𝐽 ↾_t 𝑆))‘𝑋) ∧ 𝐾 ∈ ((𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) lim_ℂ (𝐺‘𝐶)))))
87	15, 86	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → ((𝐺‘𝐶) ∈ ((int‘(𝐽 ↾_t 𝑆))‘𝑋) ∧ 𝐾 ∈ ((𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) lim_ℂ (𝐺‘𝐶))))
88	87	simprd 500	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ((𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) lim_ℂ (𝐺‘𝐶)))
89	62	mpteq2ia 5206	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))))) = (𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))))
90	89	oveq1i 7420	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))))) lim_ℂ (𝐺‘𝐶)) = ((𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) lim_ℂ (𝐺‘𝐶))
91	88, 90	eleqtrrdi 2874	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ((𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))))) lim_ℂ (𝐺‘𝐶)))
92		eqeq1 2767	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 = (𝐺‘𝐶) ↔ (𝐺‘𝑧) = (𝐺‘𝐶)))
93		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (𝐹‘𝑦) = (𝐹‘(𝐺‘𝑧)))
94	93	oveq1d 7425	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑧) → ((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))))
95		oveq1 7417	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 − (𝐺‘𝐶)) = ((𝐺‘𝑧) − (𝐺‘𝐶)))
96	94, 95	oveq12d 7428	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑧) → (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))
97	92, 96	ifbieq2d 4514	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑧) → if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) = if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))))
98		iftrue 4493	. . . . . . . 8 ⊢ ((𝐺‘𝑧) = (𝐺‘𝐶) → if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) = 𝐾)
99	98	ad2antll 741	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ (𝐺‘𝑧) = (𝐺‘𝐶))) → if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) = 𝐾)
100	59, 66, 83, 91, 97, 99	limcco 26052	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))) lim_ℂ 𝐶))
101	13	simprd 500	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
102	3	mpomulcn 25026	. . . . . . 7 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
103	5, 10, 11	dvcl 26058	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐿 ∈ ℂ)
104	1, 103	mpdan 699	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ ℂ)
105	18, 104	opelxpd 5700	. . . . . . 7 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (ℂ × ℂ))
106	54	toponunii 23073	. . . . . . . 8 ⊢ (ℂ × ℂ) = ∪ (𝐽 ×_t 𝐽)
107	106	cncnpi 23435	. . . . . . 7 ⊢ (((𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽) ∧ ⟨𝐾, 𝐿⟩ ∈ (ℂ × ℂ)) → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ (((𝐽 ×_t 𝐽) CnP 𝐽)‘⟨𝐾, 𝐿⟩))
108	102, 105, 107	sylancr 598	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ (((𝐽 ×_t 𝐽) CnP 𝐽)‘⟨𝐾, 𝐿⟩))
109	48, 50, 51, 51, 3, 55, 100, 101, 108	limccnp2 26051	. . . . 5 ⊢ (𝜑 → (𝐾(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝐿) ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) lim_ℂ 𝐶))
110		df-mpt 5193	. . . . . 6 ⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))}
111	110	oveq1i 7420	. . . . 5 ⊢ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) lim_ℂ 𝐶) = ({⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))} lim_ℂ 𝐶)
112	109, 111	eleqtrdi 2873	. . . 4 ⊢ (𝜑 → (𝐾(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝐿) ∈ ({⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))} lim_ℂ 𝐶))
113		ovmpot 7571	. . . . 5 ⊢ ((𝐾 ∈ ℂ ∧ 𝐿 ∈ ℂ) → (𝐾(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝐿) = (𝐾 · 𝐿))
114	18, 104, 113	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝐾(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝐿) = (𝐾 · 𝐿))
115		ovmpot 7571	. . . . . . . . 9 ⊢ ((if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) ∈ ℂ ∧ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ) → (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))
116	48, 50, 115	syl2anc 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))
117	116	eqeq2d 2774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) ↔ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))))
118	117	pm5.32da 589	. . . . . 6 ⊢ (𝜑 → ((𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) ↔ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))))
119	118	opabbidv 5177	. . . . 5 ⊢ (𝜑 → {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))} = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))})
120		df-mpt 5193	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))}
121	120	eqcomi 2772	. . . . . . . 8 ⊢ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))} = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))
122	121	eqeq2i 2776	. . . . . . 7 ⊢ ({⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))} = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))} ↔ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))} = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))))
123	122	biimpi 219	. . . . . 6 ⊢ ({⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))} = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))} → {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))} = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))))
124	123	oveq1d 7425	. . . . 5 ⊢ ({⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))} = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))} → ({⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))} lim_ℂ 𝐶) = ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) lim_ℂ 𝐶))
125	119, 124	syl 18	. . . 4 ⊢ (𝜑 → ({⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ 𝑤 = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))} lim_ℂ 𝐶) = ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) lim_ℂ 𝐶))
126	112, 114, 125	3eltr3d 2877	. . 3 ⊢ (𝜑 → (𝐾 · 𝐿) ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) lim_ℂ 𝐶))
127		oveq1 7417	. . . . . . . 8 ⊢ (𝐾 = if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) → (𝐾 · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))
128	127	eqeq1d 2765	. . . . . . 7 ⊢ (𝐾 = if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) → ((𝐾 · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)) ↔ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶))))
129		oveq1 7417	. . . . . . . 8 ⊢ ((((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) = if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) → ((((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))
130	129	eqeq1d 2765	. . . . . . 7 ⊢ ((((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) = if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘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)) ↔ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶))))
131	19	mul01d 11404	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐾 · 0) = 0)
132	9	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑋 ⊆ ℂ)
133	132, 23	sseldd 3938	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝑧) ∈ ℂ)
134	132, 33	sseldd 3938	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝐶) ∈ ℂ)
135	133, 134	subeq0ad 11574	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) = 0 ↔ (𝐺‘𝑧) = (𝐺‘𝐶)))
136	135	biimpar 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐺‘𝑧) − (𝐺‘𝐶)) = 0)
137	136	oveq1d 7425	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) = (0 / (𝑧 − 𝐶)))
138	49	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑌 ⊆ ℂ)
139	21	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑧 ∈ 𝑌)
140	138, 139	sseldd 3938	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑧 ∈ ℂ)
141	138, 32	sseldd 3938	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐶 ∈ ℂ)
142	140, 141	subcld 11564	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝑧 − 𝐶) ∈ ℂ)
143		eldifsni 4758	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) → 𝑧 ≠ 𝐶)
144	143	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑧 ≠ 𝐶)
145	140, 141, 144	subne0d 11573	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝑧 − 𝐶) ≠ 0)
146	142, 145	div0d 11985	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (0 / (𝑧 − 𝐶)) = 0)
147	146	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (0 / (𝑧 − 𝐶)) = 0)
148	137, 147	eqtrd 2798	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) = 0)
149	148	oveq2d 7426	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐾 · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (𝐾 · 0))
150		fveq2 6881	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑧) = (𝐺‘𝐶) → (𝐹‘(𝐺‘𝑧)) = (𝐹‘(𝐺‘𝐶)))
151	24, 34	subeq0ad 11574	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) = 0 ↔ (𝐹‘(𝐺‘𝑧)) = (𝐹‘(𝐺‘𝐶))))
152	150, 151	imbitrrid 249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((𝐺‘𝑧) = (𝐺‘𝐶) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) = 0))
153	152	imp 411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) = 0)
154	153	oveq1d 7425	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)) = (0 / (𝑧 − 𝐶)))
155	154, 147	eqtrd 2798	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)) = 0)
156	131, 149, 155	3eqtr4d 2808	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐾 · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)))
157	142	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝑧 − 𝐶) ∈ ℂ)
158	145	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝑧 − 𝐶) ≠ 0)
159	36, 42, 157, 46, 158	dmdcan2d 12016	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘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)))
160	128, 130, 156, 159	ifbothda 4526	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)))
161		fvco3 6981	. . . . . . . . 9 ⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
162	6, 21, 161	syl2an 607	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
163	6, 31	fvco3d 6982	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
164	163	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
165	162, 164	oveq12d 7428	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))))
166	165	oveq1d 7425	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶)) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)))
167	160, 166	eqtr4d 2801	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶)))
168	167	mpteq2dva 5204	. . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))))
169	168	oveq1d 7425	. . 3 ⊢ (𝜑 → ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) lim_ℂ 𝐶) = ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
170	126, 169	eleqtrd 2865	. 2 ⊢ (𝜑 → (𝐾 · 𝐿) ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
171		eqid 2763	. . 3 ⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶)))
172	16, 6	fcod 6731	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝑌⟶ℂ)
173	2, 3, 171, 5, 172, 11	eldv 26057	. 2 ⊢ (𝜑 → (𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿) ↔ (𝐶 ∈ ((int‘(𝐽 ↾_t 𝑇))‘𝑌) ∧ (𝐾 · 𝐿) ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))))
174	14, 170, 173	mpbir2and 725	1 ⊢ (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∖ cdif 3902 ⊆ wss 3905 ifcif 4487 {csn 4589 ⟨cop 4595 class class class wbr 5109 {copab 5173 ↦ cmpt 5192 × cxp 5659 dom cdm 5661 ↾ cres 5663 ∘ ccom 5665 Rel wrel 5666 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ∈ cmpo 7412 ℂcc 11093 0cc0 11095 · cmul 11100 − cmin 11436 / cdiv 11866 ↾_t crest 17468 TopOpenctopn 17469 ℂ_fldccnfld 21522 TopOnctopon 23067 intcnt 23174 Cn ccn 23381 CnP ccnp 23382 ×_t ctx 23717 lim_ℂ climc 26021 D cdv 26022

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-icc 13374 df-fz 13531 df-fzo 13679 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-ntr 23177 df-cn 23384 df-cnp 23385 df-tx 23719 df-hmeo 23912 df-xms 24477 df-ms 24478 df-tms 24479 df-cncf 25037 df-limc 26025 df-dv 26026

This theorem is referenced by: dvco 26106 dvcof 26107 dvef 26139

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