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Theorem dvcobr 25917

Description: The chain rule for derivatives at a point. For the (simpler but more limited) function version, see dvco 25919. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) Avoid ax-mulf 11118 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 2737	. . . . 5 ⊢ (𝐽 ↾_t 𝑇) = (𝐽 ↾_t 𝑇)
3		dvco.j	. . . . 5 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
4		eqid 2737	. . . . 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 3946	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
10	6, 9	fssd 6687	. . . . 5 ⊢ (𝜑 → 𝐺:𝑌⟶ℂ)
11		dvco.y	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑇)
12	2, 3, 4, 5, 10, 11	eldv 25867	. . . 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 232	. . 3 ⊢ (𝜑 → (𝐶 ∈ ((int‘(𝐽 ↾_t 𝑇))‘𝑌) ∧ 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶)))
14	13	simpld 494	. 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 25868	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺‘𝐶)(𝑆 D 𝐹)𝐾) → 𝐾 ∈ ℂ)
18	15, 17	mpdan 688	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℂ)
19	18	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝐾 ∈ ℂ)
20	16	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐹:𝑋⟶ℂ)
21		eldifi 4085	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) → 𝑧 ∈ 𝑌)
22		ffvelcdm 7035	. . . . . . . . . . . 12 ⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → (𝐺‘𝑧) ∈ 𝑋)
23	6, 21, 22	syl2an 597	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝑧) ∈ 𝑋)
24	20, 23	ffvelcdmd 7039	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐹‘(𝐺‘𝑧)) ∈ ℂ)
25	24	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐹‘(𝐺‘𝑧)) ∈ ℂ)
26	6	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐺:𝑌⟶𝑋)
27	5, 10, 11	dvbss 25870	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝑇 D 𝐺) ⊆ 𝑌)
28		reldv 25839	. . . . . . . . . . . . . . 15 ⊢ Rel (𝑇 D 𝐺)
29		releldm 5901	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝑇 D 𝐺) ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐶 ∈ dom (𝑇 D 𝐺))
30	28, 1, 29	sylancr 588	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ dom (𝑇 D 𝐺))
31	27, 30	sseldd 3936	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ 𝑌)
32	31	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐶 ∈ 𝑌)
33	26, 32	ffvelcdmd 7039	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝐶) ∈ 𝑋)
34	20, 33	ffvelcdmd 7039	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ)
35	34	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ)
36	25, 35	subcld 11504	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) ∈ ℂ)
37	10	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝐺:𝑌⟶ℂ)
38	21	ad2antlr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝑧 ∈ 𝑌)
39	37, 38	ffvelcdmd 7039	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐺‘𝑧) ∈ ℂ)
40	31	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝐶 ∈ 𝑌)
41	37, 40	ffvelcdmd 7039	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐺‘𝐶) ∈ ℂ)
42	39, 41	subcld 11504	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐺‘𝑧) − (𝐺‘𝐶)) ∈ ℂ)
43		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ¬ (𝐺‘𝑧) = (𝐺‘𝐶))
44	39, 41	subeq0ad 11514	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) = 0 ↔ (𝐺‘𝑧) = (𝐺‘𝐶)))
45	44	necon3abid 2969	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) ≠ 0 ↔ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)))
46	43, 45	mpbird 257	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐺‘𝑧) − (𝐺‘𝐶)) ≠ 0)
47	36, 42, 46	divcld 11929	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) ∈ ℂ)
48	19, 47	ifclda 4517	. . . . . 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 3946	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
50	10, 49, 31	dvlem 25865	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ)
51		ssidd 3959	. . . . . 6 ⊢ (𝜑 → ℂ ⊆ ℂ)
52	3	cnfldtopon 24738	. . . . . . . 8 ⊢ 𝐽 ∈ (TopOn‘ℂ)
53		txtopon 23547	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐽 ∈ (TopOn‘ℂ)) → (𝐽 ×_t 𝐽) ∈ (TopOn‘(ℂ × ℂ)))
54	52, 52, 53	mp2an 693	. . . . . . 7 ⊢ (𝐽 ×_t 𝐽) ∈ (TopOn‘(ℂ × ℂ))
55	54	toponrestid 22877	. . . . . 6 ⊢ (𝐽 ×_t 𝐽) = ((𝐽 ×_t 𝐽) ↾_t (ℂ × ℂ))
56	23	anim1i 616	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶)) → ((𝐺‘𝑧) ∈ 𝑋 ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶)))
57		eldifsn 4744	. . . . . . . . 9 ⊢ ((𝐺‘𝑧) ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↔ ((𝐺‘𝑧) ∈ 𝑋 ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶)))
58	56, 57	sylibr 234	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶)) → (𝐺‘𝑧) ∈ (𝑋 ∖ {(𝐺‘𝐶)}))
59	58	anasss 466	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶))) → (𝐺‘𝑧) ∈ (𝑋 ∖ {(𝐺‘𝐶)}))
60		eldifsni 4748	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) → 𝑦 ≠ (𝐺‘𝐶))
61		ifnefalse 4493	. . . . . . . . . 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 17	. . . . . . . . 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 481	. . . . . . . 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 7039	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝐶) ∈ 𝑋)
65	16, 9, 64	dvlem 25865	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)})) → (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))) ∈ ℂ)
66	63, 65	eqeltrd 2837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)})) → if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) ∈ ℂ)
67		limcresi 25854	. . . . . . . . 9 ⊢ (𝐺 lim_ℂ 𝐶) ⊆ ((𝐺 ↾ (𝑌 ∖ {𝐶})) lim_ℂ 𝐶)
68	6	feqmptd 6910	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)))
69	68	reseq1d 5945	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ↾ (𝑌 ∖ {𝐶})) = ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ (𝑌 ∖ {𝐶})))
70		difss 4090	. . . . . . . . . . . 12 ⊢ (𝑌 ∖ {𝐶}) ⊆ 𝑌
71		resmpt 6004	. . . . . . . . . . . 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 2788	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ↾ (𝑌 ∖ {𝐶})) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)))
74	73	oveq1d 7383	. . . . . . . . 9 ⊢ (𝜑 → ((𝐺 ↾ (𝑌 ∖ {𝐶})) lim_ℂ 𝐶) = ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
75	67, 74	sseqtrid 3978	. . . . . . . 8 ⊢ (𝜑 → (𝐺 lim_ℂ 𝐶) ⊆ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
76		eqid 2737	. . . . . . . . . . . 12 ⊢ (𝐽 ↾_t 𝑌) = (𝐽 ↾_t 𝑌)
77	76, 3	dvcnp2 25889	. . . . . . . . . . 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 1376	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (((𝐽 ↾_t 𝑌) CnP 𝐽)‘𝐶))
79	3, 76	cnplimc 25856	. . . . . . . . . . 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 585	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ∈ (((𝐽 ↾_t 𝑌) CnP 𝐽)‘𝐶) ↔ (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶))))
81	78, 80	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶)))
82	81	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶))
83	75, 82	sseldd 3936	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝐶) ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
84		eqid 2737	. . . . . . . . . . 11 ⊢ (𝐽 ↾_t 𝑆) = (𝐽 ↾_t 𝑆)
85		eqid 2737	. . . . . . . . . . 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 25867	. . . . . . . . . 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 232	. . . . . . . . 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 495	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ((𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) lim_ℂ (𝐺‘𝐶)))
89	62	mpteq2ia 5195	. . . . . . . . 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 7378	. . . . . . . 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 2848	. . . . . . 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 2741	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 = (𝐺‘𝐶) ↔ (𝐺‘𝑧) = (𝐺‘𝐶)))
93		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (𝐹‘𝑦) = (𝐹‘(𝐺‘𝑧)))
94	93	oveq1d 7383	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑧) → ((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))))
95		oveq1 7375	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 − (𝐺‘𝐶)) = ((𝐺‘𝑧) − (𝐺‘𝐶)))
96	94, 95	oveq12d 7386	. . . . . . . 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 4508	. . . . . . 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 4487	. . . . . . . 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 730	. . . . . . 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 25862	. . . . . 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 495	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
102	3	mpomulcn 24826	. . . . . . 7 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
103	5, 10, 11	dvcl 25868	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐿 ∈ ℂ)
104	1, 103	mpdan 688	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ ℂ)
105	18, 104	opelxpd 5671	. . . . . . 7 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (ℂ × ℂ))
106	54	toponunii 22872	. . . . . . . 8 ⊢ (ℂ × ℂ) = ∪ (𝐽 ×_t 𝐽)
107	106	cncnpi 23234	. . . . . . 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 588	. . . . . 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 25861	. . . . 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 5182	. . . . . 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 7378	. . . . 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 2847	. . . 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 7529	. . . . 5 ⊢ ((𝐾 ∈ ℂ ∧ 𝐿 ∈ ℂ) → (𝐾(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝐿) = (𝐾 · 𝐿))
114	18, 104, 113	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐾(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝐿) = (𝐾 · 𝐿))
115		ovmpot 7529	. . . . . . . . 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 585	. . . . . . . 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 2748	. . . . . . 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 579	. . . . . 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 5166	. . . . 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 5182	. . . . . . . . 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 2746	. . . . . . . 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 2750	. . . . . . 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 216	. . . . . 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 7383	. . . . 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 17	. . . 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 2851	. . 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 7375	. . . . . . . 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 2739	. . . . . . 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 7375	. . . . . . . 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 2739	. . . . . . 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 11344	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐾 · 0) = 0)
132	9	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑋 ⊆ ℂ)
133	132, 23	sseldd 3936	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝑧) ∈ ℂ)
134	132, 33	sseldd 3936	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝐶) ∈ ℂ)
135	133, 134	subeq0ad 11514	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) = 0 ↔ (𝐺‘𝑧) = (𝐺‘𝐶)))
136	135	biimpar 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐺‘𝑧) − (𝐺‘𝐶)) = 0)
137	136	oveq1d 7383	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) = (0 / (𝑧 − 𝐶)))
138	49	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑌 ⊆ ℂ)
139	21	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑧 ∈ 𝑌)
140	138, 139	sseldd 3936	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑧 ∈ ℂ)
141	138, 32	sseldd 3936	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐶 ∈ ℂ)
142	140, 141	subcld 11504	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝑧 − 𝐶) ∈ ℂ)
143		eldifsni 4748	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) → 𝑧 ≠ 𝐶)
144	143	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑧 ≠ 𝐶)
145	140, 141, 144	subne0d 11513	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝑧 − 𝐶) ≠ 0)
146	142, 145	div0d 11928	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (0 / (𝑧 − 𝐶)) = 0)
147	146	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (0 / (𝑧 − 𝐶)) = 0)
148	137, 147	eqtrd 2772	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) = 0)
149	148	oveq2d 7384	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐾 · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (𝐾 · 0))
150		fveq2 6842	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑧) = (𝐺‘𝐶) → (𝐹‘(𝐺‘𝑧)) = (𝐹‘(𝐺‘𝐶)))
151	24, 34	subeq0ad 11514	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) = 0 ↔ (𝐹‘(𝐺‘𝑧)) = (𝐹‘(𝐺‘𝐶))))
152	150, 151	imbitrrid 246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((𝐺‘𝑧) = (𝐺‘𝐶) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) = 0))
153	152	imp 406	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) = 0)
154	153	oveq1d 7383	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)) = (0 / (𝑧 − 𝐶)))
155	154, 147	eqtrd 2772	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)) = 0)
156	131, 149, 155	3eqtr4d 2782	. . . . . . 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 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝑧 − 𝐶) ∈ ℂ)
158	145	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝑧 − 𝐶) ≠ 0)
159	36, 42, 157, 46, 158	dmdcan2d 11959	. . . . . . 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 4520	. . . . . 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 6941	. . . . . . . . 9 ⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
162	6, 21, 161	syl2an 597	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
163	6, 31	fvco3d 6942	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
164	163	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
165	162, 164	oveq12d 7386	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))))
166	165	oveq1d 7383	. . . . . 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 2775	. . . . 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 5193	. . . 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 7383	. . 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 2839	. 2 ⊢ (𝜑 → (𝐾 · 𝐿) ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
171		eqid 2737	. . 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 6695	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝑌⟶ℂ)
173	2, 3, 171, 5, 172, 11	eldv 25867	. 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 714	1 ⊢ (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3900 ⊆ wss 3903 ifcif 4481 {csn 4582 ⟨cop 4588 class class class wbr 5100 {copab 5162 ↦ cmpt 5181 × cxp 5630 dom cdm 5632 ↾ cres 5634 ∘ ccom 5636 Rel wrel 5637 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 ℂcc 11036 0cc0 11038 · cmul 11043 − cmin 11376 / cdiv 11806 ↾_t crest 17352 TopOpenctopn 17353 ℂ_fldccnfld 21321 TopOnctopon 22866 intcnt 22973 Cn ccn 23180 CnP ccnp 23181 ×_t ctx 23516 lim_ℂ climc 25831 D cdv 25832

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-icc 13280 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-ntr 22976 df-cn 23183 df-cnp 23184 df-tx 23518 df-hmeo 23711 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-limc 25835 df-dv 25836

This theorem is referenced by: dvco 25919 dvcof 25920 dvef 25952

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