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Theorem dvcobrOLD 25906

Description: Obsolete version of dvcobr 25905 as of 10-Apr-2025. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

**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
dvcobrOLD	⊢ (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿))

Proof of Theorem dvcobrOLD Dummy variables 𝑦 𝑧 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 2736	. . . . 5 ⊢ (𝐽 ↾_t 𝑇) = (𝐽 ↾_t 𝑇)
3		dvco.j	. . . . 5 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
4		eqid 2736	. . . . 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 3944	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
10	6, 9	fssd 6679	. . . . 5 ⊢ (𝜑 → 𝐺:𝑌⟶ℂ)
11		dvco.y	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑇)
12	2, 3, 4, 5, 10, 11	eldv 25855	. . . 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	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝐶)(𝑆 D 𝐹)𝐾)
16		dvco.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
17	8, 16, 7	dvcl 25856	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐺‘𝐶)(𝑆 D 𝐹)𝐾) → 𝐾 ∈ ℂ)
18	15, 17	mpdan 687	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℂ)
19	18	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝐾 ∈ ℂ)
20	16	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐹:𝑋⟶ℂ)
21		eldifi 4083	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) → 𝑧 ∈ 𝑌)
22		ffvelcdm 7026	. . . . . . . . . 10 ⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → (𝐺‘𝑧) ∈ 𝑋)
23	6, 21, 22	syl2an 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝑧) ∈ 𝑋)
24	20, 23	ffvelcdmd 7030	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐹‘(𝐺‘𝑧)) ∈ ℂ)
25	24	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐹‘(𝐺‘𝑧)) ∈ ℂ)
26	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐺:𝑌⟶𝑋)
27	5, 10, 11	dvbss 25858	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (𝑇 D 𝐺) ⊆ 𝑌)
28		reldv 25827	. . . . . . . . . . . . 13 ⊢ Rel (𝑇 D 𝐺)
29		releldm 5893	. . . . . . . . . . . . 13 ⊢ ((Rel (𝑇 D 𝐺) ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐶 ∈ dom (𝑇 D 𝐺))
30	28, 1, 29	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ dom (𝑇 D 𝐺))
31	27, 30	sseldd 3934	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ 𝑌)
32	31	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐶 ∈ 𝑌)
33	26, 32	ffvelcdmd 7030	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝐶) ∈ 𝑋)
34	20, 33	ffvelcdmd 7030	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ)
35	34	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ)
36	25, 35	subcld 11492	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) ∈ ℂ)
37	10	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝐺:𝑌⟶ℂ)
38	21	ad2antlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝑧 ∈ 𝑌)
39	37, 38	ffvelcdmd 7030	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐺‘𝑧) ∈ ℂ)
40	31	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝐶 ∈ 𝑌)
41	37, 40	ffvelcdmd 7030	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐺‘𝐶) ∈ ℂ)
42	39, 41	subcld 11492	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐺‘𝑧) − (𝐺‘𝐶)) ∈ ℂ)
43		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ¬ (𝐺‘𝑧) = (𝐺‘𝐶))
44	39, 41	subeq0ad 11502	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) = 0 ↔ (𝐺‘𝑧) = (𝐺‘𝐶)))
45	44	necon3abid 2968	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) ≠ 0 ↔ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)))
46	43, 45	mpbird 257	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐺‘𝑧) − (𝐺‘𝐶)) ≠ 0)
47	36, 42, 46	divcld 11917	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) ∈ ℂ)
48	19, 47	ifclda 4515	. . . 4 ⊢ ((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 3944	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
50	10, 49, 31	dvlem 25853	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ)
51		ssidd 3957	. . . 4 ⊢ (𝜑 → ℂ ⊆ ℂ)
52	3	cnfldtopon 24726	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ)
53		txtopon 23535	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐽 ∈ (TopOn‘ℂ)) → (𝐽 ×_t 𝐽) ∈ (TopOn‘(ℂ × ℂ)))
54	52, 52, 53	mp2an 692	. . . . 5 ⊢ (𝐽 ×_t 𝐽) ∈ (TopOn‘(ℂ × ℂ))
55	54	toponrestid 22865	. . . 4 ⊢ (𝐽 ×_t 𝐽) = ((𝐽 ×_t 𝐽) ↾_t (ℂ × ℂ))
56	23	anim1i 615	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶)) → ((𝐺‘𝑧) ∈ 𝑋 ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶)))
57		eldifsn 4742	. . . . . . 7 ⊢ ((𝐺‘𝑧) ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↔ ((𝐺‘𝑧) ∈ 𝑋 ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶)))
58	56, 57	sylibr 234	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶)) → (𝐺‘𝑧) ∈ (𝑋 ∖ {(𝐺‘𝐶)}))
59	58	anasss 466	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶))) → (𝐺‘𝑧) ∈ (𝑋 ∖ {(𝐺‘𝐶)}))
60		eldifsni 4746	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) → 𝑦 ≠ (𝐺‘𝐶))
61		ifnefalse 4491	. . . . . . . 8 ⊢ (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	. . . . . . 7 ⊢ (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	. . . . . 6 ⊢ ((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 7030	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝐶) ∈ 𝑋)
65	16, 9, 64	dvlem 25853	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)})) → (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))) ∈ ℂ)
66	63, 65	eqeltrd 2836	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)})) → if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) ∈ ℂ)
67		limcresi 25842	. . . . . . 7 ⊢ (𝐺 lim_ℂ 𝐶) ⊆ ((𝐺 ↾ (𝑌 ∖ {𝐶})) lim_ℂ 𝐶)
68	6	feqmptd 6902	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)))
69	68	reseq1d 5937	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ↾ (𝑌 ∖ {𝐶})) = ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ (𝑌 ∖ {𝐶})))
70		difss 4088	. . . . . . . . . 10 ⊢ (𝑌 ∖ {𝐶}) ⊆ 𝑌
71		resmpt 5996	. . . . . . . . . 10 ⊢ ((𝑌 ∖ {𝐶}) ⊆ 𝑌 → ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ (𝑌 ∖ {𝐶})) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)))
72	70, 71	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ (𝑌 ∖ {𝐶})) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧))
73	69, 72	eqtrdi 2787	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ↾ (𝑌 ∖ {𝐶})) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)))
74	73	oveq1d 7373	. . . . . . 7 ⊢ (𝜑 → ((𝐺 ↾ (𝑌 ∖ {𝐶})) lim_ℂ 𝐶) = ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
75	67, 74	sseqtrid 3976	. . . . . 6 ⊢ (𝜑 → (𝐺 lim_ℂ 𝐶) ⊆ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
76		eqid 2736	. . . . . . . . . 10 ⊢ (𝐽 ↾_t 𝑌) = (𝐽 ↾_t 𝑌)
77	76, 3	dvcnp2 25877	. . . . . . . . 9 ⊢ (((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 1375	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (((𝐽 ↾_t 𝑌) CnP 𝐽)‘𝐶))
79	3, 76	cnplimc 25844	. . . . . . . . 9 ⊢ ((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 584	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∈ (((𝐽 ↾_t 𝑌) CnP 𝐽)‘𝐶) ↔ (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶))))
81	78, 80	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶)))
82	81	simprd 495	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝐶) ∈ (𝐺 lim_ℂ 𝐶))
83	75, 82	sseldd 3934	. . . . 5 ⊢ (𝜑 → (𝐺‘𝐶) ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) lim_ℂ 𝐶))
84		eqid 2736	. . . . . . . . 9 ⊢ (𝐽 ↾_t 𝑆) = (𝐽 ↾_t 𝑆)
85		eqid 2736	. . . . . . . . 9 ⊢ (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 25855	. . . . . . . 8 ⊢ (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	. . . . . . 7 ⊢ (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	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ((𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) lim_ℂ (𝐺‘𝐶)))
89	62	mpteq2ia 5193	. . . . . . 7 ⊢ (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 7368	. . . . . 6 ⊢ ((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 2847	. . . . 5 ⊢ (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 2740	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 = (𝐺‘𝐶) ↔ (𝐺‘𝑧) = (𝐺‘𝐶)))
93		fveq2 6834	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑧) → (𝐹‘𝑦) = (𝐹‘(𝐺‘𝑧)))
94	93	oveq1d 7373	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑧) → ((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))))
95		oveq1 7365	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 − (𝐺‘𝐶)) = ((𝐺‘𝑧) − (𝐺‘𝐶)))
96	94, 95	oveq12d 7376	. . . . . 6 ⊢ (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 4506	. . . . 5 ⊢ (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 4485	. . . . . 6 ⊢ ((𝐺‘𝑧) = (𝐺‘𝐶) → if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) = 𝐾)
99	98	ad2antll 729	. . . . 5 ⊢ ((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 25850	. . . 4 ⊢ (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	. . . 4 ⊢ (𝜑 → 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
102	3	mulcn 24812	. . . . 5 ⊢ · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
103	5, 10, 11	dvcl 25856	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐿 ∈ ℂ)
104	1, 103	mpdan 687	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℂ)
105	18, 104	opelxpd 5663	. . . . 5 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (ℂ × ℂ))
106	54	toponunii 22860	. . . . . 6 ⊢ (ℂ × ℂ) = ∪ (𝐽 ×_t 𝐽)
107	106	cncnpi 23222	. . . . 5 ⊢ (( · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽) ∧ ⟨𝐾, 𝐿⟩ ∈ (ℂ × ℂ)) → · ∈ (((𝐽 ×_t 𝐽) CnP 𝐽)‘⟨𝐾, 𝐿⟩))
108	102, 105, 107	sylancr 587	. . . 4 ⊢ (𝜑 → · ∈ (((𝐽 ×_t 𝐽) CnP 𝐽)‘⟨𝐾, 𝐿⟩))
109	48, 50, 51, 51, 3, 55, 100, 101, 108	limccnp2 25849	. . 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))
110		oveq1 7365	. . . . . . . 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))))
111	110	eqeq1d 2738	. . . . . . 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))))
112		oveq1 7365	. . . . . . . 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))))
113	112	eqeq1d 2738	. . . . . . 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))))
114	19	mul01d 11332	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐾 · 0) = 0)
115	9	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑋 ⊆ ℂ)
116	115, 23	sseldd 3934	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝑧) ∈ ℂ)
117	115, 33	sseldd 3934	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝐶) ∈ ℂ)
118	116, 117	subeq0ad 11502	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) = 0 ↔ (𝐺‘𝑧) = (𝐺‘𝐶)))
119	118	biimpar 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐺‘𝑧) − (𝐺‘𝐶)) = 0)
120	119	oveq1d 7373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) = (0 / (𝑧 − 𝐶)))
121	49	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑌 ⊆ ℂ)
122	21	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑧 ∈ 𝑌)
123	121, 122	sseldd 3934	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑧 ∈ ℂ)
124	121, 32	sseldd 3934	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐶 ∈ ℂ)
125	123, 124	subcld 11492	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝑧 − 𝐶) ∈ ℂ)
126		eldifsni 4746	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) → 𝑧 ≠ 𝐶)
127	126	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑧 ≠ 𝐶)
128	123, 124, 127	subne0d 11501	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝑧 − 𝐶) ≠ 0)
129	125, 128	div0d 11916	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (0 / (𝑧 − 𝐶)) = 0)
130	129	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (0 / (𝑧 − 𝐶)) = 0)
131	120, 130	eqtrd 2771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) = 0)
132	131	oveq2d 7374	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐾 · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (𝐾 · 0))
133		fveq2 6834	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑧) = (𝐺‘𝐶) → (𝐹‘(𝐺‘𝑧)) = (𝐹‘(𝐺‘𝐶)))
134	24, 34	subeq0ad 11502	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) = 0 ↔ (𝐹‘(𝐺‘𝑧)) = (𝐹‘(𝐺‘𝐶))))
135	133, 134	imbitrrid 246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((𝐺‘𝑧) = (𝐺‘𝐶) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) = 0))
136	135	imp 406	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) = 0)
137	136	oveq1d 7373	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)) = (0 / (𝑧 − 𝐶)))
138	137, 130	eqtrd 2771	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)) = 0)
139	114, 132, 138	3eqtr4d 2781	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐾 · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)))
140	125	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝑧 − 𝐶) ∈ ℂ)
141	128	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝑧 − 𝐶) ≠ 0)
142	36, 42, 140, 46, 141	dmdcan2d 11947	. . . . . . 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)))
143	111, 113, 139, 142	ifbothda 4518	. . . . . 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)))
144		fvco3 6933	. . . . . . . . 9 ⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
145	6, 21, 144	syl2an 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
146		fvco3 6933	. . . . . . . . . 10 ⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝐶 ∈ 𝑌) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
147	6, 31, 146	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
148	147	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
149	145, 148	oveq12d 7376	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))))
150	149	oveq1d 7373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶)) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)))
151	143, 150	eqtr4d 2774	. . . . 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)))
152	151	mpteq2dva 5191	. . . 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))))
153	152	oveq1d 7373	. . 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))
154	109, 153	eleqtrd 2838	. 2 ⊢ (𝜑 → (𝐾 · 𝐿) ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
155		eqid 2736	. . 3 ⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶)))
156		fco 6686	. . . 4 ⊢ ((𝐹:𝑋⟶ℂ ∧ 𝐺:𝑌⟶𝑋) → (𝐹 ∘ 𝐺):𝑌⟶ℂ)
157	16, 6, 156	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝑌⟶ℂ)
158	2, 3, 155, 5, 157, 11	eldv 25855	. 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))))
159	14, 154, 158	mpbir2and 713	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 1541 ∈ wcel 2113 ≠ wne 2932 ∖ cdif 3898 ⊆ wss 3901 ifcif 4479 {csn 4580 ⟨cop 4586 class class class wbr 5098 ↦ cmpt 5179 × cxp 5622 dom cdm 5624 ↾ cres 5626 ∘ ccom 5628 Rel wrel 5629 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 0cc0 11026 · cmul 11031 − cmin 11364 / cdiv 11794 ↾_t crest 17340 TopOpenctopn 17341 ℂ_fldccnfld 21309 TopOnctopon 22854 intcnt 22961 Cn ccn 23168 CnP ccnp 23169 ×_t ctx 23504 lim_ℂ climc 25819 D cdv 25820

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105 ax-mulf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-icc 13268 df-fz 13424 df-fzo 13571 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-ntr 22964 df-cn 23171 df-cnp 23172 df-tx 23506 df-hmeo 23699 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25823 df-dv 25824

This theorem is referenced by: (None)

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