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Theorem ulmdvlem1 24154

Description: Lemma for ulmdv 24157. (Contributed by Mario Carneiro, 3-Mar-2015.)

**Hypotheses**
Ref	Expression
ulmdv.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
ulmdv.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
ulmdv.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
ulmdv.f	⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑_𝑚 𝑋))
ulmdv.g	⊢ (𝜑 → 𝐺:𝑋⟶ℂ)
ulmdv.l	⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧))
ulmdv.u	⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻)
ulmdvlem1.c	⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ 𝑋)
ulmdvlem1.r	⊢ ((𝜑 ∧ 𝜓) → 𝑅 ∈ ℝ⁺)
ulmdvlem1.u	⊢ ((𝜑 ∧ 𝜓) → 𝑈 ∈ ℝ⁺)
ulmdvlem1.v	⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ ℝ⁺)
ulmdvlem1.l	⊢ ((𝜑 ∧ 𝜓) → 𝑈 < 𝑊)
ulmdvlem1.b	⊢ ((𝜑 ∧ 𝜓) → (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈) ⊆ 𝑋)
ulmdvlem1.a	⊢ ((𝜑 ∧ 𝜓) → (abs‘(𝑌 − 𝐶)) < 𝑈)
ulmdvlem1.n	⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ 𝑍)
ulmdvlem1.1	⊢ ((𝜑 ∧ 𝜓) → ∀𝑚 ∈ (ℤ_≥‘𝑁)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑅 / 2) / 2))
ulmdvlem1.2	⊢ ((𝜑 ∧ 𝜓) → (abs‘(((𝑆 D (𝐹‘𝑁))‘𝐶) − (𝐻‘𝐶))) < (𝑅 / 2))
ulmdvlem1.y	⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ 𝑋)
ulmdvlem1.3	⊢ ((𝜑 ∧ 𝜓) → 𝑌 ≠ 𝐶)
ulmdvlem1.4	⊢ ((𝜑 ∧ 𝜓) → ((abs‘(𝑌 − 𝐶)) < 𝑊 → (abs‘(((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)) − ((𝑆 D (𝐹‘𝑁))‘𝐶))) < ((𝑅 / 2) / 2)))

**Assertion**
Ref	Expression
ulmdvlem1	⊢ ((𝜑 ∧ 𝜓) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − (𝐻‘𝐶))) < 𝑅)

Distinct variable groups: 𝑘,𝑚,𝑥,𝑧,𝐹 𝑧,𝐺 𝑘,𝑁,𝑚,𝑥 𝐶,𝑘,𝑧 𝑧,𝐻 𝑘,𝑀,𝑥 𝜑,𝑘,𝑚,𝑥,𝑧 𝑆,𝑘,𝑚,𝑥,𝑧 𝑅,𝑚,𝑥 𝑘,𝑋,𝑚,𝑥,𝑧 𝑘,𝑌,𝑧 𝑘,𝑍,𝑚,𝑥,𝑧 Allowed substitution hints: 𝜓(𝑥,𝑧,𝑘,𝑚) 𝐶(𝑥,𝑚) 𝑅(𝑧,𝑘) 𝑈(𝑥,𝑧,𝑘,𝑚) 𝐺(𝑥,𝑘,𝑚) 𝐻(𝑥,𝑘,𝑚) 𝑀(𝑧,𝑚) 𝑁(𝑧) 𝑊(𝑥,𝑧,𝑘,𝑚) 𝑌(𝑥,𝑚)

Proof of Theorem ulmdvlem1 Dummy variables 𝑛 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ulmdv.g	. . . . . 6 ⊢ (𝜑 → 𝐺:𝑋⟶ℂ)
2	1	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐺:𝑋⟶ℂ)
3		ulmdvlem1.y	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ 𝑋)
4	2, 3	ffvelrnd 6360	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐺‘𝑌) ∈ ℂ)
5		ulmdvlem1.c	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ 𝑋)
6	2, 5	ffvelrnd 6360	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐺‘𝐶) ∈ ℂ)
7	4, 6	subcld 10392	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐺‘𝑌) − (𝐺‘𝐶)) ∈ ℂ)
8		ulmdvlem1.n	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ 𝑍)
9		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁))
10	9	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑁 → (𝑆 D (𝐹‘𝑘)) = (𝑆 D (𝐹‘𝑁)))
11		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) = (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))
12		ovex 6678	. . . . . . . . . . . 12 ⊢ (𝑆 D (𝐹‘𝑁)) ∈ V
13	10, 11, 12	fvmpt 6282	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑁) = (𝑆 D (𝐹‘𝑁)))
14	8, 13	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑁) = (𝑆 D (𝐹‘𝑁)))
15		ovex 6678	. . . . . . . . . . . . . . 15 ⊢ (𝑆 D (𝐹‘𝑘)) ∈ V
16	15	rgenw 2924	. . . . . . . . . . . . . 14 ⊢ ∀𝑘 ∈ 𝑍 (𝑆 D (𝐹‘𝑘)) ∈ V
17	11	fnmpt 6020	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ 𝑍 (𝑆 D (𝐹‘𝑘)) ∈ V → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍)
18	16, 17	mp1i 13	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍)
19		ulmdv.u	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻)
20		ulmf2 24138	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍 ∧ (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻) → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑_𝑚 𝑋))
21	18, 19, 20	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑_𝑚 𝑋))
22	21	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑_𝑚 𝑋))
23	22, 8	ffvelrnd 6360	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑁) ∈ (ℂ ↑_𝑚 𝑋))
24	14, 23	eqeltrrd 2702	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑆 D (𝐹‘𝑁)) ∈ (ℂ ↑_𝑚 𝑋))
25		elmapi 7879	. . . . . . . . 9 ⊢ ((𝑆 D (𝐹‘𝑁)) ∈ (ℂ ↑_𝑚 𝑋) → (𝑆 D (𝐹‘𝑁)):𝑋⟶ℂ)
26	24, 25	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑆 D (𝐹‘𝑁)):𝑋⟶ℂ)
27		fdm 6051	. . . . . . . 8 ⊢ ((𝑆 D (𝐹‘𝑁)):𝑋⟶ℂ → dom (𝑆 D (𝐹‘𝑁)) = 𝑋)
28	26, 27	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → dom (𝑆 D (𝐹‘𝑁)) = 𝑋)
29		dvbsss 23666	. . . . . . 7 ⊢ dom (𝑆 D (𝐹‘𝑁)) ⊆ 𝑆
30	28, 29	syl6eqssr 3656	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ⊆ 𝑆)
31		ulmdv.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
32		recnprss 23668	. . . . . . . 8 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
33	31, 32	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
34	33	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑆 ⊆ ℂ)
35	30, 34	sstrd 3613	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ⊆ ℂ)
36	35, 3	sseldd 3604	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ ℂ)
37	35, 5	sseldd 3604	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ ℂ)
38	36, 37	subcld 10392	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑌 − 𝐶) ∈ ℂ)
39		ulmdvlem1.3	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑌 ≠ 𝐶)
40	36, 37, 39	subne0d 10401	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑌 − 𝐶) ≠ 0)
41	7, 38, 40	divcld 10801	. 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) ∈ ℂ)
42		ulmcl 24135	. . . . 5 ⊢ ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻 → 𝐻:𝑋⟶ℂ)
43	19, 42	syl 17	. . . 4 ⊢ (𝜑 → 𝐻:𝑋⟶ℂ)
44	43	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐻:𝑋⟶ℂ)
45	44, 5	ffvelrnd 6360	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐻‘𝐶) ∈ ℂ)
46	26, 5	ffvelrnd 6360	. 2 ⊢ ((𝜑 ∧ 𝜓) → ((𝑆 D (𝐹‘𝑁))‘𝐶) ∈ ℂ)
47		ulmdvlem1.r	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝑅 ∈ ℝ⁺)
48	47	rpred 11872	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝑅 ∈ ℝ)
49	41, 46	subcld 10392	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − ((𝑆 D (𝐹‘𝑁))‘𝐶)) ∈ ℂ)
50	49	abscld 14175	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − ((𝑆 D (𝐹‘𝑁))‘𝐶))) ∈ ℝ)
51		ulmdv.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑_𝑚 𝑋))
52	51	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝐹:𝑍⟶(ℂ ↑_𝑚 𝑋))
53	52, 8	ffvelrnd 6360	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝐹‘𝑁) ∈ (ℂ ↑_𝑚 𝑋))
54		elmapi 7879	. . . . . . . . . 10 ⊢ ((𝐹‘𝑁) ∈ (ℂ ↑_𝑚 𝑋) → (𝐹‘𝑁):𝑋⟶ℂ)
55	53, 54	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝐹‘𝑁):𝑋⟶ℂ)
56	55, 3	ffvelrnd 6360	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ((𝐹‘𝑁)‘𝑌) ∈ ℂ)
57	55, 5	ffvelrnd 6360	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ((𝐹‘𝑁)‘𝐶) ∈ ℂ)
58	56, 57	subcld 10392	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) ∈ ℂ)
59	58, 38, 40	divcld 10801	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)) ∈ ℂ)
60	41, 59	subcld 10392	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − ((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶))) ∈ ℂ)
61	60	abscld 14175	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − ((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)))) ∈ ℝ)
62	59, 46	subcld 10392	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)) − ((𝑆 D (𝐹‘𝑁))‘𝐶)) ∈ ℂ)
63	62	abscld 14175	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (abs‘(((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)) − ((𝑆 D (𝐹‘𝑁))‘𝐶))) ∈ ℝ)
64	61, 63	readdcld 10069	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((abs‘((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − ((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)))) + (abs‘(((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)) − ((𝑆 D (𝐹‘𝑁))‘𝐶)))) ∈ ℝ)
65	48	rehalfcld 11279	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑅 / 2) ∈ ℝ)
66	41, 46, 59	abs3difd 14199	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − ((𝑆 D (𝐹‘𝑁))‘𝐶))) ≤ ((abs‘((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − ((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)))) + (abs‘(((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)) − ((𝑆 D (𝐹‘𝑁))‘𝐶)))))
67	65	rehalfcld 11279	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑅 / 2) / 2) ∈ ℝ)
68	4, 56, 6, 57	sub4d 10441	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶))) = (((𝐺‘𝑌) − (𝐺‘𝐶)) − (((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶))))
69	68	oveq1d 6665	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → ((((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶))) / (𝑌 − 𝐶)) = ((((𝐺‘𝑌) − (𝐺‘𝐶)) − (((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶))) / (𝑌 − 𝐶)))
70	7, 58, 38, 40	divsubdird 10840	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → ((((𝐺‘𝑌) − (𝐺‘𝐶)) − (((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶))) / (𝑌 − 𝐶)) = ((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − ((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶))))
71	69, 70	eqtrd 2656	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ((((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶))) / (𝑌 − 𝐶)) = ((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − ((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶))))
72	71	fveq2d 6195	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (abs‘((((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶))) / (𝑌 − 𝐶))) = (abs‘((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − ((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)))))
73	4, 56	subcld 10392	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → ((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)) ∈ ℂ)
74	6, 57	subcld 10392	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶)) ∈ ℂ)
75	73, 74	subcld 10392	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶))) ∈ ℂ)
76	75, 38, 40	absdivd 14194	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (abs‘((((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶))) / (𝑌 − 𝐶))) = ((abs‘(((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶)))) / (abs‘(𝑌 − 𝐶))))
77	72, 76	eqtr3d 2658	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − ((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)))) = ((abs‘(((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶)))) / (abs‘(𝑌 − 𝐶))))
78		eqid 2622	. . . . . . . 8 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
79		ulmdv.z	. . . . . . . . . 10 ⊢ 𝑍 = (ℤ_≥‘𝑀)
80	8, 79	syl6eleq 2711	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ (ℤ_≥‘𝑀))
81		eluzelz 11697	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
82	80, 81	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℤ)
83		ulmdv.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
84	83	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℤ)
85		ulmdv.l	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧))
86	85	ralrimiva 2966	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧))
87	86	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑧 ∈ 𝑋 (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧))
88		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑌 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑌))
89	88	mpteq2dv 4745	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑌 → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) = (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑌)))
90		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑌 → (𝐺‘𝑧) = (𝐺‘𝑌))
91	89, 90	breq12d 4666	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑌 → ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧) ↔ (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑌)) ⇝ (𝐺‘𝑌)))
92	91	rspcv 3305	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ 𝑋 → (∀𝑧 ∈ 𝑋 (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑌)) ⇝ (𝐺‘𝑌)))
93	3, 87, 92	sylc 65	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑌)) ⇝ (𝐺‘𝑌))
94		fvex 6201	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘𝑀) ∈ V
95	79, 94	eqeltri 2697	. . . . . . . . . . . . 13 ⊢ 𝑍 ∈ V
96	95	mptex 6486	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌))) ∈ V
97	96	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌))) ∈ V)
98		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
99	98	fveq1d 6193	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘)‘𝑌) = ((𝐹‘𝑛)‘𝑌))
100		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑌)) = (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑌))
101		fvex 6201	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑛)‘𝑌) ∈ V
102	99, 100, 101	fvmpt 6282	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑌))‘𝑛) = ((𝐹‘𝑛)‘𝑌))
103	102	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑌))‘𝑛) = ((𝐹‘𝑛)‘𝑌))
104	52	ffvelrnda 6359	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (ℂ ↑_𝑚 𝑋))
105		elmapi 7879	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑛) ∈ (ℂ ↑_𝑚 𝑋) → (𝐹‘𝑛):𝑋⟶ℂ)
106	104, 105	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):𝑋⟶ℂ)
107	3	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → 𝑌 ∈ 𝑋)
108	106, 107	ffvelrnd 6360	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑌) ∈ ℂ)
109	103, 108	eqeltrd 2701	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑌))‘𝑛) ∈ ℂ)
110	99	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) = (((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)))
111		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌))) = (𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)))
112		ovex 6678	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) ∈ V
113	110, 111, 112	fvmpt 6282	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)))‘𝑛) = (((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)))
114	113	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)))‘𝑛) = (((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)))
115	103	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → (((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑌))‘𝑛) − ((𝐹‘𝑁)‘𝑌)) = (((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)))
116	114, 115	eqtr4d 2659	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)))‘𝑛) = (((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑌))‘𝑛) − ((𝐹‘𝑁)‘𝑌)))
117	79, 84, 93, 56, 97, 109, 116	climsubc1 14368	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌))) ⇝ ((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)))
118	95	mptex 6486	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 ↦ ((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))) ∈ V
119	118	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑘 ∈ 𝑍 ↦ ((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))) ∈ V)
120		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝐶 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝐶))
121	120	mpteq2dv 4745	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝐶 → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) = (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝐶)))
122		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝐶 → (𝐺‘𝑧) = (𝐺‘𝐶))
123	121, 122	breq12d 4666	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐶 → ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧) ↔ (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝐶)) ⇝ (𝐺‘𝐶)))
124	123	rspcv 3305	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝑋 → (∀𝑧 ∈ 𝑋 (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝐶)) ⇝ (𝐺‘𝐶)))
125	5, 87, 124	sylc 65	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝐶)) ⇝ (𝐺‘𝐶))
126	95	mptex 6486	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶))) ∈ V
127	126	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶))) ∈ V)
128	98	fveq1d 6193	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘)‘𝐶) = ((𝐹‘𝑛)‘𝐶))
129		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝐶)) = (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝐶))
130		fvex 6201	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑛)‘𝐶) ∈ V
131	128, 129, 130	fvmpt 6282	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝐶))‘𝑛) = ((𝐹‘𝑛)‘𝐶))
132	131	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝐶))‘𝑛) = ((𝐹‘𝑛)‘𝐶))
133	5	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → 𝐶 ∈ 𝑋)
134	106, 133	ffvelrnd 6360	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝐶) ∈ ℂ)
135	132, 134	eqeltrd 2701	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝐶))‘𝑛) ∈ ℂ)
136	128	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)) = (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))
137		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶))) = (𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))
138		ovex 6678	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)) ∈ V
139	136, 137, 138	fvmpt 6282	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))‘𝑛) = (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))
140	139	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))‘𝑛) = (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))
141	132	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → (((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝐶))‘𝑛) − ((𝐹‘𝑁)‘𝐶)) = (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))
142	140, 141	eqtr4d 2659	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))‘𝑛) = (((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝐶))‘𝑛) − ((𝐹‘𝑁)‘𝐶)))
143	79, 84, 125, 57, 127, 135, 142	climsubc1 14368	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶))) ⇝ ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶)))
144	56	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑁)‘𝑌) ∈ ℂ)
145	108, 144	subcld 10392	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → (((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) ∈ ℂ)
146	114, 145	eqeltrd 2701	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)))‘𝑛) ∈ ℂ)
147	57	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑁)‘𝐶) ∈ ℂ)
148	134, 147	subcld 10392	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)) ∈ ℂ)
149	140, 148	eqeltrd 2701	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))‘𝑛) ∈ ℂ)
150	110, 136	oveq12d 6668	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → ((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶))) = ((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶))))
151		eqid 2622	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 ↦ ((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))) = (𝑘 ∈ 𝑍 ↦ ((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶))))
152		ovex 6678	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶))) ∈ V
153	150, 151, 152	fvmpt 6282	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ ((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶))))‘𝑛) = ((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶))))
154	153	adantl 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ ((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶))))‘𝑛) = ((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶))))
155	114, 140	oveq12d 6668	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → (((𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)))‘𝑛) − ((𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))‘𝑛)) = ((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶))))
156	154, 155	eqtr4d 2659	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ ((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶))))‘𝑛) = (((𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)))‘𝑛) − ((𝑘 ∈ 𝑍 ↦ (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))‘𝑛)))
157	79, 84, 117, 119, 143, 146, 149, 156	climsub 14364	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑘 ∈ 𝑍 ↦ ((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))) ⇝ (((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶))))
158	95	mptex 6486	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑍 ↦ (abs‘((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶))))) ∈ V
159	158	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑘 ∈ 𝑍 ↦ (abs‘((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶))))) ∈ V)
160	145, 148	subcld 10392	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶))) ∈ ℂ)
161	154, 160	eqeltrd 2701	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ ((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶))))‘𝑛) ∈ ℂ)
162	150	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (abs‘((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))) = (abs‘((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))))
163		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 ↦ (abs‘((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶))))) = (𝑘 ∈ 𝑍 ↦ (abs‘((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))))
164		fvex 6201	. . . . . . . . . . . 12 ⊢ (abs‘((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))) ∈ V
165	162, 163, 164	fvmpt 6282	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ (abs‘((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))))‘𝑛) = (abs‘((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))))
166	165	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (abs‘((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))))‘𝑛) = (abs‘((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))))
167	154	fveq2d 6195	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → (abs‘((𝑘 ∈ 𝑍 ↦ ((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶))))‘𝑛)) = (abs‘((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))))
168	166, 167	eqtr4d 2659	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (abs‘((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))))‘𝑛) = (abs‘((𝑘 ∈ 𝑍 ↦ ((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶))))‘𝑛)))
169	79, 157, 159, 84, 161, 168	climabs 14334	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑘 ∈ 𝑍 ↦ (abs‘((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶))))) ⇝ (abs‘(((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶)))))
170	38	abscld 14175	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (abs‘(𝑌 − 𝐶)) ∈ ℝ)
171	67, 170	remulcld 10070	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶))) ∈ ℝ)
172	171	recnd 10068	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶))) ∈ ℂ)
173	79	eqimss2i 3660	. . . . . . . . . 10 ⊢ (ℤ_≥‘𝑀) ⊆ 𝑍
174	173, 95	climconst2 14279	. . . . . . . . 9 ⊢ (((((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶))) ∈ ℂ ∧ 𝑀 ∈ ℤ) → (𝑍 × {(((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶)))}) ⇝ (((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶))))
175	172, 84, 174	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑍 × {(((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶)))}) ⇝ (((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶))))
176	79	uztrn2 11705	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑛 ∈ 𝑍)
177	8, 176	sylan 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑛 ∈ 𝑍)
178	177, 165	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝑘 ∈ 𝑍 ↦ (abs‘((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))))‘𝑛) = (abs‘((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))))
179	160	abscld 14175	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ 𝑍) → (abs‘((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))) ∈ ℝ)
180	177, 179	syldan 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (abs‘((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))) ∈ ℝ)
181	178, 180	eqeltrd 2701	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝑘 ∈ 𝑍 ↦ (abs‘((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))))‘𝑛) ∈ ℝ)
182		ovex 6678	. . . . . . . . . . 11 ⊢ (((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶))) ∈ V
183	182	fvconst2 6469	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → ((𝑍 × {(((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶)))})‘𝑛) = (((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶))))
184	177, 183	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝑍 × {(((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶)))})‘𝑛) = (((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶))))
185	171	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶))) ∈ ℝ)
186	184, 185	eqeltrd 2701	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝑍 × {(((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶)))})‘𝑛) ∈ ℝ)
187	177, 106	syldan 487	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑛):𝑋⟶ℂ)
188		ffn 6045	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑛):𝑋⟶ℂ → (𝐹‘𝑛) Fn 𝑋)
189	187, 188	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑛) Fn 𝑋)
190	55	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑁):𝑋⟶ℂ)
191		ffn 6045	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑁):𝑋⟶ℂ → (𝐹‘𝑁) Fn 𝑋)
192	190, 191	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑁) Fn 𝑋)
193		ulmscl 24133	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻 → 𝑋 ∈ V)
194	19, 193	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ V)
195	194	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ V)
196	3	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑌 ∈ 𝑋)
197		fnfvof 6911	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘𝑛) Fn 𝑋 ∧ (𝐹‘𝑁) Fn 𝑋) ∧ (𝑋 ∈ V ∧ 𝑌 ∈ 𝑋)) → (((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))‘𝑌) = (((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)))
198	189, 192, 195, 196, 197	syl22anc 1327	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))‘𝑌) = (((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)))
199	5	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝐶 ∈ 𝑋)
200		fnfvof 6911	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘𝑛) Fn 𝑋 ∧ (𝐹‘𝑁) Fn 𝑋) ∧ (𝑋 ∈ V ∧ 𝐶 ∈ 𝑋)) → (((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))‘𝐶) = (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))
201	189, 192, 195, 199, 200	syl22anc 1327	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))‘𝐶) = (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))
202	198, 201	oveq12d 6668	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))‘𝑌) − (((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))‘𝐶)) = ((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶))))
203	202	fveq2d 6195	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (abs‘((((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))‘𝑌) − (((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))‘𝐶))) = (abs‘((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))))
204	30, 3	sseldd 3604	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ 𝑆)
205	30, 5	sseldd 3604	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ 𝑆)
206	204, 205	ovresd 6801	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → (𝑌((abs ∘ − ) ↾ (𝑆 × 𝑆))𝐶) = (𝑌(abs ∘ − )𝐶))
207		eqid 2622	. . . . . . . . . . . . . . . . . 18 ⊢ (abs ∘ − ) = (abs ∘ − )
208	207	cnmetdval 22574	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑌 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝑌(abs ∘ − )𝐶) = (abs‘(𝑌 − 𝐶)))
209	36, 37, 208	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → (𝑌(abs ∘ − )𝐶) = (abs‘(𝑌 − 𝐶)))
210	206, 209	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → (𝑌((abs ∘ − ) ↾ (𝑆 × 𝑆))𝐶) = (abs‘(𝑌 − 𝐶)))
211		ulmdvlem1.a	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → (abs‘(𝑌 − 𝐶)) < 𝑈)
212	210, 211	eqbrtrd 4675	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (𝑌((abs ∘ − ) ↾ (𝑆 × 𝑆))𝐶) < 𝑈)
213		cnxmet 22576	. . . . . . . . . . . . . . . 16 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
214		xmetres2 22166	. . . . . . . . . . . . . . . 16 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
215	213, 34, 214	sylancr 695	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
216		ulmdvlem1.u	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → 𝑈 ∈ ℝ⁺)
217	216	rpxrd 11873	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝑈 ∈ ℝ^*)
218		elbl3 22197	. . . . . . . . . . . . . . 15 ⊢ (((((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆) ∧ 𝑈 ∈ ℝ^*) ∧ (𝐶 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑌 ∈ (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈) ↔ (𝑌((abs ∘ − ) ↾ (𝑆 × 𝑆))𝐶) < 𝑈))
219	215, 217, 205, 204, 218	syl22anc 1327	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (𝑌 ∈ (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈) ↔ (𝑌((abs ∘ − ) ↾ (𝑆 × 𝑆))𝐶) < 𝑈))
220	212, 219	mpbird 247	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈))
221	220	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑌 ∈ (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈))
222		blcntr 22218	. . . . . . . . . . . . . 14 ⊢ ((((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆) ∧ 𝐶 ∈ 𝑆 ∧ 𝑈 ∈ ℝ⁺) → 𝐶 ∈ (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈))
223	215, 205, 216, 222	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈))
224	223	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝐶 ∈ (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈))
225	221, 224	jca 554	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑌 ∈ (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈) ∧ 𝐶 ∈ (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈)))
226	31	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑆 ∈ {ℝ, ℂ})
227		eqid 2622	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (𝑆 × 𝑆))
228	30	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ⊆ 𝑆)
229	187	ffvelrnda 6359	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → ((𝐹‘𝑛)‘𝑦) ∈ ℂ)
230	190	ffvelrnda 6359	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → ((𝐹‘𝑁)‘𝑦) ∈ ℂ)
231	229, 230	subcld 10392	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → (((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑁)‘𝑦)) ∈ ℂ)
232		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝑋 ↦ (((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑁)‘𝑦))) = (𝑦 ∈ 𝑋 ↦ (((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑁)‘𝑦)))
233	231, 232	fmptd 6385	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑦 ∈ 𝑋 ↦ (((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑁)‘𝑦))):𝑋⟶ℂ)
234		fvexd 6203	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → ((𝐹‘𝑛)‘𝑦) ∈ V)
235		fvexd 6203	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → ((𝐹‘𝑁)‘𝑦) ∈ V)
236	187	feqmptd 6249	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑛) = (𝑦 ∈ 𝑋 ↦ ((𝐹‘𝑛)‘𝑦)))
237	190	feqmptd 6249	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑁) = (𝑦 ∈ 𝑋 ↦ ((𝐹‘𝑁)‘𝑦)))
238	195, 234, 235, 236, 237	offval2 6914	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁)) = (𝑦 ∈ 𝑋 ↦ (((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑁)‘𝑦))))
239	238	feq1d 6030	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁)):𝑋⟶ℂ ↔ (𝑦 ∈ 𝑋 ↦ (((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑁)‘𝑦))):𝑋⟶ℂ))
240	233, 239	mpbird 247	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁)):𝑋⟶ℂ)
241	205	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝐶 ∈ 𝑆)
242	217	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑈 ∈ ℝ^*)
243		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈) = (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈)
244		ulmdvlem1.b	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈) ⊆ 𝑋)
245	244	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈) ⊆ 𝑋)
246	238	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑆 D ((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))) = (𝑆 D (𝑦 ∈ 𝑋 ↦ (((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑁)‘𝑦)))))
247		fvexd 6203	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → ((𝑆 D (𝐹‘𝑛))‘𝑦) ∈ V)
248	236	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑆 D (𝐹‘𝑛)) = (𝑆 D (𝑦 ∈ 𝑋 ↦ ((𝐹‘𝑛)‘𝑦))))
249	98	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 𝑛 → (𝑆 D (𝐹‘𝑘)) = (𝑆 D (𝐹‘𝑛)))
250		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑆 D (𝐹‘𝑛)) ∈ V
251	249, 11, 250	fvmpt 6282	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛) = (𝑆 D (𝐹‘𝑛)))
252	177, 251	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛) = (𝑆 D (𝐹‘𝑛)))
253	21	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑_𝑚 𝑋))
254	253, 177	ffvelrnd 6360	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛) ∈ (ℂ ↑_𝑚 𝑋))
255	252, 254	eqeltrrd 2702	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑆 D (𝐹‘𝑛)) ∈ (ℂ ↑_𝑚 𝑋))
256		elmapi 7879	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 D (𝐹‘𝑛)) ∈ (ℂ ↑_𝑚 𝑋) → (𝑆 D (𝐹‘𝑛)):𝑋⟶ℂ)
257	255, 256	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑆 D (𝐹‘𝑛)):𝑋⟶ℂ)
258	257	feqmptd 6249	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑆 D (𝐹‘𝑛)) = (𝑦 ∈ 𝑋 ↦ ((𝑆 D (𝐹‘𝑛))‘𝑦)))
259	248, 258	eqtr3d 2658	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑆 D (𝑦 ∈ 𝑋 ↦ ((𝐹‘𝑛)‘𝑦))) = (𝑦 ∈ 𝑋 ↦ ((𝑆 D (𝐹‘𝑛))‘𝑦)))
260		fvexd 6203	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → ((𝑆 D (𝐹‘𝑁))‘𝑦) ∈ V)
261	237	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑆 D (𝐹‘𝑁)) = (𝑆 D (𝑦 ∈ 𝑋 ↦ ((𝐹‘𝑁)‘𝑦))))
262	26	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑆 D (𝐹‘𝑁)):𝑋⟶ℂ)
263	262	feqmptd 6249	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑆 D (𝐹‘𝑁)) = (𝑦 ∈ 𝑋 ↦ ((𝑆 D (𝐹‘𝑁))‘𝑦)))
264	261, 263	eqtr3d 2658	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑆 D (𝑦 ∈ 𝑋 ↦ ((𝐹‘𝑁)‘𝑦))) = (𝑦 ∈ 𝑋 ↦ ((𝑆 D (𝐹‘𝑁))‘𝑦)))
265	226, 229, 247, 259, 230, 260, 264	dvmptsub 23730	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑆 D (𝑦 ∈ 𝑋 ↦ (((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑁)‘𝑦)))) = (𝑦 ∈ 𝑋 ↦ (((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦))))
266	246, 265	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑆 D ((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))) = (𝑦 ∈ 𝑋 ↦ (((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦))))
267	266	dmeqd 5326	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → dom (𝑆 D ((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))) = dom (𝑦 ∈ 𝑋 ↦ (((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦))))
268		ovex 6678	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦)) ∈ V
269		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑋 ↦ (((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦))) = (𝑦 ∈ 𝑋 ↦ (((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦)))
270	268, 269	dmmpti 6023	. . . . . . . . . . . . . 14 ⊢ dom (𝑦 ∈ 𝑋 ↦ (((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦))) = 𝑋
271	267, 270	syl6eq 2672	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → dom (𝑆 D ((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))) = 𝑋)
272	245, 271	sseqtr4d 3642	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈) ⊆ dom (𝑆 D ((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))))
273	67	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝑅 / 2) / 2) ∈ ℝ)
274	245	sselda 3603	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈)) → 𝑦 ∈ 𝑋)
275	266	fveq1d 6193	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝑆 D ((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁)))‘𝑦) = ((𝑦 ∈ 𝑋 ↦ (((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦)))‘𝑦))
276	269	fvmpt2 6291	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝑋 ∧ (((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦)) ∈ V) → ((𝑦 ∈ 𝑋 ↦ (((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦)))‘𝑦) = (((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦)))
277	268, 276	mpan2 707	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑋 → ((𝑦 ∈ 𝑋 ↦ (((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦)))‘𝑦) = (((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦)))
278	275, 277	sylan9eq 2676	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → ((𝑆 D ((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁)))‘𝑦) = (((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦)))
279	278	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → (abs‘((𝑆 D ((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁)))‘𝑦)) = (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦))))
280	268	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → (((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦)) ∈ V)
281	226, 231, 280, 265	dvmptcl 23722	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → (((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦)) ∈ ℂ)
282	281	abscld 14175	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦))) ∈ ℝ)
283	67	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → ((𝑅 / 2) / 2) ∈ ℝ)
284	257	ffvelrnda 6359	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → ((𝑆 D (𝐹‘𝑛))‘𝑦) ∈ ℂ)
285	262	ffvelrnda 6359	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → ((𝑆 D (𝐹‘𝑁))‘𝑦) ∈ ℂ)
286	284, 285	abssubd 14192	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦))) = (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑦) − ((𝑆 D (𝐹‘𝑛))‘𝑦))))
287		ulmdvlem1.1	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑚 ∈ (ℤ_≥‘𝑁)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑅 / 2) / 2))
288		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
289	288	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 = 𝑛 → (𝑆 D (𝐹‘𝑚)) = (𝑆 D (𝐹‘𝑛)))
290	289	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑛 → ((𝑆 D (𝐹‘𝑚))‘𝑥) = ((𝑆 D (𝐹‘𝑛))‘𝑥))
291	290	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑛 → (((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥)) = (((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑛))‘𝑥)))
292	291	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) = (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑛))‘𝑥))))
293	292	breq1d 4663	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑛 → ((abs‘(((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑅 / 2) / 2) ↔ (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑛))‘𝑥))) < ((𝑅 / 2) / 2)))
294	293	ralbidv 2986	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = 𝑛 → (∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑅 / 2) / 2) ↔ ∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑛))‘𝑥))) < ((𝑅 / 2) / 2)))
295	294	rspccva 3308	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑚 ∈ (ℤ_≥‘𝑁)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑅 / 2) / 2) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑛))‘𝑥))) < ((𝑅 / 2) / 2))
296	287, 295	sylan 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑛))‘𝑥))) < ((𝑅 / 2) / 2))
297		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → ((𝑆 D (𝐹‘𝑁))‘𝑥) = ((𝑆 D (𝐹‘𝑁))‘𝑦))
298		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → ((𝑆 D (𝐹‘𝑛))‘𝑥) = ((𝑆 D (𝐹‘𝑛))‘𝑦))
299	297, 298	oveq12d 6668	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → (((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑛))‘𝑥)) = (((𝑆 D (𝐹‘𝑁))‘𝑦) − ((𝑆 D (𝐹‘𝑛))‘𝑦)))
300	299	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑛))‘𝑥))) = (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑦) − ((𝑆 D (𝐹‘𝑛))‘𝑦))))
301	300	breq1d 4663	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((abs‘(((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑛))‘𝑥))) < ((𝑅 / 2) / 2) ↔ (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑦) − ((𝑆 D (𝐹‘𝑛))‘𝑦))) < ((𝑅 / 2) / 2)))
302	301	rspccva 3308	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑥) − ((𝑆 D (𝐹‘𝑛))‘𝑥))) < ((𝑅 / 2) / 2) ∧ 𝑦 ∈ 𝑋) → (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑦) − ((𝑆 D (𝐹‘𝑛))‘𝑦))) < ((𝑅 / 2) / 2))
303	296, 302	sylan 488	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → (abs‘(((𝑆 D (𝐹‘𝑁))‘𝑦) − ((𝑆 D (𝐹‘𝑛))‘𝑦))) < ((𝑅 / 2) / 2))
304	286, 303	eqbrtrd 4675	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦))) < ((𝑅 / 2) / 2))
305	282, 283, 304	ltled 10185	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑦) − ((𝑆 D (𝐹‘𝑁))‘𝑦))) ≤ ((𝑅 / 2) / 2))
306	279, 305	eqbrtrd 4675	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ 𝑋) → (abs‘((𝑆 D ((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁)))‘𝑦)) ≤ ((𝑅 / 2) / 2))
307	274, 306	syldan 487	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑦 ∈ (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈)) → (abs‘((𝑆 D ((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁)))‘𝑦)) ≤ ((𝑅 / 2) / 2))
308	226, 227, 228, 240, 241, 242, 243, 272, 273, 307	dvlip2 23758	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ (𝑌 ∈ (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈) ∧ 𝐶 ∈ (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈))) → (abs‘((((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))‘𝑌) − (((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))‘𝐶))) ≤ (((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶))))
309	225, 308	mpdan 702	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (abs‘((((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))‘𝑌) − (((𝐹‘𝑛) ∘_𝑓 − (𝐹‘𝑁))‘𝐶))) ≤ (((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶))))
310	203, 309	eqbrtrrd 4677	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (abs‘((((𝐹‘𝑛)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑛)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))) ≤ (((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶))))
311	310, 178, 184	3brtr4d 4685	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝑘 ∈ 𝑍 ↦ (abs‘((((𝐹‘𝑘)‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − (((𝐹‘𝑘)‘𝐶) − ((𝐹‘𝑁)‘𝐶)))))‘𝑛) ≤ ((𝑍 × {(((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶)))})‘𝑛))
312	78, 82, 169, 175, 181, 186, 311	climle 14370	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (abs‘(((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶)))) ≤ (((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶))))
313	75	abscld 14175	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (abs‘(((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶)))) ∈ ℝ)
314	38, 40	absrpcld 14187	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (abs‘(𝑌 − 𝐶)) ∈ ℝ⁺)
315	313, 67, 314	ledivmul2d 11926	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (((abs‘(((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶)))) / (abs‘(𝑌 − 𝐶))) ≤ ((𝑅 / 2) / 2) ↔ (abs‘(((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶)))) ≤ (((𝑅 / 2) / 2) · (abs‘(𝑌 − 𝐶)))))
316	312, 315	mpbird 247	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((abs‘(((𝐺‘𝑌) − ((𝐹‘𝑁)‘𝑌)) − ((𝐺‘𝐶) − ((𝐹‘𝑁)‘𝐶)))) / (abs‘(𝑌 − 𝐶))) ≤ ((𝑅 / 2) / 2))
317	77, 316	eqbrtrd 4675	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − ((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)))) ≤ ((𝑅 / 2) / 2))
318	216	rpred 11872	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑈 ∈ ℝ)
319		ulmdvlem1.v	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ ℝ⁺)
320	319	rpred 11872	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ ℝ)
321		ulmdvlem1.l	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑈 < 𝑊)
322	170, 318, 320, 211, 321	lttrd 10198	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (abs‘(𝑌 − 𝐶)) < 𝑊)
323		ulmdvlem1.4	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((abs‘(𝑌 − 𝐶)) < 𝑊 → (abs‘(((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)) − ((𝑆 D (𝐹‘𝑁))‘𝐶))) < ((𝑅 / 2) / 2)))
324	322, 323	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (abs‘(((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)) − ((𝑆 D (𝐹‘𝑁))‘𝐶))) < ((𝑅 / 2) / 2))
325	61, 63, 67, 67, 317, 324	leltaddd 10649	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((abs‘((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − ((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)))) + (abs‘(((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)) − ((𝑆 D (𝐹‘𝑁))‘𝐶)))) < (((𝑅 / 2) / 2) + ((𝑅 / 2) / 2)))
326	65	recnd 10068	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑅 / 2) ∈ ℂ)
327	326	2halvesd 11278	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (((𝑅 / 2) / 2) + ((𝑅 / 2) / 2)) = (𝑅 / 2))
328	325, 327	breqtrd 4679	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((abs‘((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − ((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)))) + (abs‘(((((𝐹‘𝑁)‘𝑌) − ((𝐹‘𝑁)‘𝐶)) / (𝑌 − 𝐶)) − ((𝑆 D (𝐹‘𝑁))‘𝐶)))) < (𝑅 / 2))
329	50, 64, 65, 66, 328	lelttrd 10195	. 2 ⊢ ((𝜑 ∧ 𝜓) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − ((𝑆 D (𝐹‘𝑁))‘𝐶))) < (𝑅 / 2))
330		ulmdvlem1.2	. 2 ⊢ ((𝜑 ∧ 𝜓) → (abs‘(((𝑆 D (𝐹‘𝑁))‘𝐶) − (𝐻‘𝐶))) < (𝑅 / 2))
331	41, 45, 46, 48, 329, 330	abs3lemd 14200	1 ⊢ ((𝜑 ∧ 𝜓) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝐶)) / (𝑌 − 𝐶)) − (𝐻‘𝐶))) < 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ⊆ wss 3574 {csn 4177 {cpr 4179 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 dom cdm 5114 ↾ cres 5116 ∘ ccom 5118 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘_𝑓 cof 6895 ↑_𝑚 cmap 7857 ℂcc 9934 ℝcr 9935 + caddc 9939 · cmul 9941 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 2c2 11070 ℤcz 11377 ℤ_≥cuz 11687 ℝ⁺crp 11832 abscabs 13974 ⇝ cli 14215 ∞Metcxmt 19731 ballcbl 19733 D cdv 23627 ⇝_𝑢culm 24130

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-ulm 24131

This theorem is referenced by: ulmdvlem3 24156

W3C validator