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Theorem supcvg 15798

Description: Extract a sequence 𝑓 in 𝑋 such that the image of the points in the bounded set 𝐴 converges to the supremum 𝑆 of the set. Similar to Equation 4 of [Kreyszig] p. 144. The proof uses countable choice ax-cc 10426. (Contributed by Mario Carneiro, 15-Feb-2013.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)

**Hypotheses**
Ref	Expression
supcvg.1	⊢ 𝑋 ∈ V
supcvg.2	⊢ 𝑆 = sup(𝐴, ℝ, < )
supcvg.3	⊢ 𝑅 = (𝑛 ∈ ℕ ↦ (𝑆 − (1 / 𝑛)))
supcvg.4	⊢ (𝜑 → 𝑋 ≠ ∅)
supcvg.5	⊢ (𝜑 → 𝐹:𝑋–onto→𝐴)
supcvg.6	⊢ (𝜑 → 𝐴 ⊆ ℝ)
supcvg.7	⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)

**Assertion**
Ref	Expression
supcvg	⊢ (𝜑 → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ (𝐹 ∘ 𝑓) ⇝ 𝑆))

Distinct variable groups: 𝑥,𝑓,𝐹 𝑓,𝑛,𝜑 𝑅,𝑓,𝑥 𝑓,𝑋,𝑥 𝑥,𝑦,𝐴 𝑆,𝑛 Allowed substitution hints: 𝜑(𝑥,𝑦) 𝐴(𝑓,𝑛) 𝑅(𝑦,𝑛) 𝑆(𝑥,𝑦,𝑓) 𝐹(𝑦,𝑛) 𝑋(𝑦,𝑛)

Proof of Theorem supcvg Dummy variables 𝑘 𝑚 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7413	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
2	1	oveq2d 7421	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑆 − (1 / 𝑛)) = (𝑆 − (1 / 𝑘)))
3		supcvg.3	. . . . . . . . . . 11 ⊢ 𝑅 = (𝑛 ∈ ℕ ↦ (𝑆 − (1 / 𝑛)))
4		ovex 7438	. . . . . . . . . . 11 ⊢ (𝑆 − (1 / 𝑘)) ∈ V
5	2, 3, 4	fvmpt 6995	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝑅‘𝑘) = (𝑆 − (1 / 𝑘)))
6	5	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑅‘𝑘) = (𝑆 − (1 / 𝑘)))
7		supcvg.2	. . . . . . . . . . 11 ⊢ 𝑆 = sup(𝐴, ℝ, < )
8		supcvg.6	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
9		supcvg.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ≠ ∅)
10		supcvg.5	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝑋–onto→𝐴)
11		fof 6802	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝑋–onto→𝐴 → 𝐹:𝑋⟶𝐴)
12	10, 11	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:𝑋⟶𝐴)
13		feq3 6697	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = ∅ → (𝐹:𝑋⟶𝐴 ↔ 𝐹:𝑋⟶∅))
14	12, 13	syl5ibcom 244	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 = ∅ → 𝐹:𝑋⟶∅))
15		f00 6770	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑋⟶∅ ↔ (𝐹 = ∅ ∧ 𝑋 = ∅))
16	15	simprbi 497	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑋⟶∅ → 𝑋 = ∅)
17	14, 16	syl6 35	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 = ∅ → 𝑋 = ∅))
18	17	necon3d 2961	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 ≠ ∅ → 𝐴 ≠ ∅))
19	9, 18	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≠ ∅)
20		supcvg.7	. . . . . . . . . . . 12 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
21	8, 19, 20	suprcld 12173	. . . . . . . . . . 11 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
22	7, 21	eqeltrid 2837	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ ℝ)
23		nnrp 12981	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ⁺)
24	23	rpreccld 13022	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ⁺)
25		ltsubrp 13006	. . . . . . . . . 10 ⊢ ((𝑆 ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ⁺) → (𝑆 − (1 / 𝑘)) < 𝑆)
26	22, 24, 25	syl2an 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆 − (1 / 𝑘)) < 𝑆)
27	6, 26	eqbrtrd 5169	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑅‘𝑘) < 𝑆)
28	27, 7	breqtrdi 5188	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑅‘𝑘) < sup(𝐴, ℝ, < ))
29	8, 19, 20	3jca 1128	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
30		nnrecre 12250	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
31		resubcl 11520	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ ℝ ∧ (1 / 𝑛) ∈ ℝ) → (𝑆 − (1 / 𝑛)) ∈ ℝ)
32	22, 30, 31	syl2an 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆 − (1 / 𝑛)) ∈ ℝ)
33	32, 3	fmptd 7110	. . . . . . . . 9 ⊢ (𝜑 → 𝑅:ℕ⟶ℝ)
34	33	ffvelcdmda 7083	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑅‘𝑘) ∈ ℝ)
35		suprlub 12174	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝑅‘𝑘) ∈ ℝ) → ((𝑅‘𝑘) < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 (𝑅‘𝑘) < 𝑧))
36	29, 34, 35	syl2an2r 683	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑅‘𝑘) < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 (𝑅‘𝑘) < 𝑧))
37	28, 36	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∃𝑧 ∈ 𝐴 (𝑅‘𝑘) < 𝑧)
38	8	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ⊆ ℝ)
39	38	sselda 3981	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
40		ltle 11298	. . . . . . . 8 ⊢ (((𝑅‘𝑘) ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑅‘𝑘) < 𝑧 → (𝑅‘𝑘) ≤ 𝑧))
41	34, 39, 40	syl2an2r 683	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑧 ∈ 𝐴) → ((𝑅‘𝑘) < 𝑧 → (𝑅‘𝑘) ≤ 𝑧))
42	41	reximdva 3168	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∃𝑧 ∈ 𝐴 (𝑅‘𝑘) < 𝑧 → ∃𝑧 ∈ 𝐴 (𝑅‘𝑘) ≤ 𝑧))
43	37, 42	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∃𝑧 ∈ 𝐴 (𝑅‘𝑘) ≤ 𝑧)
44		forn 6805	. . . . . . . . 9 ⊢ (𝐹:𝑋–onto→𝐴 → ran 𝐹 = 𝐴)
45	10, 44	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐴)
46	45	rexeqdv 3326	. . . . . . 7 ⊢ (𝜑 → (∃𝑧 ∈ ran 𝐹(𝑅‘𝑘) ≤ 𝑧 ↔ ∃𝑧 ∈ 𝐴 (𝑅‘𝑘) ≤ 𝑧))
47		ffn 6714	. . . . . . . 8 ⊢ (𝐹:𝑋⟶𝐴 → 𝐹 Fn 𝑋)
48		breq2 5151	. . . . . . . . 9 ⊢ (𝑧 = (𝐹‘𝑥) → ((𝑅‘𝑘) ≤ 𝑧 ↔ (𝑅‘𝑘) ≤ (𝐹‘𝑥)))
49	48	rexrn 7085	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → (∃𝑧 ∈ ran 𝐹(𝑅‘𝑘) ≤ 𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑅‘𝑘) ≤ (𝐹‘𝑥)))
50	12, 47, 49	3syl 18	. . . . . . 7 ⊢ (𝜑 → (∃𝑧 ∈ ran 𝐹(𝑅‘𝑘) ≤ 𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑅‘𝑘) ≤ (𝐹‘𝑥)))
51	46, 50	bitr3d 280	. . . . . 6 ⊢ (𝜑 → (∃𝑧 ∈ 𝐴 (𝑅‘𝑘) ≤ 𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑅‘𝑘) ≤ (𝐹‘𝑥)))
52	51	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∃𝑧 ∈ 𝐴 (𝑅‘𝑘) ≤ 𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑅‘𝑘) ≤ (𝐹‘𝑥)))
53	43, 52	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∃𝑥 ∈ 𝑋 (𝑅‘𝑘) ≤ (𝐹‘𝑥))
54	53	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ∃𝑥 ∈ 𝑋 (𝑅‘𝑘) ≤ (𝐹‘𝑥))
55		supcvg.1	. . . 4 ⊢ 𝑋 ∈ V
56		nnenom 13941	. . . 4 ⊢ ℕ ≈ ω
57		fveq2 6888	. . . . 5 ⊢ (𝑥 = (𝑓‘𝑘) → (𝐹‘𝑥) = (𝐹‘(𝑓‘𝑘)))
58	57	breq2d 5159	. . . 4 ⊢ (𝑥 = (𝑓‘𝑘) → ((𝑅‘𝑘) ≤ (𝐹‘𝑥) ↔ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))))
59	55, 56, 58	axcc4 10430	. . 3 ⊢ (∀𝑘 ∈ ℕ ∃𝑥 ∈ 𝑋 (𝑅‘𝑘) ≤ (𝐹‘𝑥) → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))))
60	54, 59	syl 17	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))))
61		nnuz 12861	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
62		1zzd 12589	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) → 1 ∈ ℤ)
63		1zzd 12589	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
64	22	recnd 11238	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ ℂ)
65		1z 12588	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
66	61	eqimss2i 4042	. . . . . . . . . . 11 ⊢ (ℤ_≥‘1) ⊆ ℕ
67		nnex 12214	. . . . . . . . . . 11 ⊢ ℕ ∈ V
68	66, 67	climconst2 15488	. . . . . . . . . 10 ⊢ ((𝑆 ∈ ℂ ∧ 1 ∈ ℤ) → (ℕ × {𝑆}) ⇝ 𝑆)
69	64, 65, 68	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → (ℕ × {𝑆}) ⇝ 𝑆)
70	67	mptex 7221	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ (𝑆 − (1 / 𝑛))) ∈ V
71	3, 70	eqeltri 2829	. . . . . . . . . 10 ⊢ 𝑅 ∈ V
72	71	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ V)
73		ax-1cn 11164	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
74		divcnv 15795	. . . . . . . . . 10 ⊢ (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
75	73, 74	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
76		fvconst2g 7199	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((ℕ × {𝑆})‘𝑘) = 𝑆)
77	22, 76	sylan 580	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℕ × {𝑆})‘𝑘) = 𝑆)
78	64	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑆 ∈ ℂ)
79	77, 78	eqeltrd 2833	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℕ × {𝑆})‘𝑘) ∈ ℂ)
80		eqid 2732	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
81		ovex 7438	. . . . . . . . . . . 12 ⊢ (1 / 𝑘) ∈ V
82	1, 80, 81	fvmpt 6995	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘))
83	82	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘))
84		nnrecre 12250	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
85	84	recnd 11238	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℂ)
86	85	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℂ)
87	83, 86	eqeltrd 2833	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℂ)
88	77, 83	oveq12d 7423	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((ℕ × {𝑆})‘𝑘) − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)) = (𝑆 − (1 / 𝑘)))
89	6, 88	eqtr4d 2775	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑅‘𝑘) = (((ℕ × {𝑆})‘𝑘) − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)))
90	61, 63, 69, 72, 75, 79, 87, 89	climsub 15574	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ⇝ (𝑆 − 0))
91	64	subid1d 11556	. . . . . . . 8 ⊢ (𝜑 → (𝑆 − 0) = 𝑆)
92	90, 91	breqtrd 5173	. . . . . . 7 ⊢ (𝜑 → 𝑅 ⇝ 𝑆)
93	92	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) → 𝑅 ⇝ 𝑆)
94	12	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) → 𝐹:𝑋⟶𝐴)
95		fex 7224	. . . . . . . 8 ⊢ ((𝐹:𝑋⟶𝐴 ∧ 𝑋 ∈ V) → 𝐹 ∈ V)
96	94, 55, 95	sylancl 586	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) → 𝐹 ∈ V)
97		vex 3478	. . . . . . 7 ⊢ 𝑓 ∈ V
98		coexg 7916	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ 𝑓 ∈ V) → (𝐹 ∘ 𝑓) ∈ V)
99	96, 97, 98	sylancl 586	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) → (𝐹 ∘ 𝑓) ∈ V)
100	33	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) → 𝑅:ℕ⟶ℝ)
101	100	ffvelcdmda 7083	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) ∧ 𝑚 ∈ ℕ) → (𝑅‘𝑚) ∈ ℝ)
102	12, 8	fssd 6732	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
103		fco 6738	. . . . . . . . 9 ⊢ ((𝐹:𝑋⟶ℝ ∧ 𝑓:ℕ⟶𝑋) → (𝐹 ∘ 𝑓):ℕ⟶ℝ)
104	102, 103	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:ℕ⟶𝑋) → (𝐹 ∘ 𝑓):ℕ⟶ℝ)
105	104	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) → (𝐹 ∘ 𝑓):ℕ⟶ℝ)
106	105	ffvelcdmda 7083	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) ∧ 𝑚 ∈ ℕ) → ((𝐹 ∘ 𝑓)‘𝑚) ∈ ℝ)
107		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝑅‘𝑘) = (𝑅‘𝑚))
108		2fveq3 6893	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝐹‘(𝑓‘𝑘)) = (𝐹‘(𝑓‘𝑚)))
109	107, 108	breq12d 5160	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → ((𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘)) ↔ (𝑅‘𝑚) ≤ (𝐹‘(𝑓‘𝑚))))
110	109	rspccva 3611	. . . . . . . 8 ⊢ ((∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘)) ∧ 𝑚 ∈ ℕ) → (𝑅‘𝑚) ≤ (𝐹‘(𝑓‘𝑚)))
111	110	adantll 712	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) ∧ 𝑚 ∈ ℕ) → (𝑅‘𝑚) ≤ (𝐹‘(𝑓‘𝑚)))
112		simplr 767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) → 𝑓:ℕ⟶𝑋)
113		fvco3 6987	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶𝑋 ∧ 𝑚 ∈ ℕ) → ((𝐹 ∘ 𝑓)‘𝑚) = (𝐹‘(𝑓‘𝑚)))
114	112, 113	sylan 580	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) ∧ 𝑚 ∈ ℕ) → ((𝐹 ∘ 𝑓)‘𝑚) = (𝐹‘(𝑓‘𝑚)))
115	111, 114	breqtrrd 5175	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) ∧ 𝑚 ∈ ℕ) → (𝑅‘𝑚) ≤ ((𝐹 ∘ 𝑓)‘𝑚))
116	29	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) ∧ 𝑚 ∈ ℕ) → (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
117	112	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) ∧ 𝑚 ∈ ℕ) → (𝑓‘𝑚) ∈ 𝑋)
118	94	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) ∧ (𝑓‘𝑚) ∈ 𝑋) → (𝐹‘(𝑓‘𝑚)) ∈ 𝐴)
119	117, 118	syldan 591	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) ∧ 𝑚 ∈ ℕ) → (𝐹‘(𝑓‘𝑚)) ∈ 𝐴)
120		suprub 12171	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝐹‘(𝑓‘𝑚)) ∈ 𝐴) → (𝐹‘(𝑓‘𝑚)) ≤ sup(𝐴, ℝ, < ))
121	116, 119, 120	syl2anc 584	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) ∧ 𝑚 ∈ ℕ) → (𝐹‘(𝑓‘𝑚)) ≤ sup(𝐴, ℝ, < ))
122	121, 7	breqtrrdi 5189	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) ∧ 𝑚 ∈ ℕ) → (𝐹‘(𝑓‘𝑚)) ≤ 𝑆)
123	114, 122	eqbrtrd 5169	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) ∧ 𝑚 ∈ ℕ) → ((𝐹 ∘ 𝑓)‘𝑚) ≤ 𝑆)
124	61, 62, 93, 99, 101, 106, 115, 123	climsqz 15581	. . . . 5 ⊢ (((𝜑 ∧ 𝑓:ℕ⟶𝑋) ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) → (𝐹 ∘ 𝑓) ⇝ 𝑆)
125	124	ex 413	. . . 4 ⊢ ((𝜑 ∧ 𝑓:ℕ⟶𝑋) → (∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘)) → (𝐹 ∘ 𝑓) ⇝ 𝑆))
126	125	imdistanda 572	. . 3 ⊢ (𝜑 → ((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) → (𝑓:ℕ⟶𝑋 ∧ (𝐹 ∘ 𝑓) ⇝ 𝑆)))
127	126	eximdv 1920	. 2 ⊢ (𝜑 → (∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝑅‘𝑘) ≤ (𝐹‘(𝑓‘𝑘))) → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ (𝐹 ∘ 𝑓) ⇝ 𝑆)))
128	60, 127	mpd 15	1 ⊢ (𝜑 → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ (𝐹 ∘ 𝑓) ⇝ 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ⊆ wss 3947 ∅c0 4321 {csn 4627 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 ran crn 5676 ∘ ccom 5679 Fn wfn 6535 ⟶wf 6536 –onto→wfo 6538 ‘cfv 6540 (class class class)co 7405 supcsup 9431 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 ℤcz 12554 ℤ_≥cuz 12818 ℝ⁺crp 12970 ⇝ cli 15424

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cc 10426 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fl 13753 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429

This theorem is referenced by: (None)

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