caun0 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > caun0		Structured version Visualization version GIF version

Theorem caun0 23271

Description: A metric with a Cauchy sequence cannot be empty. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.)

**Assertion**
Ref	Expression
caun0	⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝑋 ≠ ∅)

Proof of Theorem caun0 Dummy variables 𝑗 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1rp 12021	. . . 4 ⊢ 1 ∈ ℝ⁺
2	1	ne0ii 4058	. . 3 ⊢ ℝ⁺ ≠ ∅
3		iscau2 23267	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))))
4	3	simplbda 655	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
5		r19.2z 4196	. . 3 ⊢ ((ℝ⁺ ≠ ∅ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)) → ∃𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
6	2, 4, 5	sylancr 698	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → ∃𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
7		uzid 11886	. . . . . 6 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
8		ne0i 4056	. . . . . 6 ⊢ (𝑗 ∈ (ℤ_≥‘𝑗) → (ℤ_≥‘𝑗) ≠ ∅)
9		r19.2z 4196	. . . . . . 7 ⊢ (((ℤ_≥‘𝑗) ≠ ∅ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)) → ∃𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
10	9	ex 449	. . . . . 6 ⊢ ((ℤ_≥‘𝑗) ≠ ∅ → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → ∃𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)))
11	7, 8, 10	3syl 18	. . . . 5 ⊢ (𝑗 ∈ ℤ → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → ∃𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)))
12		ne0i 4056	. . . . . . 7 ⊢ ((𝐹‘𝑘) ∈ 𝑋 → 𝑋 ≠ ∅)
13	12	3ad2ant2 1128	. . . . . 6 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → 𝑋 ≠ ∅)
14	13	rexlimivw 3159	. . . . 5 ⊢ (∃𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → 𝑋 ≠ ∅)
15	11, 14	syl6 35	. . . 4 ⊢ (𝑗 ∈ ℤ → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → 𝑋 ≠ ∅))
16	15	rexlimiv 3157	. . 3 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → 𝑋 ≠ ∅)
17	16	rexlimivw 3159	. 2 ⊢ (∃𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → 𝑋 ≠ ∅)
18	6, 17	syl 17	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝑋 ≠ ∅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 ∈ wcel 2131 ≠ wne 2924 ∀wral 3042 ∃wrex 3043 ∅c0 4050 class class class wbr 4796 dom cdm 5258 ‘cfv 6041 (class class class)co 6805 ↑_pm cpm 8016 ℂcc 10118 1c1 10121 < clt 10258 ℤcz 11561 ℤ_≥cuz 11871 ℝ⁺crp 12017 ∞Metcxmt 19925 Caucca 23243

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-op 4320 df-uni 4581 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-id 5166 df-po 5179 df-so 5180 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-1st 7325 df-2nd 7326 df-er 7903 df-map 8017 df-pm 8018 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-z 11562 df-uz 11872 df-rp 12018 df-xadd 12132 df-psmet 19932 df-xmet 19933 df-bl 19935 df-cau 23246

This theorem is referenced by: cmetcau 23279

W3C validator