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| Description: A convergent function is Cauchy. (Contributed by Glauco Siliprandi, 15-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| cvgcau.1 | ⊢ Ⅎ𝑗𝐹 | 
| cvgcau.2 | ⊢ Ⅎ𝑘𝐹 | 
| cvgcau.3 | ⊢ (𝜑 → 𝑀 ∈ 𝑍) | 
| cvgcau.4 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) | 
| cvgcau.5 | ⊢ 𝑍 = (ℤ≥‘𝑀) | 
| cvgcau.6 | ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) | 
| cvgcau.7 | ⊢ (𝜑 → 𝑋 ∈ ℝ+) | 
| Ref | Expression | 
|---|---|
| cvgcau | ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | breq2 5147 | . . . 4 ⊢ (𝑥 = 𝑋 → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) | |
| 2 | 1 | anbi2d 630 | . . 3 ⊢ (𝑥 = 𝑋 → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋))) | 
| 3 | 2 | rexralbidv 3223 | . 2 ⊢ (𝑥 = 𝑋 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋))) | 
| 4 | cvgcau.6 | . . 3 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) | |
| 5 | cvgcau.5 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 6 | cvgcau.3 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑍) | |
| 7 | 5, 6 | eluzelz2d 45424 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | 
| 8 | cvgcau.4 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
| 9 | cvgcau.1 | . . . . 5 ⊢ Ⅎ𝑗𝐹 | |
| 10 | cvgcau.2 | . . . . 5 ⊢ Ⅎ𝑘𝐹 | |
| 11 | 9, 10, 5 | caucvgbf 45500 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ dom ⇝ ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))) | 
| 12 | 7, 8, 11 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))) | 
| 13 | 4, 12 | mpbid 232 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) | 
| 14 | cvgcau.7 | . 2 ⊢ (𝜑 → 𝑋 ∈ ℝ+) | |
| 15 | 3, 13, 14 | rspcdva 3623 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 ∃wrex 3070 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 < clt 11295 − cmin 11492 ℤcz 12613 ℤ≥cuz 12878 ℝ+crp 13034 abscabs 15273 ⇝ cli 15520 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-ico 13393 df-fl 13832 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 | 
| This theorem is referenced by: cvgcaule 45502 | 
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