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Mirrors > Home > HSE Home > Th. List > hlimf | Structured version Visualization version GIF version |
Description: Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlimf | ⊢ ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2771 | . . . . . . 7 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
2 | eqid 2771 | . . . . . . 7 ⊢ (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
3 | 1, 2 | hhxmet 28372 | . . . . . 6 ⊢ (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∈ (∞Met‘ ℋ) |
4 | eqid 2771 | . . . . . . 7 ⊢ (MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) = (MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) | |
5 | 4 | methaus 22545 | . . . . . 6 ⊢ ((IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∈ (∞Met‘ ℋ) → (MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) ∈ Haus) |
6 | lmfun 21406 | . . . . . 6 ⊢ ((MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) ∈ Haus → Fun (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)))) | |
7 | 3, 5, 6 | mp2b 10 | . . . . 5 ⊢ Fun (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) |
8 | funres 6071 | . . . . 5 ⊢ (Fun (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) → Fun ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑𝑚 ℕ))) | |
9 | 7, 8 | ax-mp 5 | . . . 4 ⊢ Fun ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑𝑚 ℕ)) |
10 | 1, 2, 4 | hhlm 28396 | . . . . 5 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑𝑚 ℕ)) |
11 | 10 | funeqi 6051 | . . . 4 ⊢ (Fun ⇝𝑣 ↔ Fun ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑𝑚 ℕ))) |
12 | 9, 11 | mpbir 221 | . . 3 ⊢ Fun ⇝𝑣 |
13 | funfn 6060 | . . 3 ⊢ (Fun ⇝𝑣 ↔ ⇝𝑣 Fn dom ⇝𝑣 ) | |
14 | 12, 13 | mpbi 220 | . 2 ⊢ ⇝𝑣 Fn dom ⇝𝑣 |
15 | funfvbrb 6475 | . . . . 5 ⊢ (Fun ⇝𝑣 → (𝑥 ∈ dom ⇝𝑣 ↔ 𝑥 ⇝𝑣 ( ⇝𝑣 ‘𝑥))) | |
16 | 12, 15 | ax-mp 5 | . . . 4 ⊢ (𝑥 ∈ dom ⇝𝑣 ↔ 𝑥 ⇝𝑣 ( ⇝𝑣 ‘𝑥)) |
17 | fvex 6344 | . . . . 5 ⊢ ( ⇝𝑣 ‘𝑥) ∈ V | |
18 | 17 | hlimveci 28387 | . . . 4 ⊢ (𝑥 ⇝𝑣 ( ⇝𝑣 ‘𝑥) → ( ⇝𝑣 ‘𝑥) ∈ ℋ) |
19 | 16, 18 | sylbi 207 | . . 3 ⊢ (𝑥 ∈ dom ⇝𝑣 → ( ⇝𝑣 ‘𝑥) ∈ ℋ) |
20 | 19 | rgen 3071 | . 2 ⊢ ∀𝑥 ∈ dom ⇝𝑣 ( ⇝𝑣 ‘𝑥) ∈ ℋ |
21 | ffnfv 6533 | . 2 ⊢ ( ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ ↔ ( ⇝𝑣 Fn dom ⇝𝑣 ∧ ∀𝑥 ∈ dom ⇝𝑣 ( ⇝𝑣 ‘𝑥) ∈ ℋ)) | |
22 | 14, 20, 21 | mpbir2an 690 | 1 ⊢ ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∈ wcel 2145 ∀wral 3061 〈cop 4323 class class class wbr 4787 dom cdm 5250 ↾ cres 5252 Fun wfun 6024 Fn wfn 6025 ⟶wf 6026 ‘cfv 6030 (class class class)co 6796 ↑𝑚 cmap 8013 ℕcn 11226 ∞Metcxmt 19946 MetOpencmopn 19951 ⇝𝑡clm 21251 Hauscha 21333 IndMetcims 27786 ℋchil 28116 +ℎ cva 28117 ·ℎ csm 28118 normℎcno 28120 ⇝𝑣 chli 28124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220 ax-addf 10221 ax-mulf 10222 ax-hilex 28196 ax-hfvadd 28197 ax-hvcom 28198 ax-hvass 28199 ax-hv0cl 28200 ax-hvaddid 28201 ax-hfvmul 28202 ax-hvmulid 28203 ax-hvmulass 28204 ax-hvdistr1 28205 ax-hvdistr2 28206 ax-hvmul0 28207 ax-hfi 28276 ax-his1 28279 ax-his2 28280 ax-his3 28281 ax-his4 28282 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-er 7900 df-map 8015 df-pm 8016 df-en 8114 df-dom 8115 df-sdom 8116 df-sup 8508 df-inf 8509 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-4 11287 df-n0 11500 df-z 11585 df-uz 11894 df-q 11997 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-icc 12387 df-seq 13009 df-exp 13068 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-topgen 16312 df-psmet 19953 df-xmet 19954 df-met 19955 df-bl 19956 df-mopn 19957 df-top 20919 df-topon 20936 df-bases 20971 df-lm 21254 df-haus 21340 df-grpo 27687 df-gid 27688 df-ginv 27689 df-gdiv 27690 df-ablo 27739 df-vc 27754 df-nv 27787 df-va 27790 df-ba 27791 df-sm 27792 df-0v 27793 df-vs 27794 df-nmcv 27795 df-ims 27796 df-hnorm 28165 df-hvsub 28168 df-hlim 28169 |
This theorem is referenced by: hlimuni 28435 hhsscms 28476 occllem 28502 occl 28503 chscllem2 28837 chscllem4 28839 |
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