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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 | β’ βπ£ :dom βπ£ βΆ β |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . . . . . 7 β’ β¨β¨ +β , Β·β β©, normββ© = β¨β¨ +β , Β·β β©, normββ© | |
2 | eqid 2732 | . . . . . . 7 β’ (IndMetββ¨β¨ +β , Β·β β©, normββ©) = (IndMetββ¨β¨ +β , Β·β β©, normββ©) | |
3 | 1, 2 | hhxmet 30423 | . . . . . 6 β’ (IndMetββ¨β¨ +β , Β·β β©, normββ©) β (βMetβ β) |
4 | eqid 2732 | . . . . . . 7 β’ (MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©)) = (MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©)) | |
5 | 4 | methaus 24028 | . . . . . 6 β’ ((IndMetββ¨β¨ +β , Β·β β©, normββ©) β (βMetβ β) β (MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©)) β Haus) |
6 | lmfun 22884 | . . . . . 6 β’ ((MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©)) β Haus β Fun (βπ‘β(MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©)))) | |
7 | 3, 5, 6 | mp2b 10 | . . . . 5 β’ Fun (βπ‘β(MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©))) |
8 | funres 6590 | . . . . 5 β’ (Fun (βπ‘β(MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©))) β Fun ((βπ‘β(MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©))) βΎ ( β βm β))) | |
9 | 7, 8 | ax-mp 5 | . . . 4 β’ Fun ((βπ‘β(MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©))) βΎ ( β βm β)) |
10 | 1, 2, 4 | hhlm 30447 | . . . . 5 β’ βπ£ = ((βπ‘β(MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©))) βΎ ( β βm β)) |
11 | 10 | funeqi 6569 | . . . 4 β’ (Fun βπ£ β Fun ((βπ‘β(MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©))) βΎ ( β βm β))) |
12 | 9, 11 | mpbir 230 | . . 3 β’ Fun βπ£ |
13 | funfn 6578 | . . 3 β’ (Fun βπ£ β βπ£ Fn dom βπ£ ) | |
14 | 12, 13 | mpbi 229 | . 2 β’ βπ£ Fn dom βπ£ |
15 | funfvbrb 7052 | . . . . 5 β’ (Fun βπ£ β (π₯ β dom βπ£ β π₯ βπ£ ( βπ£ βπ₯))) | |
16 | 12, 15 | ax-mp 5 | . . . 4 β’ (π₯ β dom βπ£ β π₯ βπ£ ( βπ£ βπ₯)) |
17 | fvex 6904 | . . . . 5 β’ ( βπ£ βπ₯) β V | |
18 | 17 | hlimveci 30438 | . . . 4 β’ (π₯ βπ£ ( βπ£ βπ₯) β ( βπ£ βπ₯) β β) |
19 | 16, 18 | sylbi 216 | . . 3 β’ (π₯ β dom βπ£ β ( βπ£ βπ₯) β β) |
20 | 19 | rgen 3063 | . 2 β’ βπ₯ β dom βπ£ ( βπ£ βπ₯) β β |
21 | ffnfv 7117 | . 2 β’ ( βπ£ :dom βπ£ βΆ β β ( βπ£ Fn dom βπ£ β§ βπ₯ β dom βπ£ ( βπ£ βπ₯) β β)) | |
22 | 14, 20, 21 | mpbir2an 709 | 1 β’ βπ£ :dom βπ£ βΆ β |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β wcel 2106 βwral 3061 β¨cop 4634 class class class wbr 5148 dom cdm 5676 βΎ cres 5678 Fun wfun 6537 Fn wfn 6538 βΆwf 6539 βcfv 6543 (class class class)co 7408 βm cmap 8819 βcn 12211 βMetcxmet 20928 MetOpencmopn 20933 βπ‘clm 22729 Hauscha 22811 IndMetcims 29839 βchba 30167 +β cva 30168 Β·β csm 30169 normβcno 30171 βπ£ chli 30175 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 ax-hilex 30247 ax-hfvadd 30248 ax-hvcom 30249 ax-hvass 30250 ax-hv0cl 30251 ax-hvaddid 30252 ax-hfvmul 30253 ax-hvmulid 30254 ax-hvmulass 30255 ax-hvdistr1 30256 ax-hvdistr2 30257 ax-hvmul0 30258 ax-hfi 30327 ax-his1 30330 ax-his2 30331 ax-his3 30332 ax-his4 30333 |
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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-n0 12472 df-z 12558 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-icc 13330 df-seq 13966 df-exp 14027 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-topgen 17388 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-top 22395 df-topon 22412 df-bases 22448 df-lm 22732 df-haus 22818 df-grpo 29741 df-gid 29742 df-ginv 29743 df-gdiv 29744 df-ablo 29793 df-vc 29807 df-nv 29840 df-va 29843 df-ba 29844 df-sm 29845 df-0v 29846 df-vs 29847 df-nmcv 29848 df-ims 29849 df-hnorm 30216 df-hvsub 30219 df-hlim 30220 |
This theorem is referenced by: hlimuni 30486 hhsscms 30526 occllem 30551 occl 30552 chscllem2 30886 chscllem4 30888 |
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