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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 2728 | . . . . . . 7 β’ β¨β¨ +β , Β·β β©, normββ© = β¨β¨ +β , Β·β β©, normββ© | |
2 | eqid 2728 | . . . . . . 7 β’ (IndMetββ¨β¨ +β , Β·β β©, normββ©) = (IndMetββ¨β¨ +β , Β·β β©, normββ©) | |
3 | 1, 2 | hhxmet 30978 | . . . . . 6 β’ (IndMetββ¨β¨ +β , Β·β β©, normββ©) β (βMetβ β) |
4 | eqid 2728 | . . . . . . 7 β’ (MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©)) = (MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©)) | |
5 | 4 | methaus 24422 | . . . . . 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 23278 | . . . . . 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 6589 | . . . . 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 31002 | . . . . 5 β’ βπ£ = ((βπ‘β(MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©))) βΎ ( β βm β)) |
11 | 10 | funeqi 6568 | . . . 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 6577 | . . 3 β’ (Fun βπ£ β βπ£ Fn dom βπ£ ) | |
14 | 12, 13 | mpbi 229 | . 2 β’ βπ£ Fn dom βπ£ |
15 | funfvbrb 7054 | . . . . 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 30993 | . . . 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 3059 | . 2 β’ βπ₯ β dom βπ£ ( βπ£ βπ₯) β β |
21 | ffnfv 7123 | . 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 710 | 1 β’ βπ£ :dom βπ£ βΆ β |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β wcel 2099 βwral 3057 β¨cop 4630 class class class wbr 5142 dom cdm 5672 βΎ cres 5674 Fun wfun 6536 Fn wfn 6537 βΆwf 6538 βcfv 6542 (class class class)co 7414 βm cmap 8838 βcn 12236 βMetcxmet 21257 MetOpencmopn 21262 βπ‘clm 23123 Hauscha 23205 IndMetcims 30394 βchba 30722 +β cva 30723 Β·β csm 30724 normβcno 30726 βπ£ chli 30730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211 ax-mulf 11212 ax-hilex 30802 ax-hfvadd 30803 ax-hvcom 30804 ax-hvass 30805 ax-hv0cl 30806 ax-hvaddid 30807 ax-hfvmul 30808 ax-hvmulid 30809 ax-hvmulass 30810 ax-hvdistr1 30811 ax-hvdistr2 30812 ax-hvmul0 30813 ax-hfi 30882 ax-his1 30885 ax-his2 30886 ax-his3 30887 ax-his4 30888 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8840 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9459 df-inf 9460 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-n0 12497 df-z 12583 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-icc 13357 df-seq 13993 df-exp 14053 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-topgen 17418 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-top 22789 df-topon 22806 df-bases 22842 df-lm 23126 df-haus 23212 df-grpo 30296 df-gid 30297 df-ginv 30298 df-gdiv 30299 df-ablo 30348 df-vc 30362 df-nv 30395 df-va 30398 df-ba 30399 df-sm 30400 df-0v 30401 df-vs 30402 df-nmcv 30403 df-ims 30404 df-hnorm 30771 df-hvsub 30774 df-hlim 30775 |
This theorem is referenced by: hlimuni 31041 hhsscms 31081 occllem 31106 occl 31107 chscllem2 31441 chscllem4 31443 |
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