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Mirrors > Home > MPE Home > Th. List > nmcnc | Structured version Visualization version GIF version |
Description: The norm of a normed complex vector space is a continuous function to β. (For β, see nmcvcn 29679.) (Contributed by NM, 12-Aug-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmcnc.1 | β’ π = (normCVβπ) |
nmcnc.2 | β’ πΆ = (IndMetβπ) |
nmcnc.j | β’ π½ = (MetOpenβπΆ) |
nmcnc.k | β’ πΎ = (TopOpenββfld) |
Ref | Expression |
---|---|
nmcnc | β’ (π β NrmCVec β π β (π½ Cn πΎ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmcnc.k | . . . 4 β’ πΎ = (TopOpenββfld) | |
2 | 1 | cnfldtop 24163 | . . 3 β’ πΎ β Top |
3 | cnrest2r 22654 | . . 3 β’ (πΎ β Top β (π½ Cn (πΎ βΎt β)) β (π½ Cn πΎ)) | |
4 | 2, 3 | ax-mp 5 | . 2 β’ (π½ Cn (πΎ βΎt β)) β (π½ Cn πΎ) |
5 | nmcnc.1 | . . 3 β’ π = (normCVβπ) | |
6 | nmcnc.2 | . . 3 β’ πΆ = (IndMetβπ) | |
7 | nmcnc.j | . . 3 β’ π½ = (MetOpenβπΆ) | |
8 | 1 | tgioo2 24182 | . . . 4 β’ (topGenβran (,)) = (πΎ βΎt β) |
9 | 8 | eqcomi 2742 | . . 3 β’ (πΎ βΎt β) = (topGenβran (,)) |
10 | 5, 6, 7, 9 | nmcvcn 29679 | . 2 β’ (π β NrmCVec β π β (π½ Cn (πΎ βΎt β))) |
11 | 4, 10 | sselid 3943 | 1 β’ (π β NrmCVec β π β (π½ Cn πΎ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3911 ran crn 5635 βcfv 6497 (class class class)co 7358 βcr 11055 (,)cioo 13270 βΎt crest 17307 TopOpenctopn 17308 topGenctg 17324 MetOpencmopn 20802 βfldccnfld 20812 Topctop 22258 Cn ccn 22591 NrmCVeccnv 29568 normCVcnmcv 29574 IndMetcims 29575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 ax-addf 11135 ax-mulf 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fi 9352 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-q 12879 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13274 df-fz 13431 df-seq 13913 df-exp 13974 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-mulr 17152 df-starv 17153 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-rest 17309 df-topn 17310 df-topgen 17330 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cn 22594 df-cnp 22595 df-xms 23689 df-ms 23690 df-grpo 29477 df-gid 29478 df-ginv 29479 df-gdiv 29480 df-ablo 29529 df-vc 29543 df-nv 29576 df-va 29579 df-ba 29580 df-sm 29581 df-0v 29582 df-vs 29583 df-nmcv 29584 df-ims 29585 |
This theorem is referenced by: dipcn 29704 |
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