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Mirrors > Home > MPE Home > Th. List > nlmtlm | Structured version Visualization version GIF version |
Description: A normed module is a topological module. (Contributed by Mario Carneiro, 6-Oct-2015.) |
Ref | Expression |
---|---|
nlmtlm | β’ (π β NrmMod β π β TopMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nlmngp 24064 | . . . . 5 β’ (π β NrmMod β π β NrmGrp) | |
2 | nlmlmod 24065 | . . . . . 6 β’ (π β NrmMod β π β LMod) | |
3 | lmodabl 20413 | . . . . . 6 β’ (π β LMod β π β Abel) | |
4 | 2, 3 | syl 17 | . . . . 5 β’ (π β NrmMod β π β Abel) |
5 | ngptgp 24015 | . . . . 5 β’ ((π β NrmGrp β§ π β Abel) β π β TopGrp) | |
6 | 1, 4, 5 | syl2anc 585 | . . . 4 β’ (π β NrmMod β π β TopGrp) |
7 | tgptmd 23453 | . . . 4 β’ (π β TopGrp β π β TopMnd) | |
8 | 6, 7 | syl 17 | . . 3 β’ (π β NrmMod β π β TopMnd) |
9 | eqid 2733 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
10 | 9 | nlmnrg 24066 | . . . 4 β’ (π β NrmMod β (Scalarβπ) β NrmRing) |
11 | nrgtrg 24077 | . . . 4 β’ ((Scalarβπ) β NrmRing β (Scalarβπ) β TopRing) | |
12 | 10, 11 | syl 17 | . . 3 β’ (π β NrmMod β (Scalarβπ) β TopRing) |
13 | 8, 2, 12 | 3jca 1129 | . 2 β’ (π β NrmMod β (π β TopMnd β§ π β LMod β§ (Scalarβπ) β TopRing)) |
14 | eqid 2733 | . . 3 β’ ( Β·sf βπ) = ( Β·sf βπ) | |
15 | eqid 2733 | . . 3 β’ (TopOpenβπ) = (TopOpenβπ) | |
16 | eqid 2733 | . . 3 β’ (TopOpenβ(Scalarβπ)) = (TopOpenβ(Scalarβπ)) | |
17 | 9, 14, 15, 16 | nlmvscn 24074 | . 2 β’ (π β NrmMod β ( Β·sf βπ) β (((TopOpenβ(Scalarβπ)) Γt (TopOpenβπ)) Cn (TopOpenβπ))) |
18 | 14, 15, 9, 16 | istlm 23559 | . 2 β’ (π β TopMod β ((π β TopMnd β§ π β LMod β§ (Scalarβπ) β TopRing) β§ ( Β·sf βπ) β (((TopOpenβ(Scalarβπ)) Γt (TopOpenβπ)) Cn (TopOpenβπ)))) |
19 | 13, 17, 18 | sylanbrc 584 | 1 β’ (π β NrmMod β π β TopMod) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 β wcel 2107 βcfv 6500 (class class class)co 7361 Scalarcsca 17144 TopOpenctopn 17311 Abelcabl 19571 LModclmod 20365 Β·sf cscaf 20366 Cn ccn 22598 Γt ctx 22934 TopMndctmd 23444 TopGrpctgp 23445 TopRingctrg 23530 TopModctlm 23532 NrmGrpcngp 23956 NrmRingcnrg 23958 NrmModcnlm 23959 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
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 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-fi 9355 df-sup 9386 df-inf 9387 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-ico 13279 df-icc 13280 df-fz 13434 df-fzo 13577 df-seq 13916 df-exp 13977 df-hash 14240 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-hom 17165 df-cco 17166 df-rest 17312 df-topn 17313 df-0g 17331 df-gsum 17332 df-topgen 17333 df-pt 17334 df-prds 17337 df-xrs 17392 df-qtop 17397 df-imas 17398 df-xps 17400 df-mre 17474 df-mrc 17475 df-acs 17477 df-plusf 18504 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-grp 18759 df-minusg 18760 df-sbg 18761 df-mulg 18881 df-subg 18933 df-cntz 19105 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-subrg 20262 df-abv 20319 df-lmod 20367 df-scaf 20368 df-sra 20678 df-rgmod 20679 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-cn 22601 df-cnp 22602 df-tx 22936 df-hmeo 23129 df-tmd 23446 df-tgp 23447 df-trg 23534 df-tlm 23536 df-xms 23696 df-ms 23697 df-tms 23698 df-nm 23961 df-ngp 23962 df-nrg 23964 df-nlm 23965 |
This theorem is referenced by: nvctvc 24087 |
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