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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 24538 | . . . . 5 β’ (π β NrmMod β π β NrmGrp) | |
| 2 | nlmlmod 24539 | . . . . . 6 β’ (π β NrmMod β π β LMod) | |
| 3 | lmodabl 20751 | . . . . . 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 24489 | . . . . 5 β’ ((π β NrmGrp β§ π β Abel) β π β TopGrp) | |
| 6 | 1, 4, 5 | syl2anc 583 | . . . 4 β’ (π β NrmMod β π β TopGrp) |
| 7 | tgptmd 23927 | . . . 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 2724 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
| 10 | 9 | nlmnrg 24540 | . . . 4 β’ (π β NrmMod β (Scalarβπ) β NrmRing) |
| 11 | nrgtrg 24551 | . . . 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 1125 | . 2 β’ (π β NrmMod β (π β TopMnd β§ π β LMod β§ (Scalarβπ) β TopRing)) |
| 14 | eqid 2724 | . . 3 β’ ( Β·sf βπ) = ( Β·sf βπ) | |
| 15 | eqid 2724 | . . 3 β’ (TopOpenβπ) = (TopOpenβπ) | |
| 16 | eqid 2724 | . . 3 β’ (TopOpenβ(Scalarβπ)) = (TopOpenβ(Scalarβπ)) | |
| 17 | 9, 14, 15, 16 | nlmvscn 24548 | . 2 β’ (π β NrmMod β ( Β·sf βπ) β (((TopOpenβ(Scalarβπ)) Γt (TopOpenβπ)) Cn (TopOpenβπ))) |
| 18 | 14, 15, 9, 16 | istlm 24033 | . 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 582 | 1 β’ (π β NrmMod β π β TopMod) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1084 β wcel 2098 βcfv 6534 (class class class)co 7402 Scalarcsca 17205 TopOpenctopn 17372 Abelcabl 19697 LModclmod 20702 Β·sf cscaf 20703 Cn ccn 23072 Γt ctx 23408 TopMndctmd 23918 TopGrpctgp 23919 TopRingctrg 24004 TopModctlm 24006 NrmGrpcngp 24430 NrmRingcnrg 24432 NrmModcnlm 24433 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 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-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-ico 13331 df-icc 13332 df-fz 13486 df-fzo 13629 df-seq 13968 df-exp 14029 df-hash 14292 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-plusf 18568 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-grp 18862 df-minusg 18863 df-sbg 18864 df-mulg 18992 df-subg 19046 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-subrng 20442 df-subrg 20467 df-abv 20656 df-lmod 20704 df-scaf 20705 df-sra 21017 df-rgmod 21018 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-top 22740 df-topon 22757 df-topsp 22779 df-bases 22793 df-cn 23075 df-cnp 23076 df-tx 23410 df-hmeo 23603 df-tmd 23920 df-tgp 23921 df-trg 24008 df-tlm 24010 df-xms 24170 df-ms 24171 df-tms 24172 df-nm 24435 df-ngp 24436 df-nrg 24438 df-nlm 24439 |
| This theorem is referenced by: nvctvc 24561 |
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