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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 24607 | . . . . 5 β’ (π β NrmMod β π β NrmGrp) | |
| 2 | nlmlmod 24608 | . . . . . 6 β’ (π β NrmMod β π β LMod) | |
| 3 | lmodabl 20792 | . . . . . 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 24558 | . . . . 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 23996 | . . . 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 2728 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
| 10 | 9 | nlmnrg 24609 | . . . 4 β’ (π β NrmMod β (Scalarβπ) β NrmRing) |
| 11 | nrgtrg 24620 | . . . 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 1126 | . 2 β’ (π β NrmMod β (π β TopMnd β§ π β LMod β§ (Scalarβπ) β TopRing)) |
| 14 | eqid 2728 | . . 3 β’ ( Β·sf βπ) = ( Β·sf βπ) | |
| 15 | eqid 2728 | . . 3 β’ (TopOpenβπ) = (TopOpenβπ) | |
| 16 | eqid 2728 | . . 3 β’ (TopOpenβ(Scalarβπ)) = (TopOpenβ(Scalarβπ)) | |
| 17 | 9, 14, 15, 16 | nlmvscn 24617 | . 2 β’ (π β NrmMod β ( Β·sf βπ) β (((TopOpenβ(Scalarβπ)) Γt (TopOpenβπ)) Cn (TopOpenβπ))) |
| 18 | 14, 15, 9, 16 | istlm 24102 | . 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 1085 β wcel 2099 βcfv 6548 (class class class)co 7420 Scalarcsca 17236 TopOpenctopn 17403 Abelcabl 19736 LModclmod 20743 Β·sf cscaf 20744 Cn ccn 23141 Γt ctx 23477 TopMndctmd 23987 TopGrpctgp 23988 TopRingctrg 24073 TopModctlm 24075 NrmGrpcngp 24499 NrmRingcnrg 24501 NrmModcnlm 24502 |
| 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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
| 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 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-pt 17426 df-prds 17429 df-xrs 17484 df-qtop 17489 df-imas 17490 df-xps 17492 df-mre 17566 df-mrc 17567 df-acs 17569 df-plusf 18599 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-mulg 19024 df-subg 19078 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-subrng 20483 df-subrg 20508 df-abv 20697 df-lmod 20745 df-scaf 20746 df-sra 21058 df-rgmod 21059 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cn 23144 df-cnp 23145 df-tx 23479 df-hmeo 23672 df-tmd 23989 df-tgp 23990 df-trg 24077 df-tlm 24079 df-xms 24239 df-ms 24240 df-tms 24241 df-nm 24504 df-ngp 24505 df-nrg 24507 df-nlm 24508 |
| This theorem is referenced by: nvctvc 24630 |
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