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Mirrors > Home > MPE Home > Th. List > smcn | Structured version Visualization version GIF version |
Description: Scalar multiplication is jointly continuous in both arguments. (Contributed by NM, 16-Jun-2009.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
smcn.c | ⊢ 𝐶 = (IndMet‘𝑈) |
smcn.j | ⊢ 𝐽 = (MetOpen‘𝐶) |
smcn.s | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
smcn.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
smcn | ⊢ (𝑈 ∈ NrmCVec → 𝑆 ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smcn.s | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
2 | fveq2 6717 | . . . 4 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) | |
3 | 1, 2 | syl5eq 2790 | . . 3 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → 𝑆 = ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) |
4 | smcn.j | . . . . . 6 ⊢ 𝐽 = (MetOpen‘𝐶) | |
5 | smcn.c | . . . . . . . 8 ⊢ 𝐶 = (IndMet‘𝑈) | |
6 | fveq2 6717 | . . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → (IndMet‘𝑈) = (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) | |
7 | 5, 6 | syl5eq 2790 | . . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → 𝐶 = (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) |
8 | 7 | fveq2d 6721 | . . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → (MetOpen‘𝐶) = (MetOpen‘(IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)))) |
9 | 4, 8 | syl5eq 2790 | . . . . 5 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → 𝐽 = (MetOpen‘(IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)))) |
10 | 9 | oveq2d 7229 | . . . 4 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → (𝐾 ×t 𝐽) = (𝐾 ×t (MetOpen‘(IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))))) |
11 | 10, 9 | oveq12d 7231 | . . 3 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → ((𝐾 ×t 𝐽) Cn 𝐽) = ((𝐾 ×t (MetOpen‘(IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)))) Cn (MetOpen‘(IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))))) |
12 | 3, 11 | eleq12d 2832 | . 2 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → (𝑆 ∈ ((𝐾 ×t 𝐽) Cn 𝐽) ↔ ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) ∈ ((𝐾 ×t (MetOpen‘(IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)))) Cn (MetOpen‘(IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)))))) |
13 | eqid 2737 | . . 3 ⊢ (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
14 | eqid 2737 | . . 3 ⊢ (MetOpen‘(IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) = (MetOpen‘(IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) | |
15 | eqid 2737 | . . 3 ⊢ ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
16 | smcn.k | . . 3 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
17 | eqid 2737 | . . 3 ⊢ (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
18 | eqid 2737 | . . 3 ⊢ (normCV‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (normCV‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
19 | elimnvu 28765 | . . 3 ⊢ if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) ∈ NrmCVec | |
20 | eqid 2737 | . . 3 ⊢ (1 / (1 + (((((normCV‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))‘𝑦) + (abs‘𝑥)) + 1) / 𝑟))) = (1 / (1 + (((((normCV‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))‘𝑦) + (abs‘𝑥)) + 1) / 𝑟))) | |
21 | 13, 14, 15, 16, 17, 18, 19, 20 | smcnlem 28778 | . 2 ⊢ ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) ∈ ((𝐾 ×t (MetOpen‘(IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)))) Cn (MetOpen‘(IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)))) |
22 | 12, 21 | dedth 4497 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑆 ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ifcif 4439 〈cop 4547 ‘cfv 6380 (class class class)co 7213 1c1 10730 + caddc 10732 · cmul 10734 / cdiv 11489 abscabs 14797 TopOpenctopn 16926 MetOpencmopn 20353 ℂfldccnfld 20363 Cn ccn 22121 ×t ctx 22457 NrmCVeccnv 28665 BaseSetcba 28667 ·𝑠OLD cns 28668 normCVcnmcv 28671 IndMetcims 28672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-icc 12942 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cn 22124 df-cnp 22125 df-tx 22459 df-hmeo 22652 df-xms 23218 df-tms 23220 df-grpo 28574 df-gid 28575 df-ginv 28576 df-gdiv 28577 df-ablo 28626 df-vc 28640 df-nv 28673 df-va 28676 df-ba 28677 df-sm 28678 df-0v 28679 df-vs 28680 df-nmcv 28681 df-ims 28682 |
This theorem is referenced by: vmcn 28780 dipcn 28801 ipasslem7 28917 |
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