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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 6811 | . . . 4 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) | |
3 | 1, 2 | eqtrid 2788 | . . 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 6811 | . . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → (IndMet‘𝑈) = (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) | |
7 | 5, 6 | eqtrid 2788 | . . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → 𝐶 = (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) |
8 | 7 | fveq2d 6815 | . . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → (MetOpen‘𝐶) = (MetOpen‘(IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)))) |
9 | 4, 8 | eqtrid 2788 | . . . . 5 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → 𝐽 = (MetOpen‘(IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)))) |
10 | 9 | oveq2d 7332 | . . . 4 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → (𝐾 ×t 𝐽) = (𝐾 ×t (MetOpen‘(IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))))) |
11 | 10, 9 | oveq12d 7334 | . . 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 2831 | . 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 2736 | . . 3 ⊢ (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
14 | eqid 2736 | . . 3 ⊢ (MetOpen‘(IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) = (MetOpen‘(IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) | |
15 | eqid 2736 | . . 3 ⊢ ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
16 | smcn.k | . . 3 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
17 | eqid 2736 | . . 3 ⊢ (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
18 | eqid 2736 | . . 3 ⊢ (normCV‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (normCV‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
19 | elimnvu 29178 | . . 3 ⊢ if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) ∈ NrmCVec | |
20 | eqid 2736 | . . 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 29191 | . 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 4528 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑆 ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ifcif 4470 〈cop 4576 ‘cfv 6465 (class class class)co 7316 1c1 10951 + caddc 10953 · cmul 10955 / cdiv 11711 abscabs 15021 TopOpenctopn 17206 MetOpencmopn 20667 ℂfldccnfld 20677 Cn ccn 22455 ×t ctx 22791 NrmCVeccnv 29078 BaseSetcba 29080 ·𝑠OLD cns 29081 normCVcnmcv 29084 IndMetcims 29085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5223 ax-sep 5237 ax-nul 5244 ax-pow 5302 ax-pr 5366 ax-un 7629 ax-cnex 11006 ax-resscn 11007 ax-1cn 11008 ax-icn 11009 ax-addcl 11010 ax-addrcl 11011 ax-mulcl 11012 ax-mulrcl 11013 ax-mulcom 11014 ax-addass 11015 ax-mulass 11016 ax-distr 11017 ax-i2m1 11018 ax-1ne0 11019 ax-1rid 11020 ax-rnegex 11021 ax-rrecex 11022 ax-cnre 11023 ax-pre-lttri 11024 ax-pre-lttrn 11025 ax-pre-ltadd 11026 ax-pre-mulgt0 11027 ax-pre-sup 11028 ax-addf 11029 ax-mulf 11030 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3442 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4471 df-pw 4546 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4850 df-int 4892 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5170 df-tr 5204 df-id 5506 df-eprel 5512 df-po 5520 df-so 5521 df-fr 5562 df-se 5563 df-we 5564 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-pred 6224 df-ord 6291 df-on 6292 df-lim 6293 df-suc 6294 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-isom 6474 df-riota 7273 df-ov 7319 df-oprab 7320 df-mpo 7321 df-of 7574 df-om 7759 df-1st 7877 df-2nd 7878 df-supp 8026 df-frecs 8145 df-wrecs 8176 df-recs 8250 df-rdg 8289 df-1o 8345 df-2o 8346 df-er 8547 df-map 8666 df-ixp 8735 df-en 8783 df-dom 8784 df-sdom 8785 df-fin 8786 df-fsupp 9205 df-fi 9246 df-sup 9277 df-inf 9278 df-oi 9345 df-card 9774 df-pnf 11090 df-mnf 11091 df-xr 11092 df-ltxr 11093 df-le 11094 df-sub 11286 df-neg 11287 df-div 11712 df-nn 12053 df-2 12115 df-3 12116 df-4 12117 df-5 12118 df-6 12119 df-7 12120 df-8 12121 df-9 12122 df-n0 12313 df-z 12399 df-dec 12517 df-uz 12662 df-q 12768 df-rp 12810 df-xneg 12927 df-xadd 12928 df-xmul 12929 df-icc 13165 df-fz 13319 df-fzo 13462 df-seq 13801 df-exp 13862 df-hash 14124 df-cj 14886 df-re 14887 df-im 14888 df-sqrt 15022 df-abs 15023 df-struct 16922 df-sets 16939 df-slot 16957 df-ndx 16969 df-base 16987 df-ress 17016 df-plusg 17049 df-mulr 17050 df-starv 17051 df-sca 17052 df-vsca 17053 df-ip 17054 df-tset 17055 df-ple 17056 df-ds 17058 df-unif 17059 df-hom 17060 df-cco 17061 df-rest 17207 df-topn 17208 df-0g 17226 df-gsum 17227 df-topgen 17228 df-pt 17229 df-prds 17232 df-xrs 17287 df-qtop 17292 df-imas 17293 df-xps 17295 df-mre 17369 df-mrc 17370 df-acs 17372 df-mgm 18400 df-sgrp 18449 df-mnd 18460 df-submnd 18505 df-mulg 18774 df-cntz 18996 df-cmn 19460 df-psmet 20669 df-xmet 20670 df-met 20671 df-bl 20672 df-mopn 20673 df-cnfld 20678 df-top 22123 df-topon 22140 df-topsp 22162 df-bases 22176 df-cn 22458 df-cnp 22459 df-tx 22793 df-hmeo 22986 df-xms 23553 df-tms 23555 df-grpo 28987 df-gid 28988 df-ginv 28989 df-gdiv 28990 df-ablo 29039 df-vc 29053 df-nv 29086 df-va 29089 df-ba 29090 df-sm 29091 df-0v 29092 df-vs 29093 df-nmcv 29094 df-ims 29095 |
This theorem is referenced by: vmcn 29193 dipcn 29214 ipasslem7 29330 |
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