Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dipcn | Structured version Visualization version GIF version |
Description: Inner product is jointly continuous in both arguments. (Contributed by NM, 21-Aug-2007.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dipcn.p | ⊢ 𝑃 = (·𝑖OLD‘𝑈) |
dipcn.c | ⊢ 𝐶 = (IndMet‘𝑈) |
dipcn.j | ⊢ 𝐽 = (MetOpen‘𝐶) |
dipcn.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
dipcn | ⊢ (𝑈 ∈ NrmCVec → 𝑃 ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2819 | . . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
2 | eqid 2819 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
3 | eqid 2819 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
4 | eqid 2819 | . . 3 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
5 | dipcn.p | . . 3 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
6 | 1, 2, 3, 4, 5 | dipfval 28471 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑃 = (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝑦)))↑2)) / 4))) |
7 | dipcn.c | . . . . 5 ⊢ 𝐶 = (IndMet‘𝑈) | |
8 | 1, 7 | imsxmet 28461 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝐶 ∈ (∞Met‘(BaseSet‘𝑈))) |
9 | dipcn.j | . . . . 5 ⊢ 𝐽 = (MetOpen‘𝐶) | |
10 | 9 | mopntopon 23041 | . . . 4 ⊢ (𝐶 ∈ (∞Met‘(BaseSet‘𝑈)) → 𝐽 ∈ (TopOn‘(BaseSet‘𝑈))) |
11 | 8, 10 | syl 17 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐽 ∈ (TopOn‘(BaseSet‘𝑈))) |
12 | dipcn.k | . . . 4 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
13 | fzfid 13333 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → (1...4) ∈ Fin) | |
14 | 11 | adantr 483 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → 𝐽 ∈ (TopOn‘(BaseSet‘𝑈))) |
15 | 12 | cnfldtopon 23383 | . . . . . . 7 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
16 | 15 | a1i 11 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → 𝐾 ∈ (TopOn‘ℂ)) |
17 | ax-icn 10588 | . . . . . . 7 ⊢ i ∈ ℂ | |
18 | elfznn 12928 | . . . . . . . . 9 ⊢ (𝑘 ∈ (1...4) → 𝑘 ∈ ℕ) | |
19 | 18 | adantl 484 | . . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → 𝑘 ∈ ℕ) |
20 | 19 | nnnn0d 11947 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → 𝑘 ∈ ℕ0) |
21 | expcl 13439 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (i↑𝑘) ∈ ℂ) | |
22 | 17, 20, 21 | sylancr 589 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (i↑𝑘) ∈ ℂ) |
23 | 14, 14, 16, 22 | cnmpt2c 22270 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ (i↑𝑘)) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
24 | 14, 14 | cnmpt1st 22268 | . . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
25 | 14, 14 | cnmpt2nd 22269 | . . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
26 | 7, 9, 3, 12 | smcn 28467 | . . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → ( ·𝑠OLD ‘𝑈) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
27 | 26 | adantr 483 | . . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → ( ·𝑠OLD ‘𝑈) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
28 | 14, 14, 23, 25, 27 | cnmpt22f 22275 | . . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ ((i↑𝑘)( ·𝑠OLD ‘𝑈)𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
29 | 7, 9, 2 | vacn 28463 | . . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → ( +𝑣 ‘𝑈) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
30 | 29 | adantr 483 | . . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → ( +𝑣 ‘𝑈) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
31 | 14, 14, 24, 28, 30 | cnmpt22f 22275 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ (𝑥( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
32 | 4, 7, 9, 12 | nmcnc 28465 | . . . . . . . 8 ⊢ (𝑈 ∈ NrmCVec → (normCV‘𝑈) ∈ (𝐽 Cn 𝐾)) |
33 | 32 | adantr 483 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (normCV‘𝑈) ∈ (𝐽 Cn 𝐾)) |
34 | 14, 14, 31, 33 | cnmpt21f 22272 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ ((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝑦)))) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
35 | 12 | sqcn 23474 | . . . . . . 7 ⊢ (𝑧 ∈ ℂ ↦ (𝑧↑2)) ∈ (𝐾 Cn 𝐾) |
36 | 35 | a1i 11 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑧 ∈ ℂ ↦ (𝑧↑2)) ∈ (𝐾 Cn 𝐾)) |
37 | oveq1 7155 | . . . . . 6 ⊢ (𝑧 = ((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝑦))) → (𝑧↑2) = (((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝑦)))↑2)) | |
38 | 14, 14, 34, 16, 36, 37 | cnmpt21 22271 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ (((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝑦)))↑2)) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
39 | 12 | mulcn 23467 | . . . . . 6 ⊢ · ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
40 | 39 | a1i 11 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → · ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
41 | 14, 14, 23, 38, 40 | cnmpt22f 22275 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ ((i↑𝑘) · (((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝑦)))↑2))) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
42 | 12, 11, 13, 11, 41 | fsum2cn 23471 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝑦)))↑2))) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
43 | 15 | a1i 11 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐾 ∈ (TopOn‘ℂ)) |
44 | 4cn 11714 | . . . . 5 ⊢ 4 ∈ ℂ | |
45 | 4ne0 11737 | . . . . 5 ⊢ 4 ≠ 0 | |
46 | 12 | divccn 23473 | . . . . 5 ⊢ ((4 ∈ ℂ ∧ 4 ≠ 0) → (𝑧 ∈ ℂ ↦ (𝑧 / 4)) ∈ (𝐾 Cn 𝐾)) |
47 | 44, 45, 46 | mp2an 690 | . . . 4 ⊢ (𝑧 ∈ ℂ ↦ (𝑧 / 4)) ∈ (𝐾 Cn 𝐾) |
48 | 47 | a1i 11 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑧 ∈ ℂ ↦ (𝑧 / 4)) ∈ (𝐾 Cn 𝐾)) |
49 | oveq1 7155 | . . 3 ⊢ (𝑧 = Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝑦)))↑2)) → (𝑧 / 4) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝑦)))↑2)) / 4)) | |
50 | 11, 11, 42, 43, 48, 49 | cnmpt21 22271 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)((i↑𝑘)( ·𝑠OLD ‘𝑈)𝑦)))↑2)) / 4)) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
51 | 6, 50 | eqeltrd 2911 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑃 ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7148 ∈ cmpo 7150 ℂcc 10527 0cc0 10529 1c1 10530 ici 10531 · cmul 10534 / cdiv 11289 ℕcn 11630 2c2 11684 4c4 11686 ℕ0cn0 11889 ...cfz 12884 ↑cexp 13421 Σcsu 15034 TopOpenctopn 16687 ∞Metcxmet 20522 MetOpencmopn 20527 ℂfldccnfld 20537 TopOnctopon 21510 Cn ccn 21824 ×t ctx 22160 NrmCVeccnv 28353 +𝑣 cpv 28354 BaseSetcba 28355 ·𝑠OLD cns 28356 normCVcnmcv 28359 IndMetcims 28360 ·𝑖OLDcdip 28469 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-fi 8867 df-sup 8898 df-inf 8899 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-ioo 12734 df-icc 12737 df-fz 12885 df-fzo 13026 df-seq 13362 df-exp 13422 df-hash 13683 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-cnfld 20538 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-cn 21827 df-cnp 21828 df-tx 22162 df-hmeo 22355 df-xms 22922 df-ms 22923 df-tms 22924 df-grpo 28262 df-gid 28263 df-ginv 28264 df-gdiv 28265 df-ablo 28314 df-vc 28328 df-nv 28361 df-va 28364 df-ba 28365 df-sm 28366 df-0v 28367 df-vs 28368 df-nmcv 28369 df-ims 28370 df-dip 28470 |
This theorem is referenced by: ipasslem7 28605 occllem 29072 |
Copyright terms: Public domain | W3C validator |