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Mirrors > Home > MPE Home > Th. List > cnmpt1ip | Structured version Visualization version GIF version |
Description: Continuity of inner product; analogue of cnmpt12f 21946 which cannot be used directly because ·𝑖 is not a function. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
cnmpt1ip.j | ⊢ 𝐽 = (TopOpen‘𝑊) |
cnmpt1ip.c | ⊢ 𝐶 = (TopOpen‘ℂfld) |
cnmpt1ip.h | ⊢ , = (·𝑖‘𝑊) |
cnmpt1ip.r | ⊢ (𝜑 → 𝑊 ∈ ℂPreHil) |
cnmpt1ip.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋)) |
cnmpt1ip.a | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) |
cnmpt1ip.b | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) |
Ref | Expression |
---|---|
cnmpt1ip | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 , 𝐵)) ∈ (𝐾 Cn 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmpt1ip.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋)) | |
2 | cnmpt1ip.r | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℂPreHil) | |
3 | cphngp 23448 | . . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp) | |
4 | ngptps 22882 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ TopSp) | |
5 | 2, 3, 4 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ TopSp) |
6 | eqid 2793 | . . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
7 | cnmpt1ip.j | . . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝑊) | |
8 | 6, 7 | istps 21214 | . . . . . . 7 ⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑊))) |
9 | 5, 8 | sylib 219 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘𝑊))) |
10 | cnmpt1ip.a | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) | |
11 | cnf2 21529 | . . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝑊)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝑊)) | |
12 | 1, 9, 10, 11 | syl3anc 1362 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝑊)) |
13 | 12 | fvmptelrn 6731 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝑊)) |
14 | cnmpt1ip.b | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) | |
15 | cnf2 21529 | . . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝑊)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝑊)) | |
16 | 1, 9, 14, 15 | syl3anc 1362 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝑊)) |
17 | 16 | fvmptelrn 6731 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝑊)) |
18 | cnmpt1ip.h | . . . . 5 ⊢ , = (·𝑖‘𝑊) | |
19 | eqid 2793 | . . . . 5 ⊢ (·if‘𝑊) = (·if‘𝑊) | |
20 | 6, 18, 19 | ipfval 20463 | . . . 4 ⊢ ((𝐴 ∈ (Base‘𝑊) ∧ 𝐵 ∈ (Base‘𝑊)) → (𝐴(·if‘𝑊)𝐵) = (𝐴 , 𝐵)) |
21 | 13, 17, 20 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴(·if‘𝑊)𝐵) = (𝐴 , 𝐵)) |
22 | 21 | mpteq2dva 5049 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(·if‘𝑊)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 , 𝐵))) |
23 | cnmpt1ip.c | . . . . 5 ⊢ 𝐶 = (TopOpen‘ℂfld) | |
24 | 19, 7, 23 | ipcn 23520 | . . . 4 ⊢ (𝑊 ∈ ℂPreHil → (·if‘𝑊) ∈ ((𝐽 ×t 𝐽) Cn 𝐶)) |
25 | 2, 24 | syl 17 | . . 3 ⊢ (𝜑 → (·if‘𝑊) ∈ ((𝐽 ×t 𝐽) Cn 𝐶)) |
26 | 1, 10, 14, 25 | cnmpt12f 21946 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(·if‘𝑊)𝐵)) ∈ (𝐾 Cn 𝐶)) |
27 | 22, 26 | eqeltrrd 2882 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 , 𝐵)) ∈ (𝐾 Cn 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ↦ cmpt 5035 ⟶wf 6213 ‘cfv 6217 (class class class)co 7007 Basecbs 16300 ·𝑖cip 16387 TopOpenctopn 16512 ℂfldccnfld 20215 ·ifcipf 20439 TopOnctopon 21190 TopSpctps 21212 Cn ccn 21504 ×t ctx 21840 NrmGrpcngp 22858 ℂPreHilccph 23441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 ax-addf 10451 ax-mulf 10452 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-iin 4822 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-isom 6226 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-of 7258 df-om 7428 df-1st 7536 df-2nd 7537 df-supp 7673 df-tpos 7734 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-2o 7945 df-oadd 7948 df-er 8130 df-map 8249 df-ixp 8301 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-fsupp 8670 df-fi 8711 df-sup 8742 df-inf 8743 df-oi 8810 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-z 11819 df-dec 11937 df-uz 12083 df-q 12187 df-rp 12229 df-xneg 12346 df-xadd 12347 df-xmul 12348 df-ico 12583 df-icc 12584 df-fz 12732 df-fzo 12873 df-seq 13208 df-exp 13268 df-hash 13529 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-starv 16397 df-sca 16398 df-vsca 16399 df-ip 16400 df-tset 16401 df-ple 16402 df-ds 16404 df-unif 16405 df-hom 16406 df-cco 16407 df-rest 16513 df-topn 16514 df-0g 16532 df-gsum 16533 df-topgen 16534 df-pt 16535 df-prds 16538 df-xrs 16592 df-qtop 16597 df-imas 16598 df-xps 16600 df-mre 16674 df-mrc 16675 df-acs 16677 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-mhm 17762 df-submnd 17763 df-grp 17852 df-minusg 17853 df-sbg 17854 df-mulg 17970 df-subg 18018 df-ghm 18085 df-cntz 18176 df-cmn 18623 df-abl 18624 df-mgp 18918 df-ur 18930 df-ring 18977 df-cring 18978 df-oppr 19051 df-dvdsr 19069 df-unit 19070 df-invr 19100 df-dvr 19111 df-rnghom 19145 df-drng 19182 df-subrg 19211 df-staf 19294 df-srng 19295 df-lmod 19314 df-lmhm 19472 df-lvec 19553 df-sra 19622 df-rgmod 19623 df-psmet 20207 df-xmet 20208 df-met 20209 df-bl 20210 df-mopn 20211 df-cnfld 20216 df-phl 20440 df-ipf 20441 df-top 21174 df-topon 21191 df-topsp 21213 df-bases 21226 df-cn 21507 df-cnp 21508 df-tx 21842 df-hmeo 22035 df-xms 22601 df-ms 22602 df-tms 22603 df-nm 22863 df-ngp 22864 df-tng 22865 df-nlm 22867 df-clm 23338 df-cph 23443 df-tcph 23444 |
This theorem is referenced by: csscld 23523 clsocv 23524 |
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