![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cnmpt2ip | Structured version Visualization version GIF version |
Description: Continuity of inner product; analogue of cnmpt22f 23049 which cannot be used directly because Β·π is not a function. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
cnmpt1ip.j | β’ π½ = (TopOpenβπ) |
cnmpt1ip.c | β’ πΆ = (TopOpenββfld) |
cnmpt1ip.h | β’ , = (Β·πβπ) |
cnmpt1ip.r | β’ (π β π β βPreHil) |
cnmpt1ip.k | β’ (π β πΎ β (TopOnβπ)) |
cnmpt2ip.l | β’ (π β πΏ β (TopOnβπ)) |
cnmpt2ip.a | β’ (π β (π₯ β π, π¦ β π β¦ π΄) β ((πΎ Γt πΏ) Cn π½)) |
cnmpt2ip.b | β’ (π β (π₯ β π, π¦ β π β¦ π΅) β ((πΎ Γt πΏ) Cn π½)) |
Ref | Expression |
---|---|
cnmpt2ip | β’ (π β (π₯ β π, π¦ β π β¦ (π΄ , π΅)) β ((πΎ Γt πΏ) Cn πΆ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmpt1ip.k | . . . . . . . . . 10 β’ (π β πΎ β (TopOnβπ)) | |
2 | cnmpt2ip.l | . . . . . . . . . 10 β’ (π β πΏ β (TopOnβπ)) | |
3 | txtopon 22965 | . . . . . . . . . 10 β’ ((πΎ β (TopOnβπ) β§ πΏ β (TopOnβπ)) β (πΎ Γt πΏ) β (TopOnβ(π Γ π))) | |
4 | 1, 2, 3 | syl2anc 585 | . . . . . . . . 9 β’ (π β (πΎ Γt πΏ) β (TopOnβ(π Γ π))) |
5 | cnmpt1ip.r | . . . . . . . . . . 11 β’ (π β π β βPreHil) | |
6 | cphngp 24560 | . . . . . . . . . . 11 β’ (π β βPreHil β π β NrmGrp) | |
7 | ngptps 23981 | . . . . . . . . . . 11 β’ (π β NrmGrp β π β TopSp) | |
8 | 5, 6, 7 | 3syl 18 | . . . . . . . . . 10 β’ (π β π β TopSp) |
9 | eqid 2733 | . . . . . . . . . . 11 β’ (Baseβπ) = (Baseβπ) | |
10 | cnmpt1ip.j | . . . . . . . . . . 11 β’ π½ = (TopOpenβπ) | |
11 | 9, 10 | istps 22306 | . . . . . . . . . 10 β’ (π β TopSp β π½ β (TopOnβ(Baseβπ))) |
12 | 8, 11 | sylib 217 | . . . . . . . . 9 β’ (π β π½ β (TopOnβ(Baseβπ))) |
13 | cnmpt2ip.a | . . . . . . . . 9 β’ (π β (π₯ β π, π¦ β π β¦ π΄) β ((πΎ Γt πΏ) Cn π½)) | |
14 | cnf2 22623 | . . . . . . . . 9 β’ (((πΎ Γt πΏ) β (TopOnβ(π Γ π)) β§ π½ β (TopOnβ(Baseβπ)) β§ (π₯ β π, π¦ β π β¦ π΄) β ((πΎ Γt πΏ) Cn π½)) β (π₯ β π, π¦ β π β¦ π΄):(π Γ π)βΆ(Baseβπ)) | |
15 | 4, 12, 13, 14 | syl3anc 1372 | . . . . . . . 8 β’ (π β (π₯ β π, π¦ β π β¦ π΄):(π Γ π)βΆ(Baseβπ)) |
16 | eqid 2733 | . . . . . . . . 9 β’ (π₯ β π, π¦ β π β¦ π΄) = (π₯ β π, π¦ β π β¦ π΄) | |
17 | 16 | fmpo 8004 | . . . . . . . 8 β’ (βπ₯ β π βπ¦ β π π΄ β (Baseβπ) β (π₯ β π, π¦ β π β¦ π΄):(π Γ π)βΆ(Baseβπ)) |
18 | 15, 17 | sylibr 233 | . . . . . . 7 β’ (π β βπ₯ β π βπ¦ β π π΄ β (Baseβπ)) |
19 | 18 | r19.21bi 3233 | . . . . . 6 β’ ((π β§ π₯ β π) β βπ¦ β π π΄ β (Baseβπ)) |
20 | 19 | r19.21bi 3233 | . . . . 5 β’ (((π β§ π₯ β π) β§ π¦ β π) β π΄ β (Baseβπ)) |
21 | cnmpt2ip.b | . . . . . . . . 9 β’ (π β (π₯ β π, π¦ β π β¦ π΅) β ((πΎ Γt πΏ) Cn π½)) | |
22 | cnf2 22623 | . . . . . . . . 9 β’ (((πΎ Γt πΏ) β (TopOnβ(π Γ π)) β§ π½ β (TopOnβ(Baseβπ)) β§ (π₯ β π, π¦ β π β¦ π΅) β ((πΎ Γt πΏ) Cn π½)) β (π₯ β π, π¦ β π β¦ π΅):(π Γ π)βΆ(Baseβπ)) | |
23 | 4, 12, 21, 22 | syl3anc 1372 | . . . . . . . 8 β’ (π β (π₯ β π, π¦ β π β¦ π΅):(π Γ π)βΆ(Baseβπ)) |
24 | eqid 2733 | . . . . . . . . 9 β’ (π₯ β π, π¦ β π β¦ π΅) = (π₯ β π, π¦ β π β¦ π΅) | |
25 | 24 | fmpo 8004 | . . . . . . . 8 β’ (βπ₯ β π βπ¦ β π π΅ β (Baseβπ) β (π₯ β π, π¦ β π β¦ π΅):(π Γ π)βΆ(Baseβπ)) |
26 | 23, 25 | sylibr 233 | . . . . . . 7 β’ (π β βπ₯ β π βπ¦ β π π΅ β (Baseβπ)) |
27 | 26 | r19.21bi 3233 | . . . . . 6 β’ ((π β§ π₯ β π) β βπ¦ β π π΅ β (Baseβπ)) |
28 | 27 | r19.21bi 3233 | . . . . 5 β’ (((π β§ π₯ β π) β§ π¦ β π) β π΅ β (Baseβπ)) |
29 | cnmpt1ip.h | . . . . . 6 β’ , = (Β·πβπ) | |
30 | eqid 2733 | . . . . . 6 β’ (Β·ifβπ) = (Β·ifβπ) | |
31 | 9, 29, 30 | ipfval 21076 | . . . . 5 β’ ((π΄ β (Baseβπ) β§ π΅ β (Baseβπ)) β (π΄(Β·ifβπ)π΅) = (π΄ , π΅)) |
32 | 20, 28, 31 | syl2anc 585 | . . . 4 β’ (((π β§ π₯ β π) β§ π¦ β π) β (π΄(Β·ifβπ)π΅) = (π΄ , π΅)) |
33 | 32 | 3impa 1111 | . . 3 β’ ((π β§ π₯ β π β§ π¦ β π) β (π΄(Β·ifβπ)π΅) = (π΄ , π΅)) |
34 | 33 | mpoeq3dva 7438 | . 2 β’ (π β (π₯ β π, π¦ β π β¦ (π΄(Β·ifβπ)π΅)) = (π₯ β π, π¦ β π β¦ (π΄ , π΅))) |
35 | cnmpt1ip.c | . . . . 5 β’ πΆ = (TopOpenββfld) | |
36 | 30, 10, 35 | ipcn 24633 | . . . 4 β’ (π β βPreHil β (Β·ifβπ) β ((π½ Γt π½) Cn πΆ)) |
37 | 5, 36 | syl 17 | . . 3 β’ (π β (Β·ifβπ) β ((π½ Γt π½) Cn πΆ)) |
38 | 1, 2, 13, 21, 37 | cnmpt22f 23049 | . 2 β’ (π β (π₯ β π, π¦ β π β¦ (π΄(Β·ifβπ)π΅)) β ((πΎ Γt πΏ) Cn πΆ)) |
39 | 34, 38 | eqeltrrd 2835 | 1 β’ (π β (π₯ β π, π¦ β π β¦ (π΄ , π΅)) β ((πΎ Γt πΏ) Cn πΆ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 Γ cxp 5635 βΆwf 6496 βcfv 6500 (class class class)co 7361 β cmpo 7363 Basecbs 17091 Β·πcip 17146 TopOpenctopn 17311 βfldccnfld 20819 Β·ifcipf 21052 TopOnctopon 22282 TopSpctps 22304 Cn ccn 22598 Γt ctx 22934 NrmGrpcngp 23956 βPreHilccph 24553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-fi 9355 df-sup 9386 df-inf 9387 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-ico 13279 df-icc 13280 df-fz 13434 df-fzo 13577 df-seq 13916 df-exp 13977 df-hash 14240 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-hom 17165 df-cco 17166 df-rest 17312 df-topn 17313 df-0g 17331 df-gsum 17332 df-topgen 17333 df-pt 17334 df-prds 17337 df-xrs 17392 df-qtop 17397 df-imas 17398 df-xps 17400 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mhm 18609 df-submnd 18610 df-grp 18759 df-minusg 18760 df-sbg 18761 df-mulg 18881 df-subg 18933 df-ghm 19014 df-cntz 19105 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-dvr 20120 df-rnghom 20156 df-drng 20221 df-subrg 20262 df-staf 20347 df-srng 20348 df-lmod 20367 df-lmhm 20527 df-lvec 20608 df-sra 20678 df-rgmod 20679 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-cnfld 20820 df-phl 21053 df-ipf 21054 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-cn 22601 df-cnp 22602 df-tx 22936 df-hmeo 23129 df-xms 23696 df-ms 23697 df-tms 23698 df-nm 23961 df-ngp 23962 df-tng 23963 df-nlm 23965 df-clm 24449 df-cph 24555 df-tcph 24556 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |