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Mirrors > Home > MPE Home > Th. List > cnmpt2ip | Structured version Visualization version GIF version |
Description: Continuity of inner product; analogue of cnmpt22f 23178 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 23094 | . . . . . . . . . 10 β’ ((πΎ β (TopOnβπ) β§ πΏ β (TopOnβπ)) β (πΎ Γt πΏ) β (TopOnβ(π Γ π))) | |
4 | 1, 2, 3 | syl2anc 584 | . . . . . . . . 9 β’ (π β (πΎ Γt πΏ) β (TopOnβ(π Γ π))) |
5 | cnmpt1ip.r | . . . . . . . . . . 11 β’ (π β π β βPreHil) | |
6 | cphngp 24689 | . . . . . . . . . . 11 β’ (π β βPreHil β π β NrmGrp) | |
7 | ngptps 24110 | . . . . . . . . . . 11 β’ (π β NrmGrp β π β TopSp) | |
8 | 5, 6, 7 | 3syl 18 | . . . . . . . . . 10 β’ (π β π β TopSp) |
9 | eqid 2732 | . . . . . . . . . . 11 β’ (Baseβπ) = (Baseβπ) | |
10 | cnmpt1ip.j | . . . . . . . . . . 11 β’ π½ = (TopOpenβπ) | |
11 | 9, 10 | istps 22435 | . . . . . . . . . 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 22752 | . . . . . . . . 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 1371 | . . . . . . . 8 β’ (π β (π₯ β π, π¦ β π β¦ π΄):(π Γ π)βΆ(Baseβπ)) |
16 | eqid 2732 | . . . . . . . . 9 β’ (π₯ β π, π¦ β π β¦ π΄) = (π₯ β π, π¦ β π β¦ π΄) | |
17 | 16 | fmpo 8053 | . . . . . . . 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 3248 | . . . . . 6 β’ ((π β§ π₯ β π) β βπ¦ β π π΄ β (Baseβπ)) |
20 | 19 | r19.21bi 3248 | . . . . 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 22752 | . . . . . . . . 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 1371 | . . . . . . . 8 β’ (π β (π₯ β π, π¦ β π β¦ π΅):(π Γ π)βΆ(Baseβπ)) |
24 | eqid 2732 | . . . . . . . . 9 β’ (π₯ β π, π¦ β π β¦ π΅) = (π₯ β π, π¦ β π β¦ π΅) | |
25 | 24 | fmpo 8053 | . . . . . . . 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 3248 | . . . . . 6 β’ ((π β§ π₯ β π) β βπ¦ β π π΅ β (Baseβπ)) |
28 | 27 | r19.21bi 3248 | . . . . 5 β’ (((π β§ π₯ β π) β§ π¦ β π) β π΅ β (Baseβπ)) |
29 | cnmpt1ip.h | . . . . . 6 β’ , = (Β·πβπ) | |
30 | eqid 2732 | . . . . . 6 β’ (Β·ifβπ) = (Β·ifβπ) | |
31 | 9, 29, 30 | ipfval 21201 | . . . . 5 β’ ((π΄ β (Baseβπ) β§ π΅ β (Baseβπ)) β (π΄(Β·ifβπ)π΅) = (π΄ , π΅)) |
32 | 20, 28, 31 | syl2anc 584 | . . . 4 β’ (((π β§ π₯ β π) β§ π¦ β π) β (π΄(Β·ifβπ)π΅) = (π΄ , π΅)) |
33 | 32 | 3impa 1110 | . . 3 β’ ((π β§ π₯ β π β§ π¦ β π) β (π΄(Β·ifβπ)π΅) = (π΄ , π΅)) |
34 | 33 | mpoeq3dva 7485 | . 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 24762 | . . . 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 23178 | . 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 2834 | 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 396 = wceq 1541 β wcel 2106 βwral 3061 Γ cxp 5674 βΆwf 6539 βcfv 6543 (class class class)co 7408 β cmpo 7410 Basecbs 17143 Β·πcip 17201 TopOpenctopn 17366 βfldccnfld 20943 Β·ifcipf 21177 TopOnctopon 22411 TopSpctps 22433 Cn ccn 22727 Γt ctx 23063 NrmGrpcngp 24085 βPreHilccph 24682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-tpos 8210 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-seq 13966 df-exp 14027 df-hash 14290 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mulg 18950 df-subg 19002 df-ghm 19089 df-cntz 19180 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-cring 20058 df-oppr 20149 df-dvdsr 20170 df-unit 20171 df-invr 20201 df-dvr 20214 df-rnghom 20250 df-subrg 20316 df-drng 20358 df-staf 20452 df-srng 20453 df-lmod 20472 df-lmhm 20632 df-lvec 20713 df-sra 20784 df-rgmod 20785 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-cnfld 20944 df-phl 21178 df-ipf 21179 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cn 22730 df-cnp 22731 df-tx 23065 df-hmeo 23258 df-xms 23825 df-ms 23826 df-tms 23827 df-nm 24090 df-ngp 24091 df-tng 24092 df-nlm 24094 df-clm 24578 df-cph 24684 df-tcph 24685 |
This theorem is referenced by: (None) |
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