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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 23578 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 23494 | . . . . . . . . . 10 β’ ((πΎ β (TopOnβπ) β§ πΏ β (TopOnβπ)) β (πΎ Γt πΏ) β (TopOnβ(π Γ π))) | |
4 | 1, 2, 3 | syl2anc 583 | . . . . . . . . 9 β’ (π β (πΎ Γt πΏ) β (TopOnβ(π Γ π))) |
5 | cnmpt1ip.r | . . . . . . . . . . 11 β’ (π β π β βPreHil) | |
6 | cphngp 25100 | . . . . . . . . . . 11 β’ (π β βPreHil β π β NrmGrp) | |
7 | ngptps 24510 | . . . . . . . . . . 11 β’ (π β NrmGrp β π β TopSp) | |
8 | 5, 6, 7 | 3syl 18 | . . . . . . . . . 10 β’ (π β π β TopSp) |
9 | eqid 2728 | . . . . . . . . . . 11 β’ (Baseβπ) = (Baseβπ) | |
10 | cnmpt1ip.j | . . . . . . . . . . 11 β’ π½ = (TopOpenβπ) | |
11 | 9, 10 | istps 22835 | . . . . . . . . . 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 23152 | . . . . . . . . 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 1369 | . . . . . . . 8 β’ (π β (π₯ β π, π¦ β π β¦ π΄):(π Γ π)βΆ(Baseβπ)) |
16 | eqid 2728 | . . . . . . . . 9 β’ (π₯ β π, π¦ β π β¦ π΄) = (π₯ β π, π¦ β π β¦ π΄) | |
17 | 16 | fmpo 8072 | . . . . . . . 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 3245 | . . . . . 6 β’ ((π β§ π₯ β π) β βπ¦ β π π΄ β (Baseβπ)) |
20 | 19 | r19.21bi 3245 | . . . . 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 23152 | . . . . . . . . 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 1369 | . . . . . . . 8 β’ (π β (π₯ β π, π¦ β π β¦ π΅):(π Γ π)βΆ(Baseβπ)) |
24 | eqid 2728 | . . . . . . . . 9 β’ (π₯ β π, π¦ β π β¦ π΅) = (π₯ β π, π¦ β π β¦ π΅) | |
25 | 24 | fmpo 8072 | . . . . . . . 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 3245 | . . . . . 6 β’ ((π β§ π₯ β π) β βπ¦ β π π΅ β (Baseβπ)) |
28 | 27 | r19.21bi 3245 | . . . . 5 β’ (((π β§ π₯ β π) β§ π¦ β π) β π΅ β (Baseβπ)) |
29 | cnmpt1ip.h | . . . . . 6 β’ , = (Β·πβπ) | |
30 | eqid 2728 | . . . . . 6 β’ (Β·ifβπ) = (Β·ifβπ) | |
31 | 9, 29, 30 | ipfval 21580 | . . . . 5 β’ ((π΄ β (Baseβπ) β§ π΅ β (Baseβπ)) β (π΄(Β·ifβπ)π΅) = (π΄ , π΅)) |
32 | 20, 28, 31 | syl2anc 583 | . . . 4 β’ (((π β§ π₯ β π) β§ π¦ β π) β (π΄(Β·ifβπ)π΅) = (π΄ , π΅)) |
33 | 32 | 3impa 1108 | . . 3 β’ ((π β§ π₯ β π β§ π¦ β π) β (π΄(Β·ifβπ)π΅) = (π΄ , π΅)) |
34 | 33 | mpoeq3dva 7497 | . 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 25173 | . . . 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 23578 | . 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 2830 | 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 395 = wceq 1534 β wcel 2099 βwral 3058 Γ cxp 5676 βΆwf 6544 βcfv 6548 (class class class)co 7420 β cmpo 7422 Basecbs 17179 Β·πcip 17237 TopOpenctopn 17402 βfldccnfld 21278 Β·ifcipf 21556 TopOnctopon 22811 TopSpctps 22833 Cn ccn 23127 Γt ctx 23463 NrmGrpcngp 24485 βPreHilccph 25093 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-tpos 8231 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8846 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18892 df-minusg 18893 df-sbg 18894 df-mulg 19023 df-subg 19077 df-ghm 19167 df-cntz 19267 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-rhm 20410 df-subrng 20482 df-subrg 20507 df-drng 20625 df-staf 20724 df-srng 20725 df-lmod 20744 df-lmhm 20906 df-lvec 20987 df-sra 21057 df-rgmod 21058 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-cnfld 21279 df-phl 21557 df-ipf 21558 df-top 22795 df-topon 22812 df-topsp 22834 df-bases 22848 df-cn 23130 df-cnp 23131 df-tx 23465 df-hmeo 23658 df-xms 24225 df-ms 24226 df-tms 24227 df-nm 24490 df-ngp 24491 df-tng 24492 df-nlm 24494 df-clm 24989 df-cph 25095 df-tcph 25096 |
This theorem is referenced by: (None) |
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