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Mirrors > Home > MPE Home > Th. List > ipasslem8 | Structured version Visualization version GIF version |
Description: Lemma for ipassi 30094. By ipasslem5 30088, πΉ is 0 for all β; since it is continuous and β is dense in β by qdensere2 24313, we conclude πΉ is 0 for all β. (Contributed by NM, 24-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ip1i.1 | β’ π = (BaseSetβπ) |
ip1i.2 | β’ πΊ = ( +π£ βπ) |
ip1i.4 | β’ π = ( Β·π OLD βπ) |
ip1i.7 | β’ π = (Β·πOLDβπ) |
ip1i.9 | β’ π β CPreHilOLD |
ipasslem7.a | β’ π΄ β π |
ipasslem7.b | β’ π΅ β π |
ipasslem7.f | β’ πΉ = (π€ β β β¦ (((π€ππ΄)ππ΅) β (π€ Β· (π΄ππ΅)))) |
Ref | Expression |
---|---|
ipasslem8 | β’ πΉ:ββΆ{0} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 11206 | . 2 β’ 0 β β | |
2 | qre 12937 | . . . . . 6 β’ (π₯ β β β π₯ β β) | |
3 | oveq1 7416 | . . . . . . . . 9 β’ (π€ = π₯ β (π€ππ΄) = (π₯ππ΄)) | |
4 | 3 | oveq1d 7424 | . . . . . . . 8 β’ (π€ = π₯ β ((π€ππ΄)ππ΅) = ((π₯ππ΄)ππ΅)) |
5 | oveq1 7416 | . . . . . . . 8 β’ (π€ = π₯ β (π€ Β· (π΄ππ΅)) = (π₯ Β· (π΄ππ΅))) | |
6 | 4, 5 | oveq12d 7427 | . . . . . . 7 β’ (π€ = π₯ β (((π€ππ΄)ππ΅) β (π€ Β· (π΄ππ΅))) = (((π₯ππ΄)ππ΅) β (π₯ Β· (π΄ππ΅)))) |
7 | ipasslem7.f | . . . . . . 7 β’ πΉ = (π€ β β β¦ (((π€ππ΄)ππ΅) β (π€ Β· (π΄ππ΅)))) | |
8 | ovex 7442 | . . . . . . 7 β’ (((π₯ππ΄)ππ΅) β (π₯ Β· (π΄ππ΅))) β V | |
9 | 6, 7, 8 | fvmpt 6999 | . . . . . 6 β’ (π₯ β β β (πΉβπ₯) = (((π₯ππ΄)ππ΅) β (π₯ Β· (π΄ππ΅)))) |
10 | 2, 9 | syl 17 | . . . . 5 β’ (π₯ β β β (πΉβπ₯) = (((π₯ππ΄)ππ΅) β (π₯ Β· (π΄ππ΅)))) |
11 | ipasslem7.a | . . . . . 6 β’ π΄ β π | |
12 | qcn 12947 | . . . . . . . . 9 β’ (π₯ β β β π₯ β β) | |
13 | ip1i.9 | . . . . . . . . . . 11 β’ π β CPreHilOLD | |
14 | 13 | phnvi 30069 | . . . . . . . . . 10 β’ π β NrmCVec |
15 | ip1i.1 | . . . . . . . . . . 11 β’ π = (BaseSetβπ) | |
16 | ip1i.4 | . . . . . . . . . . 11 β’ π = ( Β·π OLD βπ) | |
17 | 15, 16 | nvscl 29879 | . . . . . . . . . 10 β’ ((π β NrmCVec β§ π₯ β β β§ π΄ β π) β (π₯ππ΄) β π) |
18 | 14, 17 | mp3an1 1449 | . . . . . . . . 9 β’ ((π₯ β β β§ π΄ β π) β (π₯ππ΄) β π) |
19 | 12, 18 | sylan 581 | . . . . . . . 8 β’ ((π₯ β β β§ π΄ β π) β (π₯ππ΄) β π) |
20 | ipasslem7.b | . . . . . . . . 9 β’ π΅ β π | |
21 | ip1i.7 | . . . . . . . . . 10 β’ π = (Β·πOLDβπ) | |
22 | 15, 21 | dipcl 29965 | . . . . . . . . 9 β’ ((π β NrmCVec β§ (π₯ππ΄) β π β§ π΅ β π) β ((π₯ππ΄)ππ΅) β β) |
23 | 14, 20, 22 | mp3an13 1453 | . . . . . . . 8 β’ ((π₯ππ΄) β π β ((π₯ππ΄)ππ΅) β β) |
24 | 19, 23 | syl 17 | . . . . . . 7 β’ ((π₯ β β β§ π΄ β π) β ((π₯ππ΄)ππ΅) β β) |
25 | ip1i.2 | . . . . . . . 8 β’ πΊ = ( +π£ βπ) | |
26 | 15, 25, 16, 21, 13, 20 | ipasslem5 30088 | . . . . . . 7 β’ ((π₯ β β β§ π΄ β π) β ((π₯ππ΄)ππ΅) = (π₯ Β· (π΄ππ΅))) |
27 | 24, 26 | subeq0bd 11640 | . . . . . 6 β’ ((π₯ β β β§ π΄ β π) β (((π₯ππ΄)ππ΅) β (π₯ Β· (π΄ππ΅))) = 0) |
28 | 11, 27 | mpan2 690 | . . . . 5 β’ (π₯ β β β (((π₯ππ΄)ππ΅) β (π₯ Β· (π΄ππ΅))) = 0) |
29 | 10, 28 | eqtrd 2773 | . . . 4 β’ (π₯ β β β (πΉβπ₯) = 0) |
30 | 29 | rgen 3064 | . . 3 β’ βπ₯ β β (πΉβπ₯) = 0 |
31 | 7 | funmpt2 6588 | . . . 4 β’ Fun πΉ |
32 | qssre 12943 | . . . . 5 β’ β β β | |
33 | ovex 7442 | . . . . . 6 β’ (((π€ππ΄)ππ΅) β (π€ Β· (π΄ππ΅))) β V | |
34 | 33, 7 | dmmpti 6695 | . . . . 5 β’ dom πΉ = β |
35 | 32, 34 | sseqtrri 4020 | . . . 4 β’ β β dom πΉ |
36 | funconstss 7058 | . . . 4 β’ ((Fun πΉ β§ β β dom πΉ) β (βπ₯ β β (πΉβπ₯) = 0 β β β (β‘πΉ β {0}))) | |
37 | 31, 35, 36 | mp2an 691 | . . 3 β’ (βπ₯ β β (πΉβπ₯) = 0 β β β (β‘πΉ β {0})) |
38 | 30, 37 | mpbi 229 | . 2 β’ β β (β‘πΉ β {0}) |
39 | qdensere 24286 | . 2 β’ ((clsβ(topGenβran (,)))ββ) = β | |
40 | eqid 2733 | . . . . 5 β’ (TopOpenββfld) = (TopOpenββfld) | |
41 | 40 | cnfldhaus 24301 | . . . 4 β’ (TopOpenββfld) β Haus |
42 | haust1 22856 | . . . 4 β’ ((TopOpenββfld) β Haus β (TopOpenββfld) β Fre) | |
43 | 41, 42 | ax-mp 5 | . . 3 β’ (TopOpenββfld) β Fre |
44 | eqid 2733 | . . . 4 β’ (topGenβran (,)) = (topGenβran (,)) | |
45 | 15, 25, 16, 21, 13, 11, 20, 7, 44, 40 | ipasslem7 30089 | . . 3 β’ πΉ β ((topGenβran (,)) Cn (TopOpenββfld)) |
46 | uniretop 24279 | . . . 4 β’ β = βͺ (topGenβran (,)) | |
47 | 40 | cnfldtopon 24299 | . . . . 5 β’ (TopOpenββfld) β (TopOnββ) |
48 | 47 | toponunii 22418 | . . . 4 β’ β = βͺ (TopOpenββfld) |
49 | 46, 48 | dnsconst 22882 | . . 3 β’ ((((TopOpenββfld) β Fre β§ πΉ β ((topGenβran (,)) Cn (TopOpenββfld))) β§ (0 β β β§ β β (β‘πΉ β {0}) β§ ((clsβ(topGenβran (,)))ββ) = β)) β πΉ:ββΆ{0}) |
50 | 43, 45, 49 | mpanl12 701 | . 2 β’ ((0 β β β§ β β (β‘πΉ β {0}) β§ ((clsβ(topGenβran (,)))ββ) = β) β πΉ:ββΆ{0}) |
51 | 1, 38, 39, 50 | mp3an 1462 | 1 β’ πΉ:ββΆ{0} |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3062 β wss 3949 {csn 4629 β¦ cmpt 5232 β‘ccnv 5676 dom cdm 5677 ran crn 5678 β cima 5680 Fun wfun 6538 βΆwf 6540 βcfv 6544 (class class class)co 7409 βcc 11108 βcr 11109 0cc0 11110 Β· cmul 11115 β cmin 11444 βcq 12932 (,)cioo 13324 TopOpenctopn 17367 topGenctg 17383 βfldccnfld 20944 clsccl 22522 Cn ccn 22728 Frect1 22811 Hauscha 22812 NrmCVeccnv 29837 +π£ cpv 29838 BaseSetcba 29839 Β·π OLD cns 29840 Β·πOLDcdip 29953 CPreHilOLDccphlo 30065 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-icc 13331 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-cn 22731 df-cnp 22732 df-t1 22818 df-haus 22819 df-tx 23066 df-hmeo 23259 df-xms 23826 df-ms 23827 df-tms 23828 df-grpo 29746 df-gid 29747 df-ginv 29748 df-gdiv 29749 df-ablo 29798 df-vc 29812 df-nv 29845 df-va 29848 df-ba 29849 df-sm 29850 df-0v 29851 df-vs 29852 df-nmcv 29853 df-ims 29854 df-dip 29954 df-ph 30066 |
This theorem is referenced by: ipasslem9 30091 |
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