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Mirrors > Home > MPE Home > Th. List > ipasslem8 | Structured version Visualization version GIF version |
Description: Lemma for ipassi 30132. By ipasslem5 30126, πΉ is 0 for all β; since it is continuous and β is dense in β by qdensere2 24320, 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 11208 | . 2 β’ 0 β β | |
2 | qre 12939 | . . . . . 6 β’ (π₯ β β β π₯ β β) | |
3 | oveq1 7418 | . . . . . . . . 9 β’ (π€ = π₯ β (π€ππ΄) = (π₯ππ΄)) | |
4 | 3 | oveq1d 7426 | . . . . . . . 8 β’ (π€ = π₯ β ((π€ππ΄)ππ΅) = ((π₯ππ΄)ππ΅)) |
5 | oveq1 7418 | . . . . . . . 8 β’ (π€ = π₯ β (π€ Β· (π΄ππ΅)) = (π₯ Β· (π΄ππ΅))) | |
6 | 4, 5 | oveq12d 7429 | . . . . . . 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 7444 | . . . . . . 7 β’ (((π₯ππ΄)ππ΅) β (π₯ Β· (π΄ππ΅))) β V | |
9 | 6, 7, 8 | fvmpt 6998 | . . . . . 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 12949 | . . . . . . . . 9 β’ (π₯ β β β π₯ β β) | |
13 | ip1i.9 | . . . . . . . . . . 11 β’ π β CPreHilOLD | |
14 | 13 | phnvi 30107 | . . . . . . . . . 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 29917 | . . . . . . . . . 10 β’ ((π β NrmCVec β§ π₯ β β β§ π΄ β π) β (π₯ππ΄) β π) |
18 | 14, 17 | mp3an1 1448 | . . . . . . . . 9 β’ ((π₯ β β β§ π΄ β π) β (π₯ππ΄) β π) |
19 | 12, 18 | sylan 580 | . . . . . . . 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 30003 | . . . . . . . . 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 1452 | . . . . . . . 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 30126 | . . . . . . 7 β’ ((π₯ β β β§ π΄ β π) β ((π₯ππ΄)ππ΅) = (π₯ Β· (π΄ππ΅))) |
27 | 24, 26 | subeq0bd 11642 | . . . . . 6 β’ ((π₯ β β β§ π΄ β π) β (((π₯ππ΄)ππ΅) β (π₯ Β· (π΄ππ΅))) = 0) |
28 | 11, 27 | mpan2 689 | . . . . 5 β’ (π₯ β β β (((π₯ππ΄)ππ΅) β (π₯ Β· (π΄ππ΅))) = 0) |
29 | 10, 28 | eqtrd 2772 | . . . 4 β’ (π₯ β β β (πΉβπ₯) = 0) |
30 | 29 | rgen 3063 | . . 3 β’ βπ₯ β β (πΉβπ₯) = 0 |
31 | 7 | funmpt2 6587 | . . . 4 β’ Fun πΉ |
32 | qssre 12945 | . . . . 5 β’ β β β | |
33 | ovex 7444 | . . . . . 6 β’ (((π€ππ΄)ππ΅) β (π€ Β· (π΄ππ΅))) β V | |
34 | 33, 7 | dmmpti 6694 | . . . . 5 β’ dom πΉ = β |
35 | 32, 34 | sseqtrri 4019 | . . . 4 β’ β β dom πΉ |
36 | funconstss 7057 | . . . 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 690 | . . 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 24293 | . 2 β’ ((clsβ(topGenβran (,)))ββ) = β | |
40 | eqid 2732 | . . . . 5 β’ (TopOpenββfld) = (TopOpenββfld) | |
41 | 40 | cnfldhaus 24308 | . . . 4 β’ (TopOpenββfld) β Haus |
42 | haust1 22863 | . . . 4 β’ ((TopOpenββfld) β Haus β (TopOpenββfld) β Fre) | |
43 | 41, 42 | ax-mp 5 | . . 3 β’ (TopOpenββfld) β Fre |
44 | eqid 2732 | . . . 4 β’ (topGenβran (,)) = (topGenβran (,)) | |
45 | 15, 25, 16, 21, 13, 11, 20, 7, 44, 40 | ipasslem7 30127 | . . 3 β’ πΉ β ((topGenβran (,)) Cn (TopOpenββfld)) |
46 | uniretop 24286 | . . . 4 β’ β = βͺ (topGenβran (,)) | |
47 | 40 | cnfldtopon 24306 | . . . . 5 β’ (TopOpenββfld) β (TopOnββ) |
48 | 47 | toponunii 22425 | . . . 4 β’ β = βͺ (TopOpenββfld) |
49 | 46, 48 | dnsconst 22889 | . . 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 700 | . 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 1461 | 1 β’ πΉ:ββΆ{0} |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βwral 3061 β wss 3948 {csn 4628 β¦ cmpt 5231 β‘ccnv 5675 dom cdm 5676 ran crn 5677 β cima 5679 Fun wfun 6537 βΆwf 6539 βcfv 6543 (class class class)co 7411 βcc 11110 βcr 11111 0cc0 11112 Β· cmul 11117 β cmin 11446 βcq 12934 (,)cioo 13326 TopOpenctopn 17369 topGenctg 17385 βfldccnfld 20950 clsccl 22529 Cn ccn 22735 Frect1 22818 Hauscha 22819 NrmCVeccnv 29875 +π£ cpv 29876 BaseSetcba 29877 Β·π OLD cns 29878 Β·πOLDcdip 29991 CPreHilOLDccphlo 30103 |
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 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 |
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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-icc 13333 df-fz 13487 df-fzo 13630 df-seq 13969 df-exp 14030 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-sum 15635 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-rest 17370 df-topn 17371 df-0g 17389 df-gsum 17390 df-topgen 17391 df-pt 17392 df-prds 17395 df-xrs 17450 df-qtop 17455 df-imas 17456 df-xps 17458 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-mulg 18953 df-cntz 19183 df-cmn 19652 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-cnfld 20951 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cld 22530 df-ntr 22531 df-cls 22532 df-cn 22738 df-cnp 22739 df-t1 22825 df-haus 22826 df-tx 23073 df-hmeo 23266 df-xms 23833 df-ms 23834 df-tms 23835 df-grpo 29784 df-gid 29785 df-ginv 29786 df-gdiv 29787 df-ablo 29836 df-vc 29850 df-nv 29883 df-va 29886 df-ba 29887 df-sm 29888 df-0v 29889 df-vs 29890 df-nmcv 29891 df-ims 29892 df-dip 29992 df-ph 30104 |
This theorem is referenced by: ipasslem9 30129 |
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