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Mirrors > Home > MPE Home > Th. List > ipasslem7 | Structured version Visualization version GIF version |
Description: Lemma for ipassi 30695. Show that ((π€ππ΄)ππ΅) β (π€ Β· (π΄ππ΅)) is continuous on β. (Contributed by NM, 23-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΅)))) |
ipasslem7.j | β’ π½ = (topGenβran (,)) |
ipasslem7.k | β’ πΎ = (TopOpenββfld) |
Ref | Expression |
---|---|
ipasslem7 | β’ πΉ β (π½ Cn πΎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipasslem7.f | . 2 β’ πΉ = (π€ β β β¦ (((π€ππ΄)ππ΅) β (π€ Β· (π΄ππ΅)))) | |
2 | ipasslem7.j | . . . . 5 β’ π½ = (topGenβran (,)) | |
3 | ipasslem7.k | . . . . . 6 β’ πΎ = (TopOpenββfld) | |
4 | 3 | tgioo2 24737 | . . . . 5 β’ (topGenβran (,)) = (πΎ βΎt β) |
5 | 2, 4 | eqtri 2753 | . . . 4 β’ π½ = (πΎ βΎt β) |
6 | 3 | cnfldtopon 24717 | . . . . 5 β’ πΎ β (TopOnββ) |
7 | 6 | a1i 11 | . . . 4 β’ (β€ β πΎ β (TopOnββ)) |
8 | ax-resscn 11195 | . . . . 5 β’ β β β | |
9 | 8 | a1i 11 | . . . 4 β’ (β€ β β β β) |
10 | 7 | cnmptid 23583 | . . . . . . 7 β’ (β€ β (π€ β β β¦ π€) β (πΎ Cn πΎ)) |
11 | ip1i.9 | . . . . . . . . . . 11 β’ π β CPreHilOLD | |
12 | 11 | phnvi 30670 | . . . . . . . . . 10 β’ π β NrmCVec |
13 | ip1i.1 | . . . . . . . . . . 11 β’ π = (BaseSetβπ) | |
14 | eqid 2725 | . . . . . . . . . . 11 β’ (IndMetβπ) = (IndMetβπ) | |
15 | 13, 14 | imsxmet 30546 | . . . . . . . . . 10 β’ (π β NrmCVec β (IndMetβπ) β (βMetβπ)) |
16 | 12, 15 | ax-mp 5 | . . . . . . . . 9 β’ (IndMetβπ) β (βMetβπ) |
17 | eqid 2725 | . . . . . . . . . 10 β’ (MetOpenβ(IndMetβπ)) = (MetOpenβ(IndMetβπ)) | |
18 | 17 | mopntopon 24363 | . . . . . . . . 9 β’ ((IndMetβπ) β (βMetβπ) β (MetOpenβ(IndMetβπ)) β (TopOnβπ)) |
19 | 16, 18 | mp1i 13 | . . . . . . . 8 β’ (β€ β (MetOpenβ(IndMetβπ)) β (TopOnβπ)) |
20 | ipasslem7.a | . . . . . . . . 9 β’ π΄ β π | |
21 | 20 | a1i 11 | . . . . . . . 8 β’ (β€ β π΄ β π) |
22 | 7, 19, 21 | cnmptc 23584 | . . . . . . 7 β’ (β€ β (π€ β β β¦ π΄) β (πΎ Cn (MetOpenβ(IndMetβπ)))) |
23 | ip1i.4 | . . . . . . . . 9 β’ π = ( Β·π OLD βπ) | |
24 | 14, 17, 23, 3 | smcn 30552 | . . . . . . . 8 β’ (π β NrmCVec β π β ((πΎ Γt (MetOpenβ(IndMetβπ))) Cn (MetOpenβ(IndMetβπ)))) |
25 | 12, 24 | mp1i 13 | . . . . . . 7 β’ (β€ β π β ((πΎ Γt (MetOpenβ(IndMetβπ))) Cn (MetOpenβ(IndMetβπ)))) |
26 | 7, 10, 22, 25 | cnmpt12f 23588 | . . . . . 6 β’ (β€ β (π€ β β β¦ (π€ππ΄)) β (πΎ Cn (MetOpenβ(IndMetβπ)))) |
27 | ipasslem7.b | . . . . . . . 8 β’ π΅ β π | |
28 | 27 | a1i 11 | . . . . . . 7 β’ (β€ β π΅ β π) |
29 | 7, 19, 28 | cnmptc 23584 | . . . . . 6 β’ (β€ β (π€ β β β¦ π΅) β (πΎ Cn (MetOpenβ(IndMetβπ)))) |
30 | ip1i.7 | . . . . . . . 8 β’ π = (Β·πOLDβπ) | |
31 | 30, 14, 17, 3 | dipcn 30574 | . . . . . . 7 β’ (π β NrmCVec β π β (((MetOpenβ(IndMetβπ)) Γt (MetOpenβ(IndMetβπ))) Cn πΎ)) |
32 | 12, 31 | mp1i 13 | . . . . . 6 β’ (β€ β π β (((MetOpenβ(IndMetβπ)) Γt (MetOpenβ(IndMetβπ))) Cn πΎ)) |
33 | 7, 26, 29, 32 | cnmpt12f 23588 | . . . . 5 β’ (β€ β (π€ β β β¦ ((π€ππ΄)ππ΅)) β (πΎ Cn πΎ)) |
34 | 13, 30 | dipcl 30566 | . . . . . . . . 9 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄ππ΅) β β) |
35 | 12, 20, 27, 34 | mp3an 1457 | . . . . . . . 8 β’ (π΄ππ΅) β β |
36 | 35 | a1i 11 | . . . . . . 7 β’ (β€ β (π΄ππ΅) β β) |
37 | 7, 7, 36 | cnmptc 23584 | . . . . . 6 β’ (β€ β (π€ β β β¦ (π΄ππ΅)) β (πΎ Cn πΎ)) |
38 | 3 | mulcn 24801 | . . . . . . 7 β’ Β· β ((πΎ Γt πΎ) Cn πΎ) |
39 | 38 | a1i 11 | . . . . . 6 β’ (β€ β Β· β ((πΎ Γt πΎ) Cn πΎ)) |
40 | 7, 10, 37, 39 | cnmpt12f 23588 | . . . . 5 β’ (β€ β (π€ β β β¦ (π€ Β· (π΄ππ΅))) β (πΎ Cn πΎ)) |
41 | 3 | subcn 24800 | . . . . . 6 β’ β β ((πΎ Γt πΎ) Cn πΎ) |
42 | 41 | a1i 11 | . . . . 5 β’ (β€ β β β ((πΎ Γt πΎ) Cn πΎ)) |
43 | 7, 33, 40, 42 | cnmpt12f 23588 | . . . 4 β’ (β€ β (π€ β β β¦ (((π€ππ΄)ππ΅) β (π€ Β· (π΄ππ΅)))) β (πΎ Cn πΎ)) |
44 | 5, 7, 9, 43 | cnmpt1res 23598 | . . 3 β’ (β€ β (π€ β β β¦ (((π€ππ΄)ππ΅) β (π€ Β· (π΄ππ΅)))) β (π½ Cn πΎ)) |
45 | 44 | mptru 1540 | . 2 β’ (π€ β β β¦ (((π€ππ΄)ππ΅) β (π€ Β· (π΄ππ΅)))) β (π½ Cn πΎ) |
46 | 1, 45 | eqeltri 2821 | 1 β’ πΉ β (π½ Cn πΎ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 β€wtru 1534 β wcel 2098 β wss 3939 β¦ cmpt 5226 ran crn 5673 βcfv 6543 (class class class)co 7416 βcc 11136 βcr 11137 Β· cmul 11143 β cmin 11474 (,)cioo 13356 βΎt crest 17401 TopOpenctopn 17402 topGenctg 17418 βMetcxmet 21268 MetOpencmopn 21273 βfldccnfld 21283 TopOnctopon 22830 Cn ccn 23146 Γt ctx 23482 NrmCVeccnv 30438 +π£ cpv 30439 BaseSetcba 30440 Β·π OLD cns 30441 IndMetcims 30445 Β·πOLDcdip 30554 CPreHilOLDccphlo 30666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 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 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 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-ioo 13360 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-clim 15464 df-sum 15665 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-submnd 18740 df-mulg 19028 df-cntz 19272 df-cmn 19741 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-cnfld 21284 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cn 23149 df-cnp 23150 df-tx 23484 df-hmeo 23677 df-xms 24244 df-ms 24245 df-tms 24246 df-grpo 30347 df-gid 30348 df-ginv 30349 df-gdiv 30350 df-ablo 30399 df-vc 30413 df-nv 30446 df-va 30449 df-ba 30450 df-sm 30451 df-0v 30452 df-vs 30453 df-nmcv 30454 df-ims 30455 df-dip 30555 df-ph 30667 |
This theorem is referenced by: ipasslem8 30691 |
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