![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > imasf1oms | Structured version Visualization version GIF version |
Description: The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
imasf1obl.u | β’ (π β π = (πΉ βs π )) |
imasf1obl.v | β’ (π β π = (Baseβπ )) |
imasf1obl.f | β’ (π β πΉ:πβ1-1-ontoβπ΅) |
imasf1oms.r | β’ (π β π β MetSp) |
Ref | Expression |
---|---|
imasf1oms | β’ (π β π β MetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasf1obl.u | . . 3 β’ (π β π = (πΉ βs π )) | |
2 | imasf1obl.v | . . 3 β’ (π β π = (Baseβπ )) | |
3 | imasf1obl.f | . . 3 β’ (π β πΉ:πβ1-1-ontoβπ΅) | |
4 | imasf1oms.r | . . . 4 β’ (π β π β MetSp) | |
5 | msxms 24378 | . . . 4 β’ (π β MetSp β π β βMetSp) | |
6 | 4, 5 | syl 17 | . . 3 β’ (π β π β βMetSp) |
7 | 1, 2, 3, 6 | imasf1oxms 24416 | . 2 β’ (π β π β βMetSp) |
8 | eqid 2725 | . . . . 5 β’ ((distβπ ) βΎ (π Γ π)) = ((distβπ ) βΎ (π Γ π)) | |
9 | eqid 2725 | . . . . 5 β’ (distβπ) = (distβπ) | |
10 | eqid 2725 | . . . . . . . 8 β’ (Baseβπ ) = (Baseβπ ) | |
11 | eqid 2725 | . . . . . . . 8 β’ ((distβπ ) βΎ ((Baseβπ ) Γ (Baseβπ ))) = ((distβπ ) βΎ ((Baseβπ ) Γ (Baseβπ ))) | |
12 | 10, 11 | msmet 24381 | . . . . . . 7 β’ (π β MetSp β ((distβπ ) βΎ ((Baseβπ ) Γ (Baseβπ ))) β (Metβ(Baseβπ ))) |
13 | 4, 12 | syl 17 | . . . . . 6 β’ (π β ((distβπ ) βΎ ((Baseβπ ) Γ (Baseβπ ))) β (Metβ(Baseβπ ))) |
14 | 2 | sqxpeqd 5704 | . . . . . . 7 β’ (π β (π Γ π) = ((Baseβπ ) Γ (Baseβπ ))) |
15 | 14 | reseq2d 5979 | . . . . . 6 β’ (π β ((distβπ ) βΎ (π Γ π)) = ((distβπ ) βΎ ((Baseβπ ) Γ (Baseβπ )))) |
16 | 2 | fveq2d 6896 | . . . . . 6 β’ (π β (Metβπ) = (Metβ(Baseβπ ))) |
17 | 13, 15, 16 | 3eltr4d 2840 | . . . . 5 β’ (π β ((distβπ ) βΎ (π Γ π)) β (Metβπ)) |
18 | 1, 2, 3, 4, 8, 9, 17 | imasf1omet 24300 | . . . 4 β’ (π β (distβπ) β (Metβπ΅)) |
19 | f1ofo 6841 | . . . . . . 7 β’ (πΉ:πβ1-1-ontoβπ΅ β πΉ:πβontoβπ΅) | |
20 | 3, 19 | syl 17 | . . . . . 6 β’ (π β πΉ:πβontoβπ΅) |
21 | 1, 2, 20, 4 | imasbas 17493 | . . . . 5 β’ (π β π΅ = (Baseβπ)) |
22 | 21 | fveq2d 6896 | . . . 4 β’ (π β (Metβπ΅) = (Metβ(Baseβπ))) |
23 | 18, 22 | eleqtrd 2827 | . . 3 β’ (π β (distβπ) β (Metβ(Baseβπ))) |
24 | ssid 3995 | . . 3 β’ (Baseβπ) β (Baseβπ) | |
25 | metres2 24287 | . . 3 β’ (((distβπ) β (Metβ(Baseβπ)) β§ (Baseβπ) β (Baseβπ)) β ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) β (Metβ(Baseβπ))) | |
26 | 23, 24, 25 | sylancl 584 | . 2 β’ (π β ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) β (Metβ(Baseβπ))) |
27 | eqid 2725 | . . 3 β’ (TopOpenβπ) = (TopOpenβπ) | |
28 | eqid 2725 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
29 | eqid 2725 | . . 3 β’ ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) = ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) | |
30 | 27, 28, 29 | isms 24373 | . 2 β’ (π β MetSp β (π β βMetSp β§ ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) β (Metβ(Baseβπ)))) |
31 | 7, 26, 30 | sylanbrc 581 | 1 β’ (π β π β MetSp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3939 Γ cxp 5670 βΎ cres 5674 βontoβwfo 6541 β1-1-ontoβwf1o 6542 βcfv 6543 (class class class)co 7416 Basecbs 17179 distcds 17241 TopOpenctopn 17402 βs cimas 17485 Metcmet 21269 βMetSpcxms 24241 MetSpcms 24242 |
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-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 |
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-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 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-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 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-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-xrs 17483 df-qtop 17488 df-imas 17489 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-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-xms 24244 df-ms 24245 |
This theorem is referenced by: xpsms 24462 |
Copyright terms: Public domain | W3C validator |