![]() |
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 24315 | . . . 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 24353 | . 2 β’ (π β π β βMetSp) |
8 | eqid 2726 | . . . . 5 β’ ((distβπ ) βΎ (π Γ π)) = ((distβπ ) βΎ (π Γ π)) | |
9 | eqid 2726 | . . . . 5 β’ (distβπ) = (distβπ) | |
10 | eqid 2726 | . . . . . . . 8 β’ (Baseβπ ) = (Baseβπ ) | |
11 | eqid 2726 | . . . . . . . 8 β’ ((distβπ ) βΎ ((Baseβπ ) Γ (Baseβπ ))) = ((distβπ ) βΎ ((Baseβπ ) Γ (Baseβπ ))) | |
12 | 10, 11 | msmet 24318 | . . . . . . 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 5701 | . . . . . . 7 β’ (π β (π Γ π) = ((Baseβπ ) Γ (Baseβπ ))) |
15 | 14 | reseq2d 5975 | . . . . . 6 β’ (π β ((distβπ ) βΎ (π Γ π)) = ((distβπ ) βΎ ((Baseβπ ) Γ (Baseβπ )))) |
16 | 2 | fveq2d 6889 | . . . . . 6 β’ (π β (Metβπ) = (Metβ(Baseβπ ))) |
17 | 13, 15, 16 | 3eltr4d 2842 | . . . . 5 β’ (π β ((distβπ ) βΎ (π Γ π)) β (Metβπ)) |
18 | 1, 2, 3, 4, 8, 9, 17 | imasf1omet 24237 | . . . 4 β’ (π β (distβπ) β (Metβπ΅)) |
19 | f1ofo 6834 | . . . . . . 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 17467 | . . . . 5 β’ (π β π΅ = (Baseβπ)) |
22 | 21 | fveq2d 6889 | . . . 4 β’ (π β (Metβπ΅) = (Metβ(Baseβπ))) |
23 | 18, 22 | eleqtrd 2829 | . . 3 β’ (π β (distβπ) β (Metβ(Baseβπ))) |
24 | ssid 3999 | . . 3 β’ (Baseβπ) β (Baseβπ) | |
25 | metres2 24224 | . . 3 β’ (((distβπ) β (Metβ(Baseβπ)) β§ (Baseβπ) β (Baseβπ)) β ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) β (Metβ(Baseβπ))) | |
26 | 23, 24, 25 | sylancl 585 | . 2 β’ (π β ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) β (Metβ(Baseβπ))) |
27 | eqid 2726 | . . 3 β’ (TopOpenβπ) = (TopOpenβπ) | |
28 | eqid 2726 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
29 | eqid 2726 | . . 3 β’ ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) = ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) | |
30 | 27, 28, 29 | isms 24310 | . 2 β’ (π β MetSp β (π β βMetSp β§ ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) β (Metβ(Baseβπ)))) |
31 | 7, 26, 30 | sylanbrc 582 | 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 3943 Γ cxp 5667 βΎ cres 5671 βontoβwfo 6535 β1-1-ontoβwf1o 6536 βcfv 6537 (class class class)co 7405 Basecbs 17153 distcds 17215 TopOpenctopn 17376 βs cimas 17459 Metcmet 21226 βMetSpcxms 24178 MetSpcms 24179 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 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 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-xrs 17457 df-qtop 17462 df-imas 17463 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-xms 24181 df-ms 24182 |
This theorem is referenced by: xpsms 24399 |
Copyright terms: Public domain | W3C validator |