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Mirrors > Home > MPE Home > Th. List > tmslem | Structured version Visualization version GIF version |
Description: Lemma for tmsbas 24343, tmsds 24344, and tmstopn 24345. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tmsval.m | β’ π = {β¨(Baseβndx), πβ©, β¨(distβndx), π·β©} |
tmsval.k | β’ πΎ = (toMetSpβπ·) |
Ref | Expression |
---|---|
tmslem | β’ (π· β (βMetβπ) β (π = (BaseβπΎ) β§ π· = (distβπΎ) β§ (MetOpenβπ·) = (TopOpenβπΎ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6921 | . . . 4 β’ (π· β (βMetβπ) β π β dom βMet) | |
2 | tmsval.m | . . . . 5 β’ π = {β¨(Baseβndx), πβ©, β¨(distβndx), π·β©} | |
3 | basendxltdsndx 17340 | . . . . 5 β’ (Baseβndx) < (distβndx) | |
4 | dsndxnn 17339 | . . . . 5 β’ (distβndx) β β | |
5 | 2, 3, 4 | 2strbas1 17178 | . . . 4 β’ (π β dom βMet β π = (Baseβπ)) |
6 | 1, 5 | syl 17 | . . 3 β’ (π· β (βMetβπ) β π = (Baseβπ)) |
7 | xmetf 24186 | . . . . 5 β’ (π· β (βMetβπ) β π·:(π Γ π)βΆβ*) | |
8 | ffn 6710 | . . . . 5 β’ (π·:(π Γ π)βΆβ* β π· Fn (π Γ π)) | |
9 | fnresdm 6662 | . . . . 5 β’ (π· Fn (π Γ π) β (π· βΎ (π Γ π)) = π·) | |
10 | 7, 8, 9 | 3syl 18 | . . . 4 β’ (π· β (βMetβπ) β (π· βΎ (π Γ π)) = π·) |
11 | dsid 17338 | . . . . . 6 β’ dist = Slot (distβndx) | |
12 | 2, 3, 4, 11 | 2strop1 17179 | . . . . 5 β’ (π· β (βMetβπ) β π· = (distβπ)) |
13 | 12 | reseq1d 5973 | . . . 4 β’ (π· β (βMetβπ) β (π· βΎ (π Γ π)) = ((distβπ) βΎ (π Γ π))) |
14 | 10, 13 | eqtr3d 2768 | . . 3 β’ (π· β (βMetβπ) β π· = ((distβπ) βΎ (π Γ π))) |
15 | tmsval.k | . . . 4 β’ πΎ = (toMetSpβπ·) | |
16 | 2, 15 | tmsval 24340 | . . 3 β’ (π· β (βMetβπ) β πΎ = (π sSet β¨(TopSetβndx), (MetOpenβπ·)β©)) |
17 | 6, 14, 16 | setsmsbas 24332 | . 2 β’ (π· β (βMetβπ) β π = (BaseβπΎ)) |
18 | 6, 14, 16 | setsmsds 24334 | . . 3 β’ (π· β (βMetβπ) β (distβπ) = (distβπΎ)) |
19 | 12, 18 | eqtrd 2766 | . 2 β’ (π· β (βMetβπ) β π· = (distβπΎ)) |
20 | prex 5425 | . . . . 5 β’ {β¨(Baseβndx), πβ©, β¨(distβndx), π·β©} β V | |
21 | 2, 20 | eqeltri 2823 | . . . 4 β’ π β V |
22 | 21 | a1i 11 | . . 3 β’ (π· β (βMetβπ) β π β V) |
23 | 6, 14, 16, 22 | setsmstopn 24337 | . 2 β’ (π· β (βMetβπ) β (MetOpenβπ·) = (TopOpenβπΎ)) |
24 | 17, 19, 23 | 3jca 1125 | 1 β’ (π· β (βMetβπ) β (π = (BaseβπΎ) β§ π· = (distβπΎ) β§ (MetOpenβπ·) = (TopOpenβπΎ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3468 {cpr 4625 β¨cop 4629 Γ cxp 5667 dom cdm 5669 βΎ cres 5671 Fn wfn 6531 βΆwf 6532 βcfv 6536 β*cxr 11248 ndxcnx 17133 Basecbs 17151 distcds 17213 TopOpenctopn 17374 βMetcxmet 21221 MetOpencmopn 21226 toMetSpctms 24176 |
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 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
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-op 4630 df-uni 4903 df-iun 4992 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-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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-fz 13488 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-tset 17223 df-ds 17226 df-rest 17375 df-topn 17376 df-topgen 17396 df-psmet 21228 df-xmet 21229 df-bl 21231 df-mopn 21232 df-top 22747 df-topon 22764 df-bases 22800 df-tms 24179 |
This theorem is referenced by: tmsbas 24343 tmsds 24344 tmstopn 24345 |
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