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Mirrors > Home > MPE Home > Th. List > tmslem | Structured version Visualization version GIF version |
Description: Lemma for tmsbas 23855, tmsds 23856, and tmstopn 23857. (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 6880 | . . . 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 17274 | . . . . 5 β’ (Baseβndx) < (distβndx) | |
4 | dsndxnn 17273 | . . . . 5 β’ (distβndx) β β | |
5 | 2, 3, 4 | 2strbas1 17115 | . . . 4 β’ (π β dom βMet β π = (Baseβπ)) |
6 | 1, 5 | syl 17 | . . 3 β’ (π· β (βMetβπ) β π = (Baseβπ)) |
7 | xmetf 23698 | . . . . 5 β’ (π· β (βMetβπ) β π·:(π Γ π)βΆβ*) | |
8 | ffn 6669 | . . . . 5 β’ (π·:(π Γ π)βΆβ* β π· Fn (π Γ π)) | |
9 | fnresdm 6621 | . . . . 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 17272 | . . . . . 6 β’ dist = Slot (distβndx) | |
12 | 2, 3, 4, 11 | 2strop1 17116 | . . . . 5 β’ (π· β (βMetβπ) β π· = (distβπ)) |
13 | 12 | reseq1d 5937 | . . . 4 β’ (π· β (βMetβπ) β (π· βΎ (π Γ π)) = ((distβπ) βΎ (π Γ π))) |
14 | 10, 13 | eqtr3d 2775 | . . 3 β’ (π· β (βMetβπ) β π· = ((distβπ) βΎ (π Γ π))) |
15 | tmsval.k | . . . 4 β’ πΎ = (toMetSpβπ·) | |
16 | 2, 15 | tmsval 23852 | . . 3 β’ (π· β (βMetβπ) β πΎ = (π sSet β¨(TopSetβndx), (MetOpenβπ·)β©)) |
17 | 6, 14, 16 | setsmsbas 23844 | . 2 β’ (π· β (βMetβπ) β π = (BaseβπΎ)) |
18 | 6, 14, 16 | setsmsds 23846 | . . 3 β’ (π· β (βMetβπ) β (distβπ) = (distβπΎ)) |
19 | 12, 18 | eqtrd 2773 | . 2 β’ (π· β (βMetβπ) β π· = (distβπΎ)) |
20 | prex 5390 | . . . . 5 β’ {β¨(Baseβndx), πβ©, β¨(distβndx), π·β©} β V | |
21 | 2, 20 | eqeltri 2830 | . . . 4 β’ π β V |
22 | 21 | a1i 11 | . . 3 β’ (π· β (βMetβπ) β π β V) |
23 | 6, 14, 16, 22 | setsmstopn 23849 | . 2 β’ (π· β (βMetβπ) β (MetOpenβπ·) = (TopOpenβπΎ)) |
24 | 17, 19, 23 | 3jca 1129 | 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 1088 = wceq 1542 β wcel 2107 Vcvv 3444 {cpr 4589 β¨cop 4593 Γ cxp 5632 dom cdm 5634 βΎ cres 5636 Fn wfn 6492 βΆwf 6493 βcfv 6497 β*cxr 11193 ndxcnx 17070 Basecbs 17088 distcds 17147 TopOpenctopn 17308 βMetcxmet 20797 MetOpencmopn 20802 toMetSpctms 23688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-q 12879 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-tset 17157 df-ds 17160 df-rest 17309 df-topn 17310 df-topgen 17330 df-psmet 20804 df-xmet 20805 df-bl 20807 df-mopn 20808 df-top 22259 df-topon 22276 df-bases 22312 df-tms 23691 |
This theorem is referenced by: tmsbas 23855 tmsds 23856 tmstopn 23857 |
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