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Mirrors > Home > MPE Home > Th. List > tmslem | Structured version Visualization version GIF version |
Description: Lemma for tmsbas 23637, tmsds 23638, and tmstopn 23639. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tmsval.m | ⊢ 𝑀 = {〈(Base‘ndx), 𝑋〉, 〈(dist‘ndx), 𝐷〉} |
tmsval.k | ⊢ 𝐾 = (toMetSp‘𝐷) |
Ref | Expression |
---|---|
tmslem | ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝐷 = (dist‘𝐾) ∧ (MetOpen‘𝐷) = (TopOpen‘𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6803 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met) | |
2 | tmsval.m | . . . . 5 ⊢ 𝑀 = {〈(Base‘ndx), 𝑋〉, 〈(dist‘ndx), 𝐷〉} | |
3 | basendxltdsndx 17096 | . . . . 5 ⊢ (Base‘ndx) < (dist‘ndx) | |
4 | dsndxnn 17095 | . . . . 5 ⊢ (dist‘ndx) ∈ ℕ | |
5 | 2, 3, 4 | 2strbas1 16937 | . . . 4 ⊢ (𝑋 ∈ dom ∞Met → 𝑋 = (Base‘𝑀)) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝑀)) |
7 | xmetf 23480 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
8 | ffn 6598 | . . . . 5 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → 𝐷 Fn (𝑋 × 𝑋)) | |
9 | fnresdm 6549 | . . . . 5 ⊢ (𝐷 Fn (𝑋 × 𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = 𝐷) | |
10 | 7, 8, 9 | 3syl 18 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = 𝐷) |
11 | dsid 17094 | . . . . . 6 ⊢ dist = Slot (dist‘ndx) | |
12 | 2, 3, 4, 11 | 2strop1 16938 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝑀)) |
13 | 12 | reseq1d 5889 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
14 | 10, 13 | eqtr3d 2782 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
15 | tmsval.k | . . . 4 ⊢ 𝐾 = (toMetSp‘𝐷) | |
16 | 2, 15 | tmsval 23634 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) |
17 | 6, 14, 16 | setsmsbas 23626 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾)) |
18 | 6, 14, 16 | setsmsds 23628 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (dist‘𝑀) = (dist‘𝐾)) |
19 | 12, 18 | eqtrd 2780 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾)) |
20 | prex 5359 | . . . . 5 ⊢ {〈(Base‘ndx), 𝑋〉, 〈(dist‘ndx), 𝐷〉} ∈ V | |
21 | 2, 20 | eqeltri 2837 | . . . 4 ⊢ 𝑀 ∈ V |
22 | 21 | a1i 11 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑀 ∈ V) |
23 | 6, 14, 16, 22 | setsmstopn 23631 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) = (TopOpen‘𝐾)) |
24 | 17, 19, 23 | 3jca 1127 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝐷 = (dist‘𝐾) ∧ (MetOpen‘𝐷) = (TopOpen‘𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 Vcvv 3431 {cpr 4569 〈cop 4573 × cxp 5588 dom cdm 5590 ↾ cres 5592 Fn wfn 6427 ⟶wf 6428 ‘cfv 6432 ℝ*cxr 11009 ndxcnx 16892 Basecbs 16910 distcds 16969 TopOpenctopn 17130 ∞Metcxmet 20580 MetOpencmopn 20585 toMetSpctms 23470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-sup 9179 df-inf 9180 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-fz 13239 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-tset 16979 df-ds 16982 df-rest 17131 df-topn 17132 df-topgen 17152 df-psmet 20587 df-xmet 20588 df-bl 20590 df-mopn 20591 df-top 22041 df-topon 22058 df-bases 22094 df-tms 23473 |
This theorem is referenced by: tmsbas 23637 tmsds 23638 tmstopn 23639 |
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