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Mirrors > Home > MPE Home > Th. List > tmslem | Structured version Visualization version GIF version |
Description: Lemma for tmsbas 22786, tmsds 22787, and tmstopn 22788. (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 6525 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met) | |
2 | tmsval.m | . . . . 5 ⊢ 𝑀 = {〈(Base‘ndx), 𝑋〉, 〈(dist‘ndx), 𝐷〉} | |
3 | df-ds 16433 | . . . . 5 ⊢ dist = Slot ;12 | |
4 | 1nn 11444 | . . . . . 6 ⊢ 1 ∈ ℕ | |
5 | 2nn0 11719 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
6 | 1nn0 11718 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
7 | 1lt10 12045 | . . . . . 6 ⊢ 1 < ;10 | |
8 | 4, 5, 6, 7 | declti 11943 | . . . . 5 ⊢ 1 < ;12 |
9 | 2nn 11506 | . . . . . 6 ⊢ 2 ∈ ℕ | |
10 | 6, 9 | decnncl 11925 | . . . . 5 ⊢ ;12 ∈ ℕ |
11 | 2, 3, 8, 10 | 2strbas 16449 | . . . 4 ⊢ (𝑋 ∈ dom ∞Met → 𝑋 = (Base‘𝑀)) |
12 | 1, 11 | syl 17 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝑀)) |
13 | xmetf 22632 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
14 | ffn 6338 | . . . . 5 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → 𝐷 Fn (𝑋 × 𝑋)) | |
15 | fnresdm 6293 | . . . . 5 ⊢ (𝐷 Fn (𝑋 × 𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = 𝐷) | |
16 | 13, 14, 15 | 3syl 18 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = 𝐷) |
17 | 2, 3, 8, 10 | 2strop 16450 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝑀)) |
18 | 17 | reseq1d 5687 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
19 | 16, 18 | eqtr3d 2810 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
20 | tmsval.k | . . . 4 ⊢ 𝐾 = (toMetSp‘𝐷) | |
21 | 2, 20 | tmsval 22784 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) |
22 | 12, 19, 21 | setsmsbas 22778 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾)) |
23 | 12, 19, 21 | setsmsds 22779 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (dist‘𝑀) = (dist‘𝐾)) |
24 | 17, 23 | eqtrd 2808 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾)) |
25 | prex 5183 | . . . . 5 ⊢ {〈(Base‘ndx), 𝑋〉, 〈(dist‘ndx), 𝐷〉} ∈ V | |
26 | 2, 25 | eqeltri 2856 | . . . 4 ⊢ 𝑀 ∈ V |
27 | 26 | a1i 11 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑀 ∈ V) |
28 | 12, 19, 21, 27 | setsmstopn 22781 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) = (TopOpen‘𝐾)) |
29 | 22, 24, 28 | 3jca 1108 | 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 1068 = wceq 1507 ∈ wcel 2048 Vcvv 3409 {cpr 4437 〈cop 4441 × cxp 5398 dom cdm 5400 ↾ cres 5402 Fn wfn 6177 ⟶wf 6178 ‘cfv 6182 1c1 10328 ℝ*cxr 10465 2c2 11488 ;cdc 11904 ndxcnx 16326 Basecbs 16329 distcds 16420 TopOpenctopn 16541 ∞Metcxmet 20222 MetOpencmopn 20227 toMetSpctms 22622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-sup 8693 df-inf 8694 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-q 12156 df-rp 12198 df-xneg 12317 df-xadd 12318 df-xmul 12319 df-fz 12702 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-tset 16430 df-ds 16433 df-rest 16542 df-topn 16543 df-topgen 16563 df-psmet 20229 df-xmet 20230 df-bl 20232 df-mopn 20233 df-top 21196 df-topon 21213 df-bases 21248 df-tms 22625 |
This theorem is referenced by: tmsbas 22786 tmsds 22787 tmstopn 22788 |
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