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Mirrors > Home > MPE Home > Th. List > tmslem | Structured version Visualization version GIF version |
Description: Lemma for tmsbas 22501, tmsds 22502, and tmstopn 22503. (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 6359 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met) | |
2 | tmsval.m | . . . . 5 ⊢ 𝑀 = {〈(Base‘ndx), 𝑋〉, 〈(dist‘ndx), 𝐷〉} | |
3 | df-ds 16165 | . . . . 5 ⊢ dist = Slot ;12 | |
4 | 1nn 11231 | . . . . . 6 ⊢ 1 ∈ ℕ | |
5 | 2nn0 11509 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
6 | 1nn0 11508 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
7 | 1lt10 11880 | . . . . . 6 ⊢ 1 < ;10 | |
8 | 4, 5, 6, 7 | declti 11746 | . . . . 5 ⊢ 1 < ;12 |
9 | 2nn 11385 | . . . . . 6 ⊢ 2 ∈ ℕ | |
10 | 6, 9 | decnncl 11718 | . . . . 5 ⊢ ;12 ∈ ℕ |
11 | 2, 3, 8, 10 | 2strbas 16185 | . . . 4 ⊢ (𝑋 ∈ dom ∞Met → 𝑋 = (Base‘𝑀)) |
12 | 1, 11 | syl 17 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝑀)) |
13 | xmetf 22347 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
14 | ffn 6183 | . . . . 5 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → 𝐷 Fn (𝑋 × 𝑋)) | |
15 | fnresdm 6138 | . . . . 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 16186 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝑀)) |
18 | 17 | reseq1d 5531 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
19 | 16, 18 | eqtr3d 2807 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
20 | tmsval.k | . . . 4 ⊢ 𝐾 = (toMetSp‘𝐷) | |
21 | 2, 20 | tmsval 22499 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) |
22 | 12, 19, 21 | setsmsbas 22493 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾)) |
23 | 12, 19, 21 | setsmsds 22494 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (dist‘𝑀) = (dist‘𝐾)) |
24 | 17, 23 | eqtrd 2805 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾)) |
25 | prex 5037 | . . . . 5 ⊢ {〈(Base‘ndx), 𝑋〉, 〈(dist‘ndx), 𝐷〉} ∈ V | |
26 | 2, 25 | eqeltri 2846 | . . . 4 ⊢ 𝑀 ∈ V |
27 | 26 | a1i 11 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑀 ∈ V) |
28 | 12, 19, 21, 27 | setsmstopn 22496 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) = (TopOpen‘𝐾)) |
29 | 22, 24, 28 | 3jca 1122 | 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 1071 = wceq 1631 ∈ wcel 2145 Vcvv 3351 {cpr 4318 〈cop 4322 × cxp 5247 dom cdm 5249 ↾ cres 5251 Fn wfn 6024 ⟶wf 6025 ‘cfv 6029 1c1 10137 ℝ*cxr 10273 2c2 11270 ;cdc 11693 ndxcnx 16054 Basecbs 16057 distcds 16151 TopOpenctopn 16283 ∞Metcxmt 19939 MetOpencmopn 19944 toMetSpctmt 22337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 ax-pre-sup 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-1st 7313 df-2nd 7314 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-oadd 7715 df-er 7894 df-map 8009 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-sup 8502 df-inf 8503 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-div 10885 df-nn 11221 df-2 11279 df-3 11280 df-4 11281 df-5 11282 df-6 11283 df-7 11284 df-8 11285 df-9 11286 df-n0 11493 df-z 11578 df-dec 11694 df-uz 11887 df-q 11990 df-rp 12029 df-xneg 12144 df-xadd 12145 df-xmul 12146 df-fz 12527 df-struct 16059 df-ndx 16060 df-slot 16061 df-base 16063 df-sets 16064 df-tset 16161 df-ds 16165 df-rest 16284 df-topn 16285 df-topgen 16305 df-psmet 19946 df-xmet 19947 df-bl 19949 df-mopn 19950 df-top 20912 df-topon 20929 df-bases 20964 df-tms 22340 |
This theorem is referenced by: tmsbas 22501 tmsds 22502 tmstopn 22503 |
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