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Mirrors > Home > MPE Home > Th. List > tngds | Structured version Visualization version GIF version |
Description: The metric function of a structure augmented with a norm. (Contributed by Mario Carneiro, 3-Oct-2015.) (Proof shortened by AV, 29-Oct-2024.) |
Ref | Expression |
---|---|
tngbas.t | ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
tngds.2 | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
tngds | ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) = (dist‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dsid 17313 | . . . 4 ⊢ dist = Slot (dist‘ndx) | |
2 | dsndxntsetndx 17320 | . . . 4 ⊢ (dist‘ndx) ≠ (TopSet‘ndx) | |
3 | 1, 2 | setsnid 17124 | . . 3 ⊢ (dist‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉)) = (dist‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ − ))〉)) |
4 | tngds.2 | . . . . . 6 ⊢ − = (-g‘𝐺) | |
5 | 4 | fvexi 6892 | . . . . 5 ⊢ − ∈ V |
6 | coexg 7902 | . . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ − ∈ V) → (𝑁 ∘ − ) ∈ V) | |
7 | 5, 6 | mpan2 689 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) ∈ V) |
8 | 1 | setsid 17123 | . . . 4 ⊢ ((𝐺 ∈ V ∧ (𝑁 ∘ − ) ∈ V) → (𝑁 ∘ − ) = (dist‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉))) |
9 | 7, 8 | sylan2 593 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉))) |
10 | tngbas.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
11 | eqid 2731 | . . . . 5 ⊢ (𝑁 ∘ − ) = (𝑁 ∘ − ) | |
12 | eqid 2731 | . . . . 5 ⊢ (MetOpen‘(𝑁 ∘ − )) = (MetOpen‘(𝑁 ∘ − )) | |
13 | 10, 4, 11, 12 | tngval 24077 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → 𝑇 = ((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ − ))〉)) |
14 | 13 | fveq2d 6882 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (dist‘𝑇) = (dist‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ − ))〉))) |
15 | 3, 9, 14 | 3eqtr4a 2797 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘𝑇)) |
16 | co02 6248 | . . . . 5 ⊢ (𝑁 ∘ ∅) = ∅ | |
17 | 1 | str0 17104 | . . . . 5 ⊢ ∅ = (dist‘∅) |
18 | 16, 17 | eqtri 2759 | . . . 4 ⊢ (𝑁 ∘ ∅) = (dist‘∅) |
19 | fvprc 6870 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (-g‘𝐺) = ∅) | |
20 | 4, 19 | eqtrid 2783 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → − = ∅) |
21 | 20 | coeq2d 5854 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∘ − ) = (𝑁 ∘ ∅)) |
22 | reldmtng 24076 | . . . . . . 7 ⊢ Rel dom toNrmGrp | |
23 | 22 | ovprc1 7432 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (𝐺 toNrmGrp 𝑁) = ∅) |
24 | 10, 23 | eqtrid 2783 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝑇 = ∅) |
25 | 24 | fveq2d 6882 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (dist‘𝑇) = (dist‘∅)) |
26 | 18, 21, 25 | 3eqtr4a 2797 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∘ − ) = (dist‘𝑇)) |
27 | 26 | adantr 481 | . 2 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘𝑇)) |
28 | 15, 27 | pm2.61ian 810 | 1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) = (dist‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3473 ∅c0 4318 〈cop 4628 ∘ ccom 5673 ‘cfv 6532 (class class class)co 7393 sSet csts 17078 ndxcnx 17108 TopSetcts 17185 distcds 17188 -gcsg 18796 MetOpencmopn 20868 toNrmGrp ctng 24016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 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 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-sets 17079 df-slot 17097 df-ndx 17109 df-tset 17198 df-ds 17201 df-tng 24022 |
This theorem is referenced by: tngtset 24095 tngtopn 24096 tngnm 24097 tngngp2 24098 tngngpd 24099 nrmtngdist 24103 tngnrg 24120 cnindmet 24608 tcphds 24677 |
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