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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 17430 | . . . 4 ⊢ dist = Slot (dist‘ndx) | |
| 2 | dsndxntsetndx 17437 | . . . 4 ⊢ (dist‘ndx) ≠ (TopSet‘ndx) | |
| 3 | 1, 2 | setsnid 17245 | . . 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 6920 | . . . . 5 ⊢ − ∈ V |
| 6 | coexg 7951 | . . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ − ∈ V) → (𝑁 ∘ − ) ∈ V) | |
| 7 | 5, 6 | mpan2 691 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) ∈ V) |
| 8 | 1 | setsid 17244 | . . . 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 2737 | . . . . 5 ⊢ (𝑁 ∘ − ) = (𝑁 ∘ − ) | |
| 12 | eqid 2737 | . . . . 5 ⊢ (MetOpen‘(𝑁 ∘ − )) = (MetOpen‘(𝑁 ∘ − )) | |
| 13 | 10, 4, 11, 12 | tngval 24652 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → 𝑇 = ((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ − ))〉)) |
| 14 | 13 | fveq2d 6910 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (dist‘𝑇) = (dist‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ − ))〉))) |
| 15 | 3, 9, 14 | 3eqtr4a 2803 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘𝑇)) |
| 16 | co02 6280 | . . . . 5 ⊢ (𝑁 ∘ ∅) = ∅ | |
| 17 | 1 | str0 17226 | . . . . 5 ⊢ ∅ = (dist‘∅) |
| 18 | 16, 17 | eqtri 2765 | . . . 4 ⊢ (𝑁 ∘ ∅) = (dist‘∅) |
| 19 | fvprc 6898 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (-g‘𝐺) = ∅) | |
| 20 | 4, 19 | eqtrid 2789 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → − = ∅) |
| 21 | 20 | coeq2d 5873 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∘ − ) = (𝑁 ∘ ∅)) |
| 22 | reldmtng 24651 | . . . . . . 7 ⊢ Rel dom toNrmGrp | |
| 23 | 22 | ovprc1 7470 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (𝐺 toNrmGrp 𝑁) = ∅) |
| 24 | 10, 23 | eqtrid 2789 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝑇 = ∅) |
| 25 | 24 | fveq2d 6910 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (dist‘𝑇) = (dist‘∅)) |
| 26 | 18, 21, 25 | 3eqtr4a 2803 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∘ − ) = (dist‘𝑇)) |
| 27 | 26 | adantr 480 | . 2 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘𝑇)) |
| 28 | 15, 27 | pm2.61ian 812 | 1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) = (dist‘𝑇)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 〈cop 4632 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 sSet csts 17200 ndxcnx 17230 TopSetcts 17303 distcds 17306 -gcsg 18953 MetOpencmopn 21354 toNrmGrp ctng 24591 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-sets 17201 df-slot 17219 df-ndx 17231 df-tset 17316 df-ds 17319 df-tng 24597 |
| This theorem is referenced by: tngtset 24670 tngtopn 24671 tngnm 24672 tngngp2 24673 tngngpd 24674 nrmtngdist 24678 tngnrg 24695 cnindmet 25196 tcphds 25265 |
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