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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.) |
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 16904 | . . . 4 ⊢ dist = Slot (dist‘ndx) | |
2 | 9re 11929 | . . . . . 6 ⊢ 9 ∈ ℝ | |
3 | 1nn 11841 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
4 | 2nn0 12107 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
5 | 9nn0 12114 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
6 | 9lt10 12424 | . . . . . . 7 ⊢ 9 < ;10 | |
7 | 3, 4, 5, 6 | declti 12331 | . . . . . 6 ⊢ 9 < ;12 |
8 | 2, 7 | gtneii 10944 | . . . . 5 ⊢ ;12 ≠ 9 |
9 | dsndx 16903 | . . . . . 6 ⊢ (dist‘ndx) = ;12 | |
10 | tsetndx 16885 | . . . . . 6 ⊢ (TopSet‘ndx) = 9 | |
11 | 9, 10 | neeq12i 3007 | . . . . 5 ⊢ ((dist‘ndx) ≠ (TopSet‘ndx) ↔ ;12 ≠ 9) |
12 | 8, 11 | mpbir 234 | . . . 4 ⊢ (dist‘ndx) ≠ (TopSet‘ndx) |
13 | 1, 12 | setsnid 16759 | . . 3 ⊢ (dist‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉)) = (dist‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ − ))〉)) |
14 | tngds.2 | . . . . . 6 ⊢ − = (-g‘𝐺) | |
15 | 14 | fvexi 6731 | . . . . 5 ⊢ − ∈ V |
16 | coexg 7707 | . . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ − ∈ V) → (𝑁 ∘ − ) ∈ V) | |
17 | 15, 16 | mpan2 691 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) ∈ V) |
18 | 1 | setsid 16758 | . . . 4 ⊢ ((𝐺 ∈ V ∧ (𝑁 ∘ − ) ∈ V) → (𝑁 ∘ − ) = (dist‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉))) |
19 | 17, 18 | sylan2 596 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉))) |
20 | tngbas.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
21 | eqid 2737 | . . . . 5 ⊢ (𝑁 ∘ − ) = (𝑁 ∘ − ) | |
22 | eqid 2737 | . . . . 5 ⊢ (MetOpen‘(𝑁 ∘ − )) = (MetOpen‘(𝑁 ∘ − )) | |
23 | 20, 14, 21, 22 | tngval 23537 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → 𝑇 = ((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ − ))〉)) |
24 | 23 | fveq2d 6721 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (dist‘𝑇) = (dist‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ − ))〉))) |
25 | 13, 19, 24 | 3eqtr4a 2804 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘𝑇)) |
26 | co02 6124 | . . . . 5 ⊢ (𝑁 ∘ ∅) = ∅ | |
27 | df-ds 16824 | . . . . . 6 ⊢ dist = Slot ;12 | |
28 | 27 | str0 16742 | . . . . 5 ⊢ ∅ = (dist‘∅) |
29 | 26, 28 | eqtri 2765 | . . . 4 ⊢ (𝑁 ∘ ∅) = (dist‘∅) |
30 | fvprc 6709 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (-g‘𝐺) = ∅) | |
31 | 14, 30 | syl5eq 2790 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → − = ∅) |
32 | 31 | coeq2d 5731 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∘ − ) = (𝑁 ∘ ∅)) |
33 | reldmtng 23536 | . . . . . . 7 ⊢ Rel dom toNrmGrp | |
34 | 33 | ovprc1 7252 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (𝐺 toNrmGrp 𝑁) = ∅) |
35 | 20, 34 | syl5eq 2790 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝑇 = ∅) |
36 | 35 | fveq2d 6721 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (dist‘𝑇) = (dist‘∅)) |
37 | 29, 32, 36 | 3eqtr4a 2804 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∘ − ) = (dist‘𝑇)) |
38 | 37 | adantr 484 | . 2 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘𝑇)) |
39 | 25, 38 | pm2.61ian 812 | 1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) = (dist‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 Vcvv 3408 ∅c0 4237 〈cop 4547 ∘ ccom 5555 ‘cfv 6380 (class class class)co 7213 1c1 10730 2c2 11885 9c9 11892 ;cdc 12293 sSet csts 16716 ndxcnx 16744 TopSetcts 16808 distcds 16811 -gcsg 18367 MetOpencmopn 20353 toNrmGrp ctng 23476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-sets 16717 df-slot 16735 df-ndx 16745 df-tset 16821 df-ds 16824 df-tng 23482 |
This theorem is referenced by: tngtset 23547 tngtopn 23548 tngnm 23549 tngngp2 23550 tngngpd 23551 nrmtngdist 23555 tngnrg 23572 cnindmet 24059 tcphds 24128 |
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