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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 15886 | . . . 4 ⊢ dist = Slot (dist‘ndx) | |
2 | 9re 10984 | . . . . . 6 ⊢ 9 ∈ ℝ | |
3 | 1nn 10908 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
4 | 2nn0 11186 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
5 | 9nn0 11193 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
6 | 9lt10 11549 | . . . . . . 7 ⊢ 9 < ;10 | |
7 | 3, 4, 5, 6 | declti 11422 | . . . . . 6 ⊢ 9 < ;12 |
8 | 2, 7 | gtneii 10028 | . . . . 5 ⊢ ;12 ≠ 9 |
9 | dsndx 15885 | . . . . . 6 ⊢ (dist‘ndx) = ;12 | |
10 | tsetndx 15863 | . . . . . 6 ⊢ (TopSet‘ndx) = 9 | |
11 | 9, 10 | neeq12i 2848 | . . . . 5 ⊢ ((dist‘ndx) ≠ (TopSet‘ndx) ↔ ;12 ≠ 9) |
12 | 8, 11 | mpbir 220 | . . . 4 ⊢ (dist‘ndx) ≠ (TopSet‘ndx) |
13 | 1, 12 | setsnid 15743 | . . 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 | fvex 6113 | . . . . . 6 ⊢ (-g‘𝐺) ∈ V | |
16 | 14, 15 | eqeltri 2684 | . . . . 5 ⊢ − ∈ V |
17 | coexg 7010 | . . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ − ∈ V) → (𝑁 ∘ − ) ∈ V) | |
18 | 16, 17 | mpan2 703 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) ∈ V) |
19 | 1 | setsid 15742 | . . . 4 ⊢ ((𝐺 ∈ V ∧ (𝑁 ∘ − ) ∈ V) → (𝑁 ∘ − ) = (dist‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉))) |
20 | 18, 19 | sylan2 490 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉))) |
21 | tngbas.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
22 | eqid 2610 | . . . . 5 ⊢ (𝑁 ∘ − ) = (𝑁 ∘ − ) | |
23 | eqid 2610 | . . . . 5 ⊢ (MetOpen‘(𝑁 ∘ − )) = (MetOpen‘(𝑁 ∘ − )) | |
24 | 21, 14, 22, 23 | tngval 22253 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → 𝑇 = ((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ − ))〉)) |
25 | 24 | fveq2d 6107 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (dist‘𝑇) = (dist‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ − ))〉))) |
26 | 13, 20, 25 | 3eqtr4a 2670 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘𝑇)) |
27 | co02 5566 | . . . . 5 ⊢ (𝑁 ∘ ∅) = ∅ | |
28 | df-ds 15791 | . . . . . 6 ⊢ dist = Slot ;12 | |
29 | 28 | str0 15739 | . . . . 5 ⊢ ∅ = (dist‘∅) |
30 | 27, 29 | eqtri 2632 | . . . 4 ⊢ (𝑁 ∘ ∅) = (dist‘∅) |
31 | fvprc 6097 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (-g‘𝐺) = ∅) | |
32 | 14, 31 | syl5eq 2656 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → − = ∅) |
33 | 32 | coeq2d 5206 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∘ − ) = (𝑁 ∘ ∅)) |
34 | reldmtng 22252 | . . . . . . 7 ⊢ Rel dom toNrmGrp | |
35 | 34 | ovprc1 6582 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (𝐺 toNrmGrp 𝑁) = ∅) |
36 | 21, 35 | syl5eq 2656 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝑇 = ∅) |
37 | 36 | fveq2d 6107 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (dist‘𝑇) = (dist‘∅)) |
38 | 30, 33, 37 | 3eqtr4a 2670 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∘ − ) = (dist‘𝑇)) |
39 | 38 | adantr 480 | . 2 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘𝑇)) |
40 | 26, 39 | pm2.61ian 827 | 1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) = (dist‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∅c0 3874 〈cop 4131 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 1c1 9816 2c2 10947 9c9 10954 ;cdc 11369 ndxcnx 15692 sSet csts 15693 TopSetcts 15774 distcds 15777 -gcsg 17247 MetOpencmopn 19557 toNrmGrp ctng 22193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-ndx 15698 df-slot 15699 df-sets 15701 df-tset 15787 df-ds 15791 df-tng 22199 |
This theorem is referenced by: tngtset 22263 tngtopn 22264 tngnm 22265 tngngp2 22266 tngngpd 22267 nrmtngdist 22271 tngnrg 22288 cnindmet 22770 tchds 22838 |
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