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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 16063 | . . . 4 ⊢ dist = Slot (dist‘ndx) | |
2 | 9re 11107 | . . . . . 6 ⊢ 9 ∈ ℝ | |
3 | 1nn 11031 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
4 | 2nn0 11309 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
5 | 9nn0 11316 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
6 | 9lt10 11673 | . . . . . . 7 ⊢ 9 < ;10 | |
7 | 3, 4, 5, 6 | declti 11546 | . . . . . 6 ⊢ 9 < ;12 |
8 | 2, 7 | gtneii 10149 | . . . . 5 ⊢ ;12 ≠ 9 |
9 | dsndx 16062 | . . . . . 6 ⊢ (dist‘ndx) = ;12 | |
10 | tsetndx 16040 | . . . . . 6 ⊢ (TopSet‘ndx) = 9 | |
11 | 9, 10 | neeq12i 2860 | . . . . 5 ⊢ ((dist‘ndx) ≠ (TopSet‘ndx) ↔ ;12 ≠ 9) |
12 | 8, 11 | mpbir 221 | . . . 4 ⊢ (dist‘ndx) ≠ (TopSet‘ndx) |
13 | 1, 12 | setsnid 15915 | . . 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 6201 | . . . . . 6 ⊢ (-g‘𝐺) ∈ V | |
16 | 14, 15 | eqeltri 2697 | . . . . 5 ⊢ − ∈ V |
17 | coexg 7117 | . . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ − ∈ V) → (𝑁 ∘ − ) ∈ V) | |
18 | 16, 17 | mpan2 707 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) ∈ V) |
19 | 1 | setsid 15914 | . . . 4 ⊢ ((𝐺 ∈ V ∧ (𝑁 ∘ − ) ∈ V) → (𝑁 ∘ − ) = (dist‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉))) |
20 | 18, 19 | sylan2 491 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉))) |
21 | tngbas.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
22 | eqid 2622 | . . . . 5 ⊢ (𝑁 ∘ − ) = (𝑁 ∘ − ) | |
23 | eqid 2622 | . . . . 5 ⊢ (MetOpen‘(𝑁 ∘ − )) = (MetOpen‘(𝑁 ∘ − )) | |
24 | 21, 14, 22, 23 | tngval 22443 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → 𝑇 = ((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ − ))〉)) |
25 | 24 | fveq2d 6195 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (dist‘𝑇) = (dist‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ − ))〉))) |
26 | 13, 20, 25 | 3eqtr4a 2682 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘𝑇)) |
27 | co02 5649 | . . . . 5 ⊢ (𝑁 ∘ ∅) = ∅ | |
28 | df-ds 15964 | . . . . . 6 ⊢ dist = Slot ;12 | |
29 | 28 | str0 15911 | . . . . 5 ⊢ ∅ = (dist‘∅) |
30 | 27, 29 | eqtri 2644 | . . . 4 ⊢ (𝑁 ∘ ∅) = (dist‘∅) |
31 | fvprc 6185 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (-g‘𝐺) = ∅) | |
32 | 14, 31 | syl5eq 2668 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → − = ∅) |
33 | 32 | coeq2d 5284 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∘ − ) = (𝑁 ∘ ∅)) |
34 | reldmtng 22442 | . . . . . . 7 ⊢ Rel dom toNrmGrp | |
35 | 34 | ovprc1 6684 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (𝐺 toNrmGrp 𝑁) = ∅) |
36 | 21, 35 | syl5eq 2668 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝑇 = ∅) |
37 | 36 | fveq2d 6195 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (dist‘𝑇) = (dist‘∅)) |
38 | 30, 33, 37 | 3eqtr4a 2682 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∘ − ) = (dist‘𝑇)) |
39 | 38 | adantr 481 | . 2 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘𝑇)) |
40 | 26, 39 | pm2.61ian 831 | 1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) = (dist‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∅c0 3915 〈cop 4183 ∘ ccom 5118 ‘cfv 5888 (class class class)co 6650 1c1 9937 2c2 11070 9c9 11077 ;cdc 11493 ndxcnx 15854 sSet csts 15855 TopSetcts 15947 distcds 15950 -gcsg 17424 MetOpencmopn 19736 toNrmGrp ctng 22383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-ndx 15860 df-slot 15861 df-sets 15864 df-tset 15960 df-ds 15964 df-tng 22389 |
This theorem is referenced by: tngtset 22453 tngtopn 22454 tngnm 22455 tngngp2 22456 tngngpd 22457 nrmtngdist 22461 tngnrg 22478 cnindmet 22962 tchds 23030 |
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