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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 16678 | . . . 4 ⊢ dist = Slot (dist‘ndx) | |
2 | 9re 11739 | . . . . . 6 ⊢ 9 ∈ ℝ | |
3 | 1nn 11651 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
4 | 2nn0 11917 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
5 | 9nn0 11924 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
6 | 9lt10 12232 | . . . . . . 7 ⊢ 9 < ;10 | |
7 | 3, 4, 5, 6 | declti 12139 | . . . . . 6 ⊢ 9 < ;12 |
8 | 2, 7 | gtneii 10754 | . . . . 5 ⊢ ;12 ≠ 9 |
9 | dsndx 16677 | . . . . . 6 ⊢ (dist‘ndx) = ;12 | |
10 | tsetndx 16661 | . . . . . 6 ⊢ (TopSet‘ndx) = 9 | |
11 | 9, 10 | neeq12i 3084 | . . . . 5 ⊢ ((dist‘ndx) ≠ (TopSet‘ndx) ↔ ;12 ≠ 9) |
12 | 8, 11 | mpbir 233 | . . . 4 ⊢ (dist‘ndx) ≠ (TopSet‘ndx) |
13 | 1, 12 | setsnid 16541 | . . 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 6686 | . . . . 5 ⊢ − ∈ V |
16 | coexg 7636 | . . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ − ∈ V) → (𝑁 ∘ − ) ∈ V) | |
17 | 15, 16 | mpan2 689 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) ∈ V) |
18 | 1 | setsid 16540 | . . . 4 ⊢ ((𝐺 ∈ V ∧ (𝑁 ∘ − ) ∈ V) → (𝑁 ∘ − ) = (dist‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉))) |
19 | 17, 18 | sylan2 594 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘(𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉))) |
20 | tngbas.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
21 | eqid 2823 | . . . . 5 ⊢ (𝑁 ∘ − ) = (𝑁 ∘ − ) | |
22 | eqid 2823 | . . . . 5 ⊢ (MetOpen‘(𝑁 ∘ − )) = (MetOpen‘(𝑁 ∘ − )) | |
23 | 20, 14, 21, 22 | tngval 23250 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → 𝑇 = ((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ − ))〉)) |
24 | 23 | fveq2d 6676 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (dist‘𝑇) = (dist‘((𝐺 sSet 〈(dist‘ndx), (𝑁 ∘ − )〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑁 ∘ − ))〉))) |
25 | 13, 19, 24 | 3eqtr4a 2884 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘𝑇)) |
26 | co02 6115 | . . . . 5 ⊢ (𝑁 ∘ ∅) = ∅ | |
27 | df-ds 16589 | . . . . . 6 ⊢ dist = Slot ;12 | |
28 | 27 | str0 16537 | . . . . 5 ⊢ ∅ = (dist‘∅) |
29 | 26, 28 | eqtri 2846 | . . . 4 ⊢ (𝑁 ∘ ∅) = (dist‘∅) |
30 | fvprc 6665 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (-g‘𝐺) = ∅) | |
31 | 14, 30 | syl5eq 2870 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → − = ∅) |
32 | 31 | coeq2d 5735 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∘ − ) = (𝑁 ∘ ∅)) |
33 | reldmtng 23249 | . . . . . . 7 ⊢ Rel dom toNrmGrp | |
34 | 33 | ovprc1 7197 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (𝐺 toNrmGrp 𝑁) = ∅) |
35 | 20, 34 | syl5eq 2870 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝑇 = ∅) |
36 | 35 | fveq2d 6676 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (dist‘𝑇) = (dist‘∅)) |
37 | 29, 32, 36 | 3eqtr4a 2884 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∘ − ) = (dist‘𝑇)) |
38 | 37 | adantr 483 | . 2 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑁 ∘ − ) = (dist‘𝑇)) |
39 | 25, 38 | pm2.61ian 810 | 1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) = (dist‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∅c0 4293 〈cop 4575 ∘ ccom 5561 ‘cfv 6357 (class class class)co 7158 1c1 10540 2c2 11695 9c9 11702 ;cdc 12101 ndxcnx 16482 sSet csts 16483 TopSetcts 16573 distcds 16576 -gcsg 18107 MetOpencmopn 20537 toNrmGrp ctng 23190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-ndx 16488 df-slot 16489 df-sets 16492 df-tset 16586 df-ds 16589 df-tng 23196 |
This theorem is referenced by: tngtset 23260 tngtopn 23261 tngnm 23262 tngngp2 23263 tngngpd 23264 nrmtngdist 23268 tngnrg 23285 cnindmet 23768 tcphds 23836 |
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