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Mirrors > Home > MPE Home > Th. List > tngngpim | Structured version Visualization version GIF version |
Description: The induced metric of a normed group is a function. (Contributed by AV, 19-Oct-2021.) |
Ref | Expression |
---|---|
tngngpim.t | ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
tngngpim.n | ⊢ 𝑁 = (norm‘𝐺) |
tngngpim.x | ⊢ 𝑋 = (Base‘𝐺) |
tngngpim.d | ⊢ 𝐷 = (dist‘𝑇) |
Ref | Expression |
---|---|
tngngpim | ⊢ (𝐺 ∈ NrmGrp → 𝐷:(𝑋 × 𝑋)⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tngngpim.x | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
2 | tngngpim.n | . . 3 ⊢ 𝑁 = (norm‘𝐺) | |
3 | 1, 2 | nmf 23226 | . 2 ⊢ (𝐺 ∈ NrmGrp → 𝑁:𝑋⟶ℝ) |
4 | tngngpim.t | . . . . . 6 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
5 | 2 | oveq2i 7169 | . . . . . 6 ⊢ (𝐺 toNrmGrp 𝑁) = (𝐺 toNrmGrp (norm‘𝐺)) |
6 | 4, 5 | eqtri 2846 | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp (norm‘𝐺)) |
7 | 6 | nrmtngnrm 23269 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → (𝑇 ∈ NrmGrp ∧ (norm‘𝑇) = (norm‘𝐺))) |
8 | tngngpim.d | . . . . . . . 8 ⊢ 𝐷 = (dist‘𝑇) | |
9 | 4, 1, 8 | tngngp2 23263 | . . . . . . 7 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋)))) |
10 | simpr 487 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋)) → 𝐷 ∈ (Met‘𝑋)) | |
11 | 9, 10 | syl6bi 255 | . . . . . 6 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp → 𝐷 ∈ (Met‘𝑋))) |
12 | 11 | com12 32 | . . . . 5 ⊢ (𝑇 ∈ NrmGrp → (𝑁:𝑋⟶ℝ → 𝐷 ∈ (Met‘𝑋))) |
13 | 12 | adantr 483 | . . . 4 ⊢ ((𝑇 ∈ NrmGrp ∧ (norm‘𝑇) = (norm‘𝐺)) → (𝑁:𝑋⟶ℝ → 𝐷 ∈ (Met‘𝑋))) |
14 | 7, 13 | syl 17 | . . 3 ⊢ (𝐺 ∈ NrmGrp → (𝑁:𝑋⟶ℝ → 𝐷 ∈ (Met‘𝑋))) |
15 | metf 22942 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) | |
16 | 14, 15 | syl6 35 | . 2 ⊢ (𝐺 ∈ NrmGrp → (𝑁:𝑋⟶ℝ → 𝐷:(𝑋 × 𝑋)⟶ℝ)) |
17 | 3, 16 | mpd 15 | 1 ⊢ (𝐺 ∈ NrmGrp → 𝐷:(𝑋 × 𝑋)⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 × cxp 5555 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 Basecbs 16485 distcds 16576 Grpcgrp 18105 Metcmet 20533 normcnm 23188 NrmGrpcngp 23189 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-rep 5192 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 ax-pre-sup 10617 |
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-rmo 3148 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-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 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-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-tset 16586 df-ds 16589 df-rest 16698 df-topn 16699 df-0g 16717 df-topgen 16719 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-xms 22932 df-ms 22933 df-nm 23194 df-ngp 23195 df-tng 23196 |
This theorem is referenced by: (None) |
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