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Mirrors > Home > MPE Home > Th. List > tng0 | Structured version Visualization version GIF version |
Description: The group identity of a structure augmented with a norm. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
tngbas.t | ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
tng0.2 | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
tng0 | ⊢ (𝑁 ∈ 𝑉 → 0 = (0g‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tng0.2 | . 2 ⊢ 0 = (0g‘𝐺) | |
2 | eqidd 2826 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝐺)) | |
3 | tngbas.t | . . . 4 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
4 | eqid 2825 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | 3, 4 | tngbas 22822 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝑇)) |
6 | eqid 2825 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
7 | 3, 6 | tngplusg 22823 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (+g‘𝐺) = (+g‘𝑇)) |
8 | 7 | oveqdr 6938 | . . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑇)𝑦)) |
9 | 2, 5, 8 | grpidpropd 17621 | . 2 ⊢ (𝑁 ∈ 𝑉 → (0g‘𝐺) = (0g‘𝑇)) |
10 | 1, 9 | syl5eq 2873 | 1 ⊢ (𝑁 ∈ 𝑉 → 0 = (0g‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 +gcplusg 16312 0gc0g 16460 toNrmGrp ctng 22760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-plusg 16325 df-tset 16331 df-ds 16334 df-0g 16462 df-tng 22766 |
This theorem is referenced by: tngnm 22832 tngngp 22835 tngngp3 22837 rrx0 23572 |
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