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Mirrors > Home > MPE Home > Th. List > nghmrcl1 | Structured version Visualization version GIF version |
Description: Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
Ref | Expression |
---|---|
nghmrcl1 | ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . 4 ⊢ (𝑆 normOp 𝑇) = (𝑆 normOp 𝑇) | |
2 | 1 | isnghm 22852 | . . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ((𝑆 normOp 𝑇)‘𝐹) ∈ ℝ))) |
3 | 2 | simplbi 492 | . 2 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp)) |
4 | 3 | simpld 489 | 1 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 ‘cfv 6100 (class class class)co 6877 ℝcr 10222 GrpHom cghm 17967 NrmGrpcngp 22707 normOp cnmo 22834 NGHom cnghm 22835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 ax-pre-sup 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-op 4374 df-uni 4628 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-id 5219 df-po 5232 df-so 5233 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-1st 7400 df-2nd 7401 df-er 7981 df-en 8195 df-dom 8196 df-sdom 8197 df-sup 8589 df-inf 8590 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-ico 12427 df-nmo 22837 df-nghm 22838 |
This theorem is referenced by: nmoco 22866 nghmco 22867 nmotri 22868 nghmplusg 22869 nmods 22873 nghmcn 22874 |
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