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Theorem isngp 24096
Description: The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
isngp.n 𝑁 = (normβ€˜πΊ)
isngp.z βˆ’ = (-gβ€˜πΊ)
isngp.d 𝐷 = (distβ€˜πΊ)
Assertion
Ref Expression
isngp (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ βˆ’ ) βŠ† 𝐷))
Proof of Theorem isngp
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 elin 3963 . . 3 (𝐺 ∈ (Grp ∩ MetSp) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp))
21anbi1i 624 . 2 ((𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ βˆ’ ) βŠ† 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ βˆ’ ) βŠ† 𝐷))
3 fveq2 6888 . . . . . 6 (𝑔 = 𝐺 β†’ (normβ€˜π‘”) = (normβ€˜πΊ))
4 isngp.n . . . . . 6 𝑁 = (normβ€˜πΊ)
53, 4eqtr4di 2790 . . . . 5 (𝑔 = 𝐺 β†’ (normβ€˜π‘”) = 𝑁)
6 fveq2 6888 . . . . . 6 (𝑔 = 𝐺 β†’ (-gβ€˜π‘”) = (-gβ€˜πΊ))
7 isngp.z . . . . . 6 βˆ’ = (-gβ€˜πΊ)
86, 7eqtr4di 2790 . . . . 5 (𝑔 = 𝐺 β†’ (-gβ€˜π‘”) = βˆ’ )
95, 8coeq12d 5862 . . . 4 (𝑔 = 𝐺 β†’ ((normβ€˜π‘”) ∘ (-gβ€˜π‘”)) = (𝑁 ∘ βˆ’ ))
10 fveq2 6888 . . . . 5 (𝑔 = 𝐺 β†’ (distβ€˜π‘”) = (distβ€˜πΊ))
11 isngp.d . . . . 5 𝐷 = (distβ€˜πΊ)
1210, 11eqtr4di 2790 . . . 4 (𝑔 = 𝐺 β†’ (distβ€˜π‘”) = 𝐷)
139, 12sseq12d 4014 . . 3 (𝑔 = 𝐺 β†’ (((normβ€˜π‘”) ∘ (-gβ€˜π‘”)) βŠ† (distβ€˜π‘”) ↔ (𝑁 ∘ βˆ’ ) βŠ† 𝐷))
14 df-ngp 24083 . . 3 NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((normβ€˜π‘”) ∘ (-gβ€˜π‘”)) βŠ† (distβ€˜π‘”)}
1513, 14elrab2 3685 . 2 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ βˆ’ ) βŠ† 𝐷))
16 df-3an 1089 . 2 ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ βˆ’ ) βŠ† 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ βˆ’ ) βŠ† 𝐷))
172, 15, 163bitr4i 302 1 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ βˆ’ ) βŠ† 𝐷))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∩ cin 3946   βŠ† wss 3947   ∘ ccom 5679  β€˜cfv 6540  distcds 17202  Grpcgrp 18815  -gcsg 18817  MetSpcms 23815  normcnm 24076  NrmGrpcngp 24077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-co 5684  df-iota 6492  df-fv 6548  df-ngp 24083
This theorem is referenced by:  isngp2  24097  ngpgrp  24099  ngpms  24100  tngngp2  24160  cnngp  24287  zhmnrg  32935
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