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Theorem isngp 23968
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 3931 . . 3 (𝐺 ∈ (Grp ∩ MetSp) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp))
21anbi1i 625 . 2 ((𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ βˆ’ ) βŠ† 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ βˆ’ ) βŠ† 𝐷))
3 fveq2 6847 . . . . . 6 (𝑔 = 𝐺 β†’ (normβ€˜π‘”) = (normβ€˜πΊ))
4 isngp.n . . . . . 6 𝑁 = (normβ€˜πΊ)
53, 4eqtr4di 2795 . . . . 5 (𝑔 = 𝐺 β†’ (normβ€˜π‘”) = 𝑁)
6 fveq2 6847 . . . . . 6 (𝑔 = 𝐺 β†’ (-gβ€˜π‘”) = (-gβ€˜πΊ))
7 isngp.z . . . . . 6 βˆ’ = (-gβ€˜πΊ)
86, 7eqtr4di 2795 . . . . 5 (𝑔 = 𝐺 β†’ (-gβ€˜π‘”) = βˆ’ )
95, 8coeq12d 5825 . . . 4 (𝑔 = 𝐺 β†’ ((normβ€˜π‘”) ∘ (-gβ€˜π‘”)) = (𝑁 ∘ βˆ’ ))
10 fveq2 6847 . . . . 5 (𝑔 = 𝐺 β†’ (distβ€˜π‘”) = (distβ€˜πΊ))
11 isngp.d . . . . 5 𝐷 = (distβ€˜πΊ)
1210, 11eqtr4di 2795 . . . 4 (𝑔 = 𝐺 β†’ (distβ€˜π‘”) = 𝐷)
139, 12sseq12d 3982 . . 3 (𝑔 = 𝐺 β†’ (((normβ€˜π‘”) ∘ (-gβ€˜π‘”)) βŠ† (distβ€˜π‘”) ↔ (𝑁 ∘ βˆ’ ) βŠ† 𝐷))
14 df-ngp 23955 . . 3 NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((normβ€˜π‘”) ∘ (-gβ€˜π‘”)) βŠ† (distβ€˜π‘”)}
1513, 14elrab2 3653 . 2 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ βˆ’ ) βŠ† 𝐷))
16 df-3an 1090 . 2 ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ βˆ’ ) βŠ† 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ βˆ’ ) βŠ† 𝐷))
172, 15, 163bitr4i 303 1 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ βˆ’ ) βŠ† 𝐷))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ∩ cin 3914   βŠ† wss 3915   ∘ ccom 5642  β€˜cfv 6501  distcds 17149  Grpcgrp 18755  -gcsg 18757  MetSpcms 23687  normcnm 23948  NrmGrpcngp 23949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-co 5647  df-iota 6453  df-fv 6509  df-ngp 23955
This theorem is referenced by:  isngp2  23969  ngpgrp  23971  ngpms  23972  tngngp2  24032  cnngp  24159  zhmnrg  32588
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