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Theorem isngp 18643
 Description: The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
isngp.n
isngp.z
isngp.d
Assertion
Ref Expression
isngp NrmGrp

Proof of Theorem isngp
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elin 3530 . . 3
21anbi1i 677 . 2
3 fveq2 5728 . . . . . 6
4 isngp.n . . . . . 6
53, 4syl6eqr 2486 . . . . 5
6 fveq2 5728 . . . . . 6
7 isngp.z . . . . . 6
86, 7syl6eqr 2486 . . . . 5
95, 8coeq12d 5037 . . . 4
10 fveq2 5728 . . . . 5
11 isngp.d . . . . 5
1210, 11syl6eqr 2486 . . . 4
139, 12sseq12d 3377 . . 3
14 df-ngp 18631 . . 3 NrmGrp
1513, 14elrab2 3094 . 2 NrmGrp
16 df-3an 938 . 2
172, 15, 163bitr4i 269 1 NrmGrp
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725   cin 3319   wss 3320   ccom 4882  cfv 5454  cds 13538  cgrp 14685  csg 14688  cmt 18348  cnm 18624  NrmGrpcngp 18625 This theorem is referenced by:  isngp2  18644  ngpgrp  18646  ngpms  18647  tngngp2  18693  cnngp  18814  zhmnrg  24351 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-co 4887  df-iota 5418  df-fv 5462  df-ngp 18631
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