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Theorem ngppropd 18153
Description: Property deduction for a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
ngppropd.1  |-  ( ph  ->  B  =  ( Base `  K ) )
ngppropd.2  |-  ( ph  ->  B  =  ( Base `  L ) )
ngppropd.3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
ngppropd.4  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
ngppropd.5  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
Assertion
Ref Expression
ngppropd  |-  ( ph  ->  ( K  e. NrmGrp  <->  L  e. NrmGrp ) )
Distinct variable groups:    x, y, B    x, K, y    x, L, y    ph, x, y

Proof of Theorem ngppropd
StepHypRef Expression
1 ngppropd.1 . . . . . . . 8  |-  ( ph  ->  B  =  ( Base `  K ) )
2 ngppropd.2 . . . . . . . 8  |-  ( ph  ->  B  =  ( Base `  L ) )
3 ngppropd.4 . . . . . . . 8  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
4 ngppropd.5 . . . . . . . 8  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
51, 2, 3, 4mspropd 18020 . . . . . . 7  |-  ( ph  ->  ( K  e.  MetSp  <->  L  e.  MetSp ) )
65adantr 451 . . . . . 6  |-  ( (
ph  /\  K  e.  Grp )  ->  ( K  e.  MetSp 
<->  L  e.  MetSp ) )
71adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Grp )  ->  B  =  ( Base `  K
) )
82adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Grp )  ->  B  =  ( Base `  L
) )
9 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Grp )  ->  K  e. 
Grp )
10 ngppropd.3 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
1110adantlr 695 . . . . . . . . 9  |-  ( ( ( ph  /\  K  e.  Grp )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  K ) y )  =  ( x ( +g  `  L
) y ) )
123adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Grp )  ->  ( (
dist `  K )  |`  ( B  X.  B
) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
137, 8, 9, 11, 12nmpropd2 18117 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Grp )  ->  ( norm `  K )  =  (
norm `  L )
)
147, 8, 9, 11grpsubpropd2 14567 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Grp )  ->  ( -g `  K )  =  (
-g `  L )
)
1513, 14coeq12d 4848 . . . . . . 7  |-  ( (
ph  /\  K  e.  Grp )  ->  ( (
norm `  K )  o.  ( -g `  K
) )  =  ( ( norm `  L
)  o.  ( -g `  L ) ) )
161, 1xpeq12d 4714 . . . . . . . . . 10  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
1716reseq2d 4955 . . . . . . . . 9  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
182, 2xpeq12d 4714 . . . . . . . . . 10  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
1918reseq2d 4955 . . . . . . . . 9  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
203, 17, 193eqtr3d 2323 . . . . . . . 8  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
2120adantr 451 . . . . . . 7  |-  ( (
ph  /\  K  e.  Grp )  ->  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) )
2215, 21eqeq12d 2297 . . . . . 6  |-  ( (
ph  /\  K  e.  Grp )  ->  ( ( ( norm `  K
)  o.  ( -g `  K ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  <->  ( ( norm `  L )  o.  ( -g `  L ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ) )
236, 22anbi12d 691 . . . . 5  |-  ( (
ph  /\  K  e.  Grp )  ->  ( ( K  e.  MetSp  /\  (
( norm `  K )  o.  ( -g `  K
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )  <->  ( L  e.  MetSp  /\  ( ( norm `  L )  o.  ( -g `  L
) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) ) ) )
2423pm5.32da 622 . . . 4  |-  ( ph  ->  ( ( K  e. 
Grp  /\  ( K  e.  MetSp  /\  ( ( norm `  K )  o.  ( -g `  K
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) ) )  <->  ( K  e.  Grp  /\  ( L  e.  MetSp  /\  ( ( norm `  L )  o.  ( -g `  L
) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) ) ) ) )
251, 2, 10grppropd 14500 . . . . 5  |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
2625anbi1d 685 . . . 4  |-  ( ph  ->  ( ( K  e. 
Grp  /\  ( L  e.  MetSp  /\  ( ( norm `  L )  o.  ( -g `  L
) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) ) )  <->  ( L  e.  Grp  /\  ( L  e.  MetSp  /\  ( ( norm `  L )  o.  ( -g `  L
) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) ) ) ) )
2724, 26bitrd 244 . . 3  |-  ( ph  ->  ( ( K  e. 
Grp  /\  ( K  e.  MetSp  /\  ( ( norm `  K )  o.  ( -g `  K
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) ) )  <->  ( L  e.  Grp  /\  ( L  e.  MetSp  /\  ( ( norm `  L )  o.  ( -g `  L
) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) ) ) ) )
28 3anass 938 . . 3  |-  ( ( K  e.  Grp  /\  K  e.  MetSp  /\  (
( norm `  K )  o.  ( -g `  K
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )  <->  ( K  e.  Grp  /\  ( K  e.  MetSp  /\  ( ( norm `  K )  o.  ( -g `  K
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) ) ) )
29 3anass 938 . . 3  |-  ( ( L  e.  Grp  /\  L  e.  MetSp  /\  (
( norm `  L )  o.  ( -g `  L
) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )  <->  ( L  e.  Grp  /\  ( L  e.  MetSp  /\  ( ( norm `  L )  o.  ( -g `  L
) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) ) ) )
3027, 28, 293bitr4g 279 . 2  |-  ( ph  ->  ( ( K  e. 
Grp  /\  K  e.  MetSp  /\  ( ( norm `  K
)  o.  ( -g `  K ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )  <->  ( L  e.  Grp  /\  L  e. 
MetSp  /\  ( ( norm `  L )  o.  ( -g `  L ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ) ) )
31 eqid 2283 . . 3  |-  ( norm `  K )  =  (
norm `  K )
32 eqid 2283 . . 3  |-  ( -g `  K )  =  (
-g `  K )
33 eqid 2283 . . 3  |-  ( dist `  K )  =  (
dist `  K )
34 eqid 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
35 eqid 2283 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
3631, 32, 33, 34, 35isngp2 18119 . 2  |-  ( K  e. NrmGrp 
<->  ( K  e.  Grp  /\  K  e.  MetSp  /\  (
( norm `  K )  o.  ( -g `  K
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) ) )
37 eqid 2283 . . 3  |-  ( norm `  L )  =  (
norm `  L )
38 eqid 2283 . . 3  |-  ( -g `  L )  =  (
-g `  L )
39 eqid 2283 . . 3  |-  ( dist `  L )  =  (
dist `  L )
40 eqid 2283 . . 3  |-  ( Base `  L )  =  (
Base `  L )
41 eqid 2283 . . 3  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
4237, 38, 39, 40, 41isngp2 18119 . 2  |-  ( L  e. NrmGrp 
<->  ( L  e.  Grp  /\  L  e.  MetSp  /\  (
( norm `  L )  o.  ( -g `  L
) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) ) )
4330, 36, 423bitr4g 279 1  |-  ( ph  ->  ( K  e. NrmGrp  <->  L  e. NrmGrp ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    X. cxp 4687    |` cres 4691    o. ccom 4693   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   distcds 13217   TopOpenctopn 13326   Grpcgrp 14362   -gcsg 14365   MetSpcmt 17883   normcnm 18099  NrmGrpcngp 18100
This theorem is referenced by:  sranlm  18195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-topgen 13344  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-xms 17885  df-ms 17886  df-nm 18105  df-ngp 18106
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