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Theorem ngppropd 18680
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 18506 . . . . . . 7  |-  ( ph  ->  ( K  e.  MetSp  <->  L  e.  MetSp ) )
65adantr 453 . . . . . 6  |-  ( (
ph  /\  K  e.  Grp )  ->  ( K  e.  MetSp 
<->  L  e.  MetSp ) )
71adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Grp )  ->  B  =  ( Base `  K
) )
82adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Grp )  ->  B  =  ( Base `  L
) )
9 simpr 449 . . . . . . . . 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 697 . . . . . . . . 9  |-  ( ( ( ph  /\  K  e.  Grp )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  K ) y )  =  ( x ( +g  `  L
) y ) )
123adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Grp )  ->  ( (
dist `  K )  |`  ( B  X.  B
) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
137, 8, 9, 11, 12nmpropd2 18644 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Grp )  ->  ( norm `  K )  =  (
norm `  L )
)
147, 8, 9, 11grpsubpropd2 14892 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Grp )  ->  ( -g `  K )  =  (
-g `  L )
)
1513, 14coeq12d 5039 . . . . . . 7  |-  ( (
ph  /\  K  e.  Grp )  ->  ( (
norm `  K )  o.  ( -g `  K
) )  =  ( ( norm `  L
)  o.  ( -g `  L ) ) )
161, 1xpeq12d 4905 . . . . . . . . . 10  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
1716reseq2d 5148 . . . . . . . . 9  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
182, 2xpeq12d 4905 . . . . . . . . . 10  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
1918reseq2d 5148 . . . . . . . . 9  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
203, 17, 193eqtr3d 2478 . . . . . . . 8  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
2120adantr 453 . . . . . . 7  |-  ( (
ph  /\  K  e.  Grp )  ->  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) )
2215, 21eqeq12d 2452 . . . . . 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 693 . . . . 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 624 . . . 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 14825 . . . . 5  |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
2625anbi1d 687 . . . 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 246 . . 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 941 . . 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 941 . . 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 281 . 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 2438 . . 3  |-  ( norm `  K )  =  (
norm `  K )
32 eqid 2438 . . 3  |-  ( -g `  K )  =  (
-g `  K )
33 eqid 2438 . . 3  |-  ( dist `  K )  =  (
dist `  K )
34 eqid 2438 . . 3  |-  ( Base `  K )  =  (
Base `  K )
35 eqid 2438 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
3631, 32, 33, 34, 35isngp2 18646 . 2  |-  ( K  e. NrmGrp 
<->  ( K  e.  Grp  /\  K  e.  MetSp  /\  (
( norm `  K )  o.  ( -g `  K
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) ) )
37 eqid 2438 . . 3  |-  ( norm `  L )  =  (
norm `  L )
38 eqid 2438 . . 3  |-  ( -g `  L )  =  (
-g `  L )
39 eqid 2438 . . 3  |-  ( dist `  L )  =  (
dist `  L )
40 eqid 2438 . . 3  |-  ( Base `  L )  =  (
Base `  L )
41 eqid 2438 . . 3  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
4237, 38, 39, 40, 41isngp2 18646 . 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 281 1  |-  ( ph  ->  ( K  e. NrmGrp  <->  L  e. NrmGrp ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    X. cxp 4878    |` cres 4882    o. ccom 4884   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   distcds 13540   TopOpenctopn 13651   Grpcgrp 14687   -gcsg 14690   MetSpcmt 18350   normcnm 18626  NrmGrpcngp 18627
This theorem is referenced by:  sranlm  18722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-topgen 13669  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-sbg 14816  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-xms 18352  df-ms 18353  df-nm 18632  df-ngp 18633
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