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Theorem ngppropd 20228
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 20054 . . . . . . 7  |-  ( ph  ->  ( K  e.  MetSp  <->  L  e.  MetSp ) )
65adantr 465 . . . . . 6  |-  ( (
ph  /\  K  e.  Grp )  ->  ( K  e.  MetSp 
<->  L  e.  MetSp ) )
71adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Grp )  ->  B  =  ( Base `  K
) )
82adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Grp )  ->  B  =  ( Base `  L
) )
9 simpr 461 . . . . . . . . 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 714 . . . . . . . . 9  |-  ( ( ( ph  /\  K  e.  Grp )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  K ) y )  =  ( x ( +g  `  L
) y ) )
123adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Grp )  ->  ( (
dist `  K )  |`  ( B  X.  B
) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
137, 8, 9, 11, 12nmpropd2 20192 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Grp )  ->  ( norm `  K )  =  (
norm `  L )
)
147, 8, 9, 11grpsubpropd2 15632 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Grp )  ->  ( -g `  K )  =  (
-g `  L )
)
1513, 14coeq12d 5009 . . . . . . 7  |-  ( (
ph  /\  K  e.  Grp )  ->  ( (
norm `  K )  o.  ( -g `  K
) )  =  ( ( norm `  L
)  o.  ( -g `  L ) ) )
161, 1xpeq12d 4870 . . . . . . . . . 10  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
1716reseq2d 5115 . . . . . . . . 9  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
182, 2xpeq12d 4870 . . . . . . . . . 10  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
1918reseq2d 5115 . . . . . . . . 9  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
203, 17, 193eqtr3d 2483 . . . . . . . 8  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
2120adantr 465 . . . . . . 7  |-  ( (
ph  /\  K  e.  Grp )  ->  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) )
2215, 21eqeq12d 2457 . . . . . 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 710 . . . . 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 641 . . . 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 15561 . . . . 5  |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
2625anbi1d 704 . . . 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 253 . . 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 969 . . 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 969 . . 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 288 . 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 2443 . . 3  |-  ( norm `  K )  =  (
norm `  K )
32 eqid 2443 . . 3  |-  ( -g `  K )  =  (
-g `  K )
33 eqid 2443 . . 3  |-  ( dist `  K )  =  (
dist `  K )
34 eqid 2443 . . 3  |-  ( Base `  K )  =  (
Base `  K )
35 eqid 2443 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
3631, 32, 33, 34, 35isngp2 20194 . 2  |-  ( K  e. NrmGrp 
<->  ( K  e.  Grp  /\  K  e.  MetSp  /\  (
( norm `  K )  o.  ( -g `  K
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) ) )
37 eqid 2443 . . 3  |-  ( norm `  L )  =  (
norm `  L )
38 eqid 2443 . . 3  |-  ( -g `  L )  =  (
-g `  L )
39 eqid 2443 . . 3  |-  ( dist `  L )  =  (
dist `  L )
40 eqid 2443 . . 3  |-  ( Base `  L )  =  (
Base `  L )
41 eqid 2443 . . 3  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
4237, 38, 39, 40, 41isngp2 20194 . 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 288 1  |-  ( ph  ->  ( K  e. NrmGrp  <->  L  e. NrmGrp ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    X. cxp 4843    |` cres 4847    o. ccom 4849   ` cfv 5423  (class class class)co 6096   Basecbs 14179   +g cplusg 14243   distcds 14252   TopOpenctopn 14365   Grpcgrp 15415   -gcsg 15418   MetSpcmt 19898   normcnm 20174  NrmGrpcngp 20175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-0g 14385  df-topgen 14387  df-mnd 15420  df-grp 15550  df-minusg 15551  df-sbg 15552  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-xms 19900  df-ms 19901  df-nm 20180  df-ngp 20181
This theorem is referenced by:  sranlm  20270
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