MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ngppropd Structured version   Unicode version

Theorem ngppropd 21576
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 21420 . . . . . . 7  |-  ( ph  ->  ( K  e.  MetSp  <->  L  e.  MetSp ) )
65adantr 466 . . . . . 6  |-  ( (
ph  /\  K  e.  Grp )  ->  ( K  e.  MetSp 
<->  L  e.  MetSp ) )
71adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Grp )  ->  B  =  ( Base `  K
) )
82adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Grp )  ->  B  =  ( Base `  L
) )
9 simpr 462 . . . . . . . . 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 719 . . . . . . . . 9  |-  ( ( ( ph  /\  K  e.  Grp )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  K ) y )  =  ( x ( +g  `  L
) y ) )
123adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Grp )  ->  ( (
dist `  K )  |`  ( B  X.  B
) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
137, 8, 9, 11, 12nmpropd2 21540 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Grp )  ->  ( norm `  K )  =  (
norm `  L )
)
147, 8, 9, 11grpsubpropd2 16708 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Grp )  ->  ( -g `  K )  =  (
-g `  L )
)
1513, 14coeq12d 5019 . . . . . . 7  |-  ( (
ph  /\  K  e.  Grp )  ->  ( (
norm `  K )  o.  ( -g `  K
) )  =  ( ( norm `  L
)  o.  ( -g `  L ) ) )
161sqxpeqd 4880 . . . . . . . . . 10  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
1716reseq2d 5125 . . . . . . . . 9  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
182sqxpeqd 4880 . . . . . . . . . 10  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
1918reseq2d 5125 . . . . . . . . 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 466 . . . . . . 7  |-  ( (
ph  /\  K  e.  Grp )  ->  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) )
2215, 21eqeq12d 2451 . . . . . 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 715 . . . . 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 645 . . . 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 16635 . . . . 5  |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
2625anbi1d 709 . . . 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 256 . . 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 986 . . 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 986 . . 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 291 . 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 2429 . . 3  |-  ( norm `  K )  =  (
norm `  K )
32 eqid 2429 . . 3  |-  ( -g `  K )  =  (
-g `  K )
33 eqid 2429 . . 3  |-  ( dist `  K )  =  (
dist `  K )
34 eqid 2429 . . 3  |-  ( Base `  K )  =  (
Base `  K )
35 eqid 2429 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
3631, 32, 33, 34, 35isngp2 21542 . 2  |-  ( K  e. NrmGrp 
<->  ( K  e.  Grp  /\  K  e.  MetSp  /\  (
( norm `  K )  o.  ( -g `  K
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) ) )
37 eqid 2429 . . 3  |-  ( norm `  L )  =  (
norm `  L )
38 eqid 2429 . . 3  |-  ( -g `  L )  =  (
-g `  L )
39 eqid 2429 . . 3  |-  ( dist `  L )  =  (
dist `  L )
40 eqid 2429 . . 3  |-  ( Base `  L )  =  (
Base `  L )
41 eqid 2429 . . 3  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
4237, 38, 39, 40, 41isngp2 21542 . 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 291 1  |-  ( ph  ->  ( K  e. NrmGrp  <->  L  e. NrmGrp ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    X. cxp 4852    |` cres 4856    o. ccom 4858   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15152   distcds 15161   TopOpenctopn 15279   Grpcgrp 16620   -gcsg 16622   MetSpcmt 21264   normcnm 21522  NrmGrpcngp 21523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-0g 15299  df-topgen 15301  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-sbg 16626  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-xms 21266  df-ms 21267  df-nm 21528  df-ngp 21529
This theorem is referenced by:  sranlm  21618
  Copyright terms: Public domain W3C validator