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Theorem ngppropd 20914
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 20740 . . . . . . 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 20878 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Grp )  ->  ( norm `  K )  =  (
norm `  L )
)
147, 8, 9, 11grpsubpropd2 15951 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Grp )  ->  ( -g `  K )  =  (
-g `  L )
)
1513, 14coeq12d 5167 . . . . . . 7  |-  ( (
ph  /\  K  e.  Grp )  ->  ( (
norm `  K )  o.  ( -g `  K
) )  =  ( ( norm `  L
)  o.  ( -g `  L ) ) )
161, 1xpeq12d 5024 . . . . . . . . . 10  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
1716reseq2d 5273 . . . . . . . . 9  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
182, 2xpeq12d 5024 . . . . . . . . . 10  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
1918reseq2d 5273 . . . . . . . . 9  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
203, 17, 193eqtr3d 2516 . . . . . . . 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 2489 . . . . . 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 15878 . . . . 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 977 . . 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 977 . . 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 2467 . . 3  |-  ( norm `  K )  =  (
norm `  K )
32 eqid 2467 . . 3  |-  ( -g `  K )  =  (
-g `  K )
33 eqid 2467 . . 3  |-  ( dist `  K )  =  (
dist `  K )
34 eqid 2467 . . 3  |-  ( Base `  K )  =  (
Base `  K )
35 eqid 2467 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
3631, 32, 33, 34, 35isngp2 20880 . 2  |-  ( K  e. NrmGrp 
<->  ( K  e.  Grp  /\  K  e.  MetSp  /\  (
( norm `  K )  o.  ( -g `  K
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) ) )
37 eqid 2467 . . 3  |-  ( norm `  L )  =  (
norm `  L )
38 eqid 2467 . . 3  |-  ( -g `  L )  =  (
-g `  L )
39 eqid 2467 . . 3  |-  ( dist `  L )  =  (
dist `  L )
40 eqid 2467 . . 3  |-  ( Base `  L )  =  (
Base `  L )
41 eqid 2467 . . 3  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
4237, 38, 39, 40, 41isngp2 20880 . 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 973    = wceq 1379    e. wcel 1767    X. cxp 4997    |` cres 5001    o. ccom 5003   ` cfv 5588  (class class class)co 6284   Basecbs 14490   +g cplusg 14555   distcds 14564   TopOpenctopn 14677   Grpcgrp 15727   -gcsg 15730   MetSpcmt 20584   normcnm 20860  NrmGrpcngp 20861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-0g 14697  df-topgen 14699  df-mnd 15732  df-grp 15867  df-minusg 15868  df-sbg 15869  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-xms 20586  df-ms 20587  df-nm 20866  df-ngp 20867
This theorem is referenced by:  sranlm  20956
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