Theorem ngppropd 22441

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

Step	Hyp	Ref	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 ) )
5	1, 2, 3, 4	mspropd 22279	. . . . . . 7 |- ( ph -> ( K e. MetSp <-> L e. MetSp ) )
6	5	adantr 481	. . . . . 6 |- ( ( ph /\ K e. Grp ) -> ( K e. MetSp <-> L e. MetSp ) )
7	1	adantr 481	. . . . . . . . 9 |- ( ( ph /\ K e. Grp ) -> B = ( Base ` K ) )
8	2	adantr 481	. . . . . . . . 9 |- ( ( ph /\ K e. Grp ) -> B = ( Base ` L ) )
9		simpr 477	. . . . . . . . 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 ) )
11	10	adantlr 751	. . . . . . . . 9 |- ( ( ( ph /\ K e. Grp ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
12	3	adantr 481	. . . . . . . . 9 |- ( ( ph /\ K e. Grp ) -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( B X. B ) ) )
13	7, 8, 9, 11, 12	nmpropd2 22399	. . . . . . . 8 |- ( ( ph /\ K e. Grp ) -> ( norm ` K ) = ( norm ` L ) )
14	7, 8, 9, 11	grpsubpropd2 17521	. . . . . . . 8 |- ( ( ph /\ K e. Grp ) -> ( -g ` K ) = ( -g ` L ) )
15	13, 14	coeq12d 5286	. . . . . . 7 |- ( ( ph /\ K e. Grp ) -> ( ( norm ` K ) o. ( -g ` K ) ) = ( ( norm ` L ) o. ( -g ` L ) ) )
16	1	sqxpeqd 5141	. . . . . . . . . 10 |- ( ph -> ( B X. B ) = ( ( Base ` K ) X. ( Base ` K ) ) )
17	16	reseq2d 5396	. . . . . . . . 9 |- ( ph -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
18	2	sqxpeqd 5141	. . . . . . . . . 10 |- ( ph -> ( B X. B ) = ( ( Base ` L ) X. ( Base ` L ) ) )
19	18	reseq2d 5396	. . . . . . . . 9 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
20	3, 17, 19	3eqtr3d 2664	. . . . . . . 8 |- ( ph -> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
21	20	adantr 481	. . . . . . 7 |- ( ( ph /\ K e. Grp ) -> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
22	15, 21	eqeq12d 2637	. . . . . 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 ) ) ) ) )
23	6, 22	anbi12d 747	. . . . 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 ) ) ) ) ) )
24	23	pm5.32da 673	. . . 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 ) ) ) ) ) ) )
25	1, 2, 10	grppropd 17437	. . . . 5 |- ( ph -> ( K e. Grp <-> L e. Grp ) )
26	25	anbi1d 741	. . . 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 ) ) ) ) ) ) )
27	24, 26	bitrd 268	. . 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 1042	. . 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 1042	. . 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 ) ) ) ) ) )
30	27, 28, 29	3bitr4g 303	. 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 2622	. . 3 |- ( norm ` K ) = ( norm ` K )
32		eqid 2622	. . 3 |- ( -g ` K ) = ( -g ` K )
33		eqid 2622	. . 3 |- ( dist ` K ) = ( dist ` K )
34		eqid 2622	. . 3 |- ( Base ` K ) = ( Base ` K )
35		eqid 2622	. . 3 |- ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) )
36	31, 32, 33, 34, 35	isngp2 22401	. 2 |- ( K e. NrmGrp <-> ( K e. Grp /\ K e. MetSp /\ ( ( norm ` K ) o. ( -g ` K ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) )
37		eqid 2622	. . 3 |- ( norm ` L ) = ( norm ` L )
38		eqid 2622	. . 3 |- ( -g ` L ) = ( -g ` L )
39		eqid 2622	. . 3 |- ( dist ` L ) = ( dist ` L )
40		eqid 2622	. . 3 |- ( Base ` L ) = ( Base ` L )
41		eqid 2622	. . 3 |- ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) )
42	37, 38, 39, 40, 41	isngp2 22401	. 2 |- ( L e. NrmGrp <-> ( L e. Grp /\ L e. MetSp /\ ( ( norm ` L ) o. ( -g ` L ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) ) )
43	30, 36, 42	3bitr4g 303	1 |- ( ph -> ( K e. NrmGrp <-> L e. NrmGrp ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   X. cxp 5112   |` cres 5116   o. ccom 5118   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   distcds 15950   TopOpenctopn 16082   Grpcgrp 17422   -gcsg 17424   MetSpcmt 22123   normcnm 22381  NrmGrpcngp 22382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-0g 16102  df-topgen 16104  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-xms 22125  df-ms 22126  df-nm 22387  df-ngp 22388

This theorem is referenced by:  sranlm  22488