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Theorem ogrpaddltrbid 27689
Description: In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 4-Sep-2018.)
Hypotheses
Ref Expression
ogrpaddlt.0  |-  B  =  ( Base `  G
)
ogrpaddlt.1  |-  .<  =  ( lt `  G )
ogrpaddlt.2  |-  .+  =  ( +g  `  G )
ogrpaddltrd.1  |-  ( ph  ->  G  e.  V )
ogrpaddltrd.2  |-  ( ph  ->  (oppg
`  G )  e. oGrp
)
ogrpaddltrd.3  |-  ( ph  ->  X  e.  B )
ogrpaddltrd.4  |-  ( ph  ->  Y  e.  B )
ogrpaddltrd.5  |-  ( ph  ->  Z  e.  B )
Assertion
Ref Expression
ogrpaddltrbid  |-  ( ph  ->  ( X  .<  Y  <->  ( Z  .+  X )  .<  ( Z  .+  Y ) ) )

Proof of Theorem ogrpaddltrbid
StepHypRef Expression
1 ogrpaddlt.0 . . 3  |-  B  =  ( Base `  G
)
2 ogrpaddlt.1 . . 3  |-  .<  =  ( lt `  G )
3 ogrpaddlt.2 . . 3  |-  .+  =  ( +g  `  G )
4 ogrpaddltrd.1 . . . 4  |-  ( ph  ->  G  e.  V )
54adantr 465 . . 3  |-  ( (
ph  /\  X  .<  Y )  ->  G  e.  V )
6 ogrpaddltrd.2 . . . 4  |-  ( ph  ->  (oppg
`  G )  e. oGrp
)
76adantr 465 . . 3  |-  ( (
ph  /\  X  .<  Y )  ->  (oppg
`  G )  e. oGrp
)
8 ogrpaddltrd.3 . . . 4  |-  ( ph  ->  X  e.  B )
98adantr 465 . . 3  |-  ( (
ph  /\  X  .<  Y )  ->  X  e.  B )
10 ogrpaddltrd.4 . . . 4  |-  ( ph  ->  Y  e.  B )
1110adantr 465 . . 3  |-  ( (
ph  /\  X  .<  Y )  ->  Y  e.  B )
12 ogrpaddltrd.5 . . . 4  |-  ( ph  ->  Z  e.  B )
1312adantr 465 . . 3  |-  ( (
ph  /\  X  .<  Y )  ->  Z  e.  B )
14 simpr 461 . . 3  |-  ( (
ph  /\  X  .<  Y )  ->  X  .<  Y )
151, 2, 3, 5, 7, 9, 11, 13, 14ogrpaddltrd 27688 . 2  |-  ( (
ph  /\  X  .<  Y )  ->  ( Z  .+  X )  .<  ( Z  .+  Y ) )
164adantr 465 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  G  e.  V
)
176adantr 465 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  (oppg
`  G )  e. oGrp
)
18 ogrpgrp 27671 . . . . . . 7  |-  ( (oppg `  G )  e. oGrp  ->  (oppg `  G )  e.  Grp )
196, 18syl 16 . . . . . 6  |-  ( ph  ->  (oppg
`  G )  e. 
Grp )
2019adantr 465 . . . . 5  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  (oppg
`  G )  e. 
Grp )
218adantr 465 . . . . 5  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  X  e.  B
)
2212adantr 465 . . . . 5  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  Z  e.  B
)
23 eqid 2443 . . . . . . 7  |-  (oppg `  G
)  =  (oppg `  G
)
24 eqid 2443 . . . . . . 7  |-  ( +g  `  (oppg
`  G ) )  =  ( +g  `  (oppg `  G
) )
253, 23, 24oppgplus 16363 . . . . . 6  |-  ( X ( +g  `  (oppg `  G
) ) Z )  =  ( Z  .+  X )
2623, 1oppgbas 16365 . . . . . . 7  |-  B  =  ( Base `  (oppg `  G
) )
2726, 24grpcl 16042 . . . . . 6  |-  ( ( (oppg
`  G )  e. 
Grp  /\  X  e.  B  /\  Z  e.  B
)  ->  ( X
( +g  `  (oppg `  G
) ) Z )  e.  B )
2825, 27syl5eqelr 2536 . . . . 5  |-  ( ( (oppg
`  G )  e. 
Grp  /\  X  e.  B  /\  Z  e.  B
)  ->  ( Z  .+  X )  e.  B
)
2920, 21, 22, 28syl3anc 1229 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( Z  .+  X )  e.  B
)
3010adantr 465 . . . . 5  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  Y  e.  B
)
313, 23, 24oppgplus 16363 . . . . . 6  |-  ( Y ( +g  `  (oppg `  G
) ) Z )  =  ( Z  .+  Y )
3226, 24grpcl 16042 . . . . . 6  |-  ( ( (oppg
`  G )  e. 
Grp  /\  Y  e.  B  /\  Z  e.  B
)  ->  ( Y
( +g  `  (oppg `  G
) ) Z )  e.  B )
3331, 32syl5eqelr 2536 . . . . 5  |-  ( ( (oppg
`  G )  e. 
Grp  /\  Y  e.  B  /\  Z  e.  B
)  ->  ( Z  .+  Y )  e.  B
)
3420, 30, 22, 33syl3anc 1229 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( Z  .+  Y )  e.  B
)
3523oppggrpb 16372 . . . . . 6  |-  ( G  e.  Grp  <->  (oppg
`  G )  e. 
Grp )
3620, 35sylibr 212 . . . . 5  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  G  e.  Grp )
37 eqid 2443 . . . . . 6  |-  ( invg `  G )  =  ( invg `  G )
381, 37grpinvcl 16074 . . . . 5  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( invg `  G ) `  Z
)  e.  B )
3936, 22, 38syl2anc 661 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( invg `  G ) `
 Z )  e.  B )
40 simpr 461 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( Z  .+  X )  .<  ( Z  .+  Y ) )
411, 2, 3, 16, 17, 29, 34, 39, 40ogrpaddltrd 27688 . . 3  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( invg `  G
) `  Z )  .+  ( Z  .+  X
) )  .<  (
( ( invg `  G ) `  Z
)  .+  ( Z  .+  Y ) ) )
42 eqid 2443 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
431, 3, 42, 37grplinv 16075 . . . . . 6  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( ( invg `  G ) `
 Z )  .+  Z )  =  ( 0g `  G ) )
4436, 22, 43syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( invg `  G
) `  Z )  .+  Z )  =  ( 0g `  G ) )
4544oveq1d 6296 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( ( invg `  G ) `  Z
)  .+  Z )  .+  X )  =  ( ( 0g `  G
)  .+  X )
)
461, 3grpass 16043 . . . . 5  |-  ( ( G  e.  Grp  /\  ( ( ( invg `  G ) `
 Z )  e.  B  /\  Z  e.  B  /\  X  e.  B ) )  -> 
( ( ( ( invg `  G
) `  Z )  .+  Z )  .+  X
)  =  ( ( ( invg `  G ) `  Z
)  .+  ( Z  .+  X ) ) )
4736, 39, 22, 21, 46syl13anc 1231 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( ( invg `  G ) `  Z
)  .+  Z )  .+  X )  =  ( ( ( invg `  G ) `  Z
)  .+  ( Z  .+  X ) ) )
481, 3, 42grplid 16059 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( 0g `  G )  .+  X
)  =  X )
4936, 21, 48syl2anc 661 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( 0g
`  G )  .+  X )  =  X )
5045, 47, 493eqtr3d 2492 . . 3  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( invg `  G
) `  Z )  .+  ( Z  .+  X
) )  =  X )
5144oveq1d 6296 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( ( invg `  G ) `  Z
)  .+  Z )  .+  Y )  =  ( ( 0g `  G
)  .+  Y )
)
521, 3grpass 16043 . . . . 5  |-  ( ( G  e.  Grp  /\  ( ( ( invg `  G ) `
 Z )  e.  B  /\  Z  e.  B  /\  Y  e.  B ) )  -> 
( ( ( ( invg `  G
) `  Z )  .+  Z )  .+  Y
)  =  ( ( ( invg `  G ) `  Z
)  .+  ( Z  .+  Y ) ) )
5336, 39, 22, 30, 52syl13anc 1231 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( ( invg `  G ) `  Z
)  .+  Z )  .+  Y )  =  ( ( ( invg `  G ) `  Z
)  .+  ( Z  .+  Y ) ) )
541, 3, 42grplid 16059 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( 0g `  G )  .+  Y
)  =  Y )
5536, 30, 54syl2anc 661 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( 0g
`  G )  .+  Y )  =  Y )
5651, 53, 553eqtr3d 2492 . . 3  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( invg `  G
) `  Z )  .+  ( Z  .+  Y
) )  =  Y )
5741, 50, 563brtr3d 4466 . 2  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  X  .<  Y )
5815, 57impbida 832 1  |-  ( ph  ->  ( X  .<  Y  <->  ( Z  .+  X )  .<  ( Z  .+  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Basecbs 14614   +g cplusg 14679   0gc0g 14819   ltcplt 15549   Grpcgrp 16032   invgcminusg 16033  oppgcoppg 16359  oGrpcogrp 27666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-tpos 6957  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-plusg 14692  df-ple 14699  df-0g 14821  df-plt 15567  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-grp 16036  df-minusg 16037  df-oppg 16360  df-omnd 27667  df-ogrp 27668
This theorem is referenced by:  ogrpinvlt  27692
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