Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ogrpaddltrbid Structured version   Unicode version

Theorem ogrpaddltrbid 27401
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 27400 . 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 27383 . . . . . . 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 2467 . . . . . . 7  |-  (oppg `  G
)  =  (oppg `  G
)
24 eqid 2467 . . . . . . 7  |-  ( +g  `  (oppg
`  G ) )  =  ( +g  `  (oppg `  G
) )
253, 23, 24oppgplus 16189 . . . . . 6  |-  ( X ( +g  `  (oppg `  G
) ) Z )  =  ( Z  .+  X )
2623, 1oppgbas 16191 . . . . . . 7  |-  B  =  ( Base `  (oppg `  G
) )
2726, 24grpcl 15873 . . . . . 6  |-  ( ( (oppg
`  G )  e. 
Grp  /\  X  e.  B  /\  Z  e.  B
)  ->  ( X
( +g  `  (oppg `  G
) ) Z )  e.  B )
2825, 27syl5eqelr 2560 . . . . 5  |-  ( ( (oppg
`  G )  e. 
Grp  /\  X  e.  B  /\  Z  e.  B
)  ->  ( Z  .+  X )  e.  B
)
2920, 21, 22, 28syl3anc 1228 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( Z  .+  X )  e.  B
)
3010adantr 465 . . . . 5  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  Y  e.  B
)
313, 23, 24oppgplus 16189 . . . . . 6  |-  ( Y ( +g  `  (oppg `  G
) ) Z )  =  ( Z  .+  Y )
3226, 24grpcl 15873 . . . . . 6  |-  ( ( (oppg
`  G )  e. 
Grp  /\  Y  e.  B  /\  Z  e.  B
)  ->  ( Y
( +g  `  (oppg `  G
) ) Z )  e.  B )
3331, 32syl5eqelr 2560 . . . . 5  |-  ( ( (oppg
`  G )  e. 
Grp  /\  Y  e.  B  /\  Z  e.  B
)  ->  ( Z  .+  Y )  e.  B
)
3420, 30, 22, 33syl3anc 1228 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( Z  .+  Y )  e.  B
)
3523oppggrpb 16198 . . . . . 6  |-  ( G  e.  Grp  <->  (oppg
`  G )  e. 
Grp )
3620, 35sylibr 212 . . . . 5  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  G  e.  Grp )
37 eqid 2467 . . . . . 6  |-  ( invg `  G )  =  ( invg `  G )
381, 37grpinvcl 15905 . . . . 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 27400 . . 3  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( invg `  G
) `  Z )  .+  ( Z  .+  X
) )  .<  (
( ( invg `  G ) `  Z
)  .+  ( Z  .+  Y ) ) )
42 eqid 2467 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
431, 3, 42, 37grplinv 15906 . . . . . 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 6299 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( ( invg `  G ) `  Z
)  .+  Z )  .+  X )  =  ( ( 0g `  G
)  .+  X )
)
461, 3grpass 15874 . . . . 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 1230 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( ( invg `  G ) `  Z
)  .+  Z )  .+  X )  =  ( ( ( invg `  G ) `  Z
)  .+  ( Z  .+  X ) ) )
481, 3, 42grplid 15890 . . . . 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 2516 . . 3  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( invg `  G
) `  Z )  .+  ( Z  .+  X
) )  =  X )
5144oveq1d 6299 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( ( invg `  G ) `  Z
)  .+  Z )  .+  Y )  =  ( ( 0g `  G
)  .+  Y )
)
521, 3grpass 15874 . . . . 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 1230 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( ( invg `  G ) `  Z
)  .+  Z )  .+  Y )  =  ( ( ( invg `  G ) `  Z
)  .+  ( Z  .+  Y ) ) )
541, 3, 42grplid 15890 . . . . 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 2516 . . 3  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( invg `  G
) `  Z )  .+  ( Z  .+  Y
) )  =  Y )
5741, 50, 563brtr3d 4476 . 2  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  X  .<  Y )
5815, 57impbida 830 1  |-  ( ph  ->  ( X  .<  Y  <->  ( Z  .+  X )  .<  ( Z  .+  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   Basecbs 14490   +g cplusg 14555   0gc0g 14695   ltcplt 15428   Grpcgrp 15727   invgcminusg 15728  oppgcoppg 16185  oGrpcogrp 27378
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
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-tpos 6955  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-plusg 14568  df-ple 14575  df-0g 14697  df-plt 15445  df-mnd 15732  df-grp 15867  df-minusg 15868  df-oppg 16186  df-omnd 27379  df-ogrp 27380
This theorem is referenced by:  ogrpinvlt  27404
  Copyright terms: Public domain W3C validator