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

Theorem ogrpaddltrbid 26189
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 26188 . 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 26171 . . . . . . 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 15869 . . . . . 6  |-  ( X ( +g  `  (oppg `  G
) ) Z )  =  ( Z  .+  X )
2623, 1oppgbas 15871 . . . . . . 7  |-  B  =  ( Base `  (oppg `  G
) )
2726, 24grpcl 15556 . . . . . 6  |-  ( ( (oppg
`  G )  e. 
Grp  /\  X  e.  B  /\  Z  e.  B
)  ->  ( X
( +g  `  (oppg `  G
) ) Z )  e.  B )
2825, 27syl5eqelr 2528 . . . . 5  |-  ( ( (oppg
`  G )  e. 
Grp  /\  X  e.  B  /\  Z  e.  B
)  ->  ( Z  .+  X )  e.  B
)
2920, 21, 22, 28syl3anc 1218 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( Z  .+  X )  e.  B
)
3010adantr 465 . . . . 5  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  Y  e.  B
)
313, 23, 24oppgplus 15869 . . . . . 6  |-  ( Y ( +g  `  (oppg `  G
) ) Z )  =  ( Z  .+  Y )
3226, 24grpcl 15556 . . . . . 6  |-  ( ( (oppg
`  G )  e. 
Grp  /\  Y  e.  B  /\  Z  e.  B
)  ->  ( Y
( +g  `  (oppg `  G
) ) Z )  e.  B )
3331, 32syl5eqelr 2528 . . . . 5  |-  ( ( (oppg
`  G )  e. 
Grp  /\  Y  e.  B  /\  Z  e.  B
)  ->  ( Z  .+  Y )  e.  B
)
3420, 30, 22, 33syl3anc 1218 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( Z  .+  Y )  e.  B
)
3523oppggrpb 15878 . . . . . 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 15588 . . . . 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 26188 . . 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 15589 . . . . . 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 6111 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( ( invg `  G ) `  Z
)  .+  Z )  .+  X )  =  ( ( 0g `  G
)  .+  X )
)
461, 3grpass 15557 . . . . 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 1220 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( ( invg `  G ) `  Z
)  .+  Z )  .+  X )  =  ( ( ( invg `  G ) `  Z
)  .+  ( Z  .+  X ) ) )
481, 3, 42grplid 15573 . . . . 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 2483 . . 3  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( invg `  G
) `  Z )  .+  ( Z  .+  X
) )  =  X )
5144oveq1d 6111 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( ( invg `  G ) `  Z
)  .+  Z )  .+  Y )  =  ( ( 0g `  G
)  .+  Y )
)
521, 3grpass 15557 . . . . 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 1220 . . . 4  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( ( invg `  G ) `  Z
)  .+  Z )  .+  Y )  =  ( ( ( invg `  G ) `  Z
)  .+  ( Z  .+  Y ) ) )
541, 3, 42grplid 15573 . . . . 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 2483 . . 3  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  ( ( ( invg `  G
) `  Z )  .+  ( Z  .+  Y
) )  =  Y )
5741, 50, 563brtr3d 4326 . 2  |-  ( (
ph  /\  ( Z  .+  X )  .<  ( Z  .+  Y ) )  ->  X  .<  Y )
5815, 57impbida 828 1  |-  ( ph  ->  ( X  .<  Y  <->  ( Z  .+  X )  .<  ( Z  .+  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Basecbs 14179   +g cplusg 14243   0gc0g 14383   ltcplt 15116   Grpcgrp 15415   invgcminusg 15416  oppgcoppg 15865  oGrpcogrp 26166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-tpos 6750  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-plusg 14256  df-ple 14263  df-0g 14385  df-plt 15133  df-mnd 15420  df-grp 15550  df-minusg 15551  df-oppg 15866  df-omnd 26167  df-ogrp 26168
This theorem is referenced by:  ogrpinvlt  26192
  Copyright terms: Public domain W3C validator