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Theorem ogrpaddlt 26301
Description: In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
Hypotheses
Ref Expression
ogrpaddlt.0  |-  B  =  ( Base `  G
)
ogrpaddlt.1  |-  .<  =  ( lt `  G )
ogrpaddlt.2  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
ogrpaddlt  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  ( X  .+  Z )  .<  ( Y  .+  Z ) )

Proof of Theorem ogrpaddlt
StepHypRef Expression
1 simp1 988 . . . . 5  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  G  e. oGrp )
2 isogrp 26285 . . . . . 6  |-  ( G  e. oGrp 
<->  ( G  e.  Grp  /\  G  e. oMnd ) )
32simprbi 464 . . . . 5  |-  ( G  e. oGrp  ->  G  e. oMnd )
41, 3syl 16 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  G  e. oMnd )
5 simp2 989 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )
65simp1d 1000 . . . . 5  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  X  e.  B
)
75simp2d 1001 . . . . 5  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  Y  e.  B
)
8 simp3 990 . . . . 5  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  X  .<  Y )
9 eqid 2450 . . . . . . 7  |-  ( le
`  G )  =  ( le `  G
)
10 ogrpaddlt.1 . . . . . . 7  |-  .<  =  ( lt `  G )
119, 10pltle 15219 . . . . . 6  |-  ( ( G  e. oGrp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X
( le `  G
) Y ) )
1211imp 429 . . . . 5  |-  ( ( ( G  e. oGrp  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  X ( le
`  G ) Y )
131, 6, 7, 8, 12syl31anc 1222 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  X ( le
`  G ) Y )
14 ogrpaddlt.0 . . . . 5  |-  B  =  ( Base `  G
)
15 ogrpaddlt.2 . . . . 5  |-  .+  =  ( +g  `  G )
1614, 9, 15omndadd 26289 . . . 4  |-  ( ( G  e. oMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X ( le `  G ) Y )  ->  ( X  .+  Z ) ( le
`  G ) ( Y  .+  Z ) )
174, 5, 13, 16syl3anc 1219 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  ( X  .+  Z ) ( le
`  G ) ( Y  .+  Z ) )
1810pltne 15220 . . . . . 6  |-  ( ( G  e. oGrp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X  =/=  Y ) )
1918imp 429 . . . . 5  |-  ( ( ( G  e. oGrp  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  X  =/=  Y
)
201, 6, 7, 8, 19syl31anc 1222 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  X  =/=  Y
)
212simplbi 460 . . . . . . 7  |-  ( G  e. oGrp  ->  G  e.  Grp )
2214, 15grprcan 15659 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Z
)  =  ( Y 
.+  Z )  <->  X  =  Y ) )
2322biimpd 207 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Z
)  =  ( Y 
.+  Z )  ->  X  =  Y )
)
2421, 23sylan 471 . . . . . 6  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .+  Z )  =  ( Y  .+  Z
)  ->  X  =  Y ) )
2524necon3d 2669 . . . . 5  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  =/=  Y  ->  ( X  .+  Z )  =/=  ( Y  .+  Z ) ) )
26253impia 1185 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  =/=  Y
)  ->  ( X  .+  Z )  =/=  ( Y  .+  Z ) )
271, 5, 20, 26syl3anc 1219 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  ( X  .+  Z )  =/=  ( Y  .+  Z ) )
2817, 27jca 532 . 2  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  ( ( X 
.+  Z ) ( le `  G ) ( Y  .+  Z
)  /\  ( X  .+  Z )  =/=  ( Y  .+  Z ) ) )
29 ovex 6201 . . . 4  |-  ( X 
.+  Z )  e. 
_V
30 ovex 6201 . . . 4  |-  ( Y 
.+  Z )  e. 
_V
319, 10pltval 15218 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  .+  Z )  e. 
_V  /\  ( Y  .+  Z )  e.  _V )  ->  ( ( X 
.+  Z )  .< 
( Y  .+  Z
)  <->  ( ( X 
.+  Z ) ( le `  G ) ( Y  .+  Z
)  /\  ( X  .+  Z )  =/=  ( Y  .+  Z ) ) ) )
3229, 30, 31mp3an23 1307 . . 3  |-  ( G  e. oGrp  ->  ( ( X 
.+  Z )  .< 
( Y  .+  Z
)  <->  ( ( X 
.+  Z ) ( le `  G ) ( Y  .+  Z
)  /\  ( X  .+  Z )  =/=  ( Y  .+  Z ) ) ) )
33323ad2ant1 1009 . 2  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  ( ( X 
.+  Z )  .< 
( Y  .+  Z
)  <->  ( ( X 
.+  Z ) ( le `  G ) ( Y  .+  Z
)  /\  ( X  .+  Z )  =/=  ( Y  .+  Z ) ) ) )
3428, 33mpbird 232 1  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  ( X  .+  Z )  .<  ( Y  .+  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757    =/= wne 2641   _Vcvv 3054   class class class wbr 4376   ` cfv 5502  (class class class)co 6176   Basecbs 14262   +g cplusg 14326   lecple 14333   ltcplt 15199   Grpcgrp 15498  oMndcomnd 26280  oGrpcogrp 26281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-iota 5465  df-fun 5504  df-fv 5510  df-riota 6137  df-ov 6179  df-0g 14468  df-plt 15216  df-mnd 15503  df-grp 15633  df-omnd 26282  df-ogrp 26283
This theorem is referenced by:  ogrpaddltbi  26302  ogrpaddltrd  26303  ogrpinv0lt  26306  isarchi3  26324  archirngz  26326  archiabllem1b  26329  archiabllem2a  26331  archiabllem2c  26332  ofldchr  26402
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