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Theorem omndadd 28307
Description: In an ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
omndadd.0  |-  B  =  ( Base `  M
)
omndadd.1  |-  .<_  =  ( le `  M )
omndadd.2  |-  .+  =  ( +g  `  M )
Assertion
Ref Expression
omndadd  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  ( X  .+  Z )  .<_  ( Y 
.+  Z ) )

Proof of Theorem omndadd
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omndadd.0 . . . . 5  |-  B  =  ( Base `  M
)
2 omndadd.2 . . . . 5  |-  .+  =  ( +g  `  M )
3 omndadd.1 . . . . 5  |-  .<_  =  ( le `  M )
41, 2, 3isomnd 28302 . . . 4  |-  ( M  e. oMnd 
<->  ( M  e.  Mnd  /\  M  e. Toset  /\  A. a  e.  B  A. b  e.  B  A. c  e.  B  ( a  .<_  b  ->  ( a  .+  c )  .<_  ( b 
.+  c ) ) ) )
54simp3bi 1022 . . 3  |-  ( M  e. oMnd  ->  A. a  e.  B  A. b  e.  B  A. c  e.  B  ( a  .<_  b  -> 
( a  .+  c
)  .<_  ( b  .+  c ) ) )
6 breq1 4429 . . . . 5  |-  ( a  =  X  ->  (
a  .<_  b  <->  X  .<_  b ) )
7 oveq1 6312 . . . . . 6  |-  ( a  =  X  ->  (
a  .+  c )  =  ( X  .+  c ) )
87breq1d 4436 . . . . 5  |-  ( a  =  X  ->  (
( a  .+  c
)  .<_  ( b  .+  c )  <->  ( X  .+  c )  .<_  ( b 
.+  c ) ) )
96, 8imbi12d 321 . . . 4  |-  ( a  =  X  ->  (
( a  .<_  b  -> 
( a  .+  c
)  .<_  ( b  .+  c ) )  <->  ( X  .<_  b  ->  ( X  .+  c )  .<_  ( b 
.+  c ) ) ) )
10 breq2 4430 . . . . 5  |-  ( b  =  Y  ->  ( X  .<_  b  <->  X  .<_  Y ) )
11 oveq1 6312 . . . . . 6  |-  ( b  =  Y  ->  (
b  .+  c )  =  ( Y  .+  c ) )
1211breq2d 4438 . . . . 5  |-  ( b  =  Y  ->  (
( X  .+  c
)  .<_  ( b  .+  c )  <->  ( X  .+  c )  .<_  ( Y 
.+  c ) ) )
1310, 12imbi12d 321 . . . 4  |-  ( b  =  Y  ->  (
( X  .<_  b  -> 
( X  .+  c
)  .<_  ( b  .+  c ) )  <->  ( X  .<_  Y  ->  ( X  .+  c )  .<_  ( Y 
.+  c ) ) ) )
14 oveq2 6313 . . . . . 6  |-  ( c  =  Z  ->  ( X  .+  c )  =  ( X  .+  Z
) )
15 oveq2 6313 . . . . . 6  |-  ( c  =  Z  ->  ( Y  .+  c )  =  ( Y  .+  Z
) )
1614, 15breq12d 4439 . . . . 5  |-  ( c  =  Z  ->  (
( X  .+  c
)  .<_  ( Y  .+  c )  <->  ( X  .+  Z )  .<_  ( Y 
.+  Z ) ) )
1716imbi2d 317 . . . 4  |-  ( c  =  Z  ->  (
( X  .<_  Y  -> 
( X  .+  c
)  .<_  ( Y  .+  c ) )  <->  ( X  .<_  Y  ->  ( X  .+  Z )  .<_  ( Y 
.+  Z ) ) ) )
189, 13, 17rspc3v 3200 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( A. a  e.  B  A. b  e.  B  A. c  e.  B  ( a  .<_  b  ->  ( a  .+  c )  .<_  ( b 
.+  c ) )  ->  ( X  .<_  Y  ->  ( X  .+  Z )  .<_  ( Y 
.+  Z ) ) ) )
195, 18mpan9 471 . 2  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<_  Y  ->  ( X  .+  Z )  .<_  ( Y 
.+  Z ) ) )
20193impia 1202 1  |-  ( ( M  e. oMnd  /\  ( 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    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15152   lecple 15159  Tosetctos 16230   Mndcmnd 16486  oMndcomnd 28298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-nul 4556
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-iota 5565  df-fv 5609  df-ov 6308  df-omnd 28300
This theorem is referenced by:  omndaddr  28308  omndadd2d  28309  omndadd2rd  28310  submomnd  28311  omndmul2  28313  omndmul3  28314  ogrpinvOLD  28316  ogrpinv0le  28317  ogrpsub  28318  ogrpaddlt  28319  orngsqr  28406  ornglmulle  28407  orngrmulle  28408
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