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Theorem omndadd 26169
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 . . . . . 6  |-  B  =  ( Base `  M
)
2 omndadd.2 . . . . . 6  |-  .+  =  ( +g  `  M )
3 omndadd.1 . . . . . 6  |-  .<_  =  ( le `  M )
41, 2, 3isomnd 26164 . . . . 5  |-  ( 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 1005 . . . 4  |-  ( M  e. oMnd  ->  A. a  e.  B  A. b  e.  B  A. c  e.  B  ( a  .<_  b  -> 
( a  .+  c
)  .<_  ( b  .+  c ) ) )
6 breq1 4295 . . . . . 6  |-  ( a  =  X  ->  (
a  .<_  b  <->  X  .<_  b ) )
7 oveq1 6098 . . . . . . 7  |-  ( a  =  X  ->  (
a  .+  c )  =  ( X  .+  c ) )
87breq1d 4302 . . . . . 6  |-  ( a  =  X  ->  (
( a  .+  c
)  .<_  ( b  .+  c )  <->  ( X  .+  c )  .<_  ( b 
.+  c ) ) )
96, 8imbi12d 320 . . . . 5  |-  ( a  =  X  ->  (
( a  .<_  b  -> 
( a  .+  c
)  .<_  ( b  .+  c ) )  <->  ( X  .<_  b  ->  ( X  .+  c )  .<_  ( b 
.+  c ) ) ) )
10 breq2 4296 . . . . . 6  |-  ( b  =  Y  ->  ( X  .<_  b  <->  X  .<_  Y ) )
11 oveq1 6098 . . . . . . 7  |-  ( b  =  Y  ->  (
b  .+  c )  =  ( Y  .+  c ) )
1211breq2d 4304 . . . . . 6  |-  ( b  =  Y  ->  (
( X  .+  c
)  .<_  ( b  .+  c )  <->  ( X  .+  c )  .<_  ( Y 
.+  c ) ) )
1310, 12imbi12d 320 . . . . 5  |-  ( b  =  Y  ->  (
( X  .<_  b  -> 
( X  .+  c
)  .<_  ( b  .+  c ) )  <->  ( X  .<_  Y  ->  ( X  .+  c )  .<_  ( Y 
.+  c ) ) ) )
14 oveq2 6099 . . . . . . 7  |-  ( c  =  Z  ->  ( X  .+  c )  =  ( X  .+  Z
) )
15 oveq2 6099 . . . . . . 7  |-  ( c  =  Z  ->  ( Y  .+  c )  =  ( Y  .+  Z
) )
1614, 15breq12d 4305 . . . . . 6  |-  ( c  =  Z  ->  (
( X  .+  c
)  .<_  ( Y  .+  c )  <->  ( X  .+  Z )  .<_  ( Y 
.+  Z ) ) )
1716imbi2d 316 . . . . 5  |-  ( c  =  Z  ->  (
( X  .<_  Y  -> 
( X  .+  c
)  .<_  ( Y  .+  c ) )  <->  ( X  .<_  Y  ->  ( X  .+  Z )  .<_  ( Y 
.+  Z ) ) ) )
189, 13, 17rspc3v 3082 . . . 4  |-  ( ( 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, 18syl5 32 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( M  e. oMnd  ->  ( X  .<_  Y  ->  ( X  .+  Z ) 
.<_  ( Y  .+  Z
) ) ) )
2019impcom 430 . 2  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<_  Y  ->  ( X  .+  Z )  .<_  ( Y 
.+  Z ) ) )
21203impia 1184 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 965    = wceq 1369    e. wcel 1756   A.wral 2715   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   lecple 14245  Tosetctos 15203   Mndcmnd 15409  oMndcomnd 26160
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-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426  df-ov 6094  df-omnd 26162
This theorem is referenced by:  omndaddr  26170  omndadd2d  26171  omndadd2rd  26172  submomnd  26173  omndmul2  26175  omndmul3  26176  ogrpinvOLD  26178  ogrpinv0le  26179  ogrpsub  26180  ogrpaddlt  26181  orngsqr  26272  ornglmulle  26273  orngrmulle  26274
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