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Theorem omndadd 27455
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 27450 . . . . 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 1013 . . . 4  |-  ( M  e. oMnd  ->  A. a  e.  B  A. b  e.  B  A. c  e.  B  ( a  .<_  b  -> 
( a  .+  c
)  .<_  ( b  .+  c ) ) )
6 breq1 4450 . . . . . 6  |-  ( a  =  X  ->  (
a  .<_  b  <->  X  .<_  b ) )
7 oveq1 6292 . . . . . . 7  |-  ( a  =  X  ->  (
a  .+  c )  =  ( X  .+  c ) )
87breq1d 4457 . . . . . 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 4451 . . . . . 6  |-  ( b  =  Y  ->  ( X  .<_  b  <->  X  .<_  Y ) )
11 oveq1 6292 . . . . . . 7  |-  ( b  =  Y  ->  (
b  .+  c )  =  ( Y  .+  c ) )
1211breq2d 4459 . . . . . 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 6293 . . . . . . 7  |-  ( c  =  Z  ->  ( X  .+  c )  =  ( X  .+  Z
) )
15 oveq2 6293 . . . . . . 7  |-  ( c  =  Z  ->  ( Y  .+  c )  =  ( Y  .+  Z
) )
1614, 15breq12d 4460 . . . . . 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 3226 . . . 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 1193 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   +g cplusg 14558   lecple 14565  Tosetctos 15523   Mndcmnd 15729  oMndcomnd 27446
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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6288  df-omnd 27448
This theorem is referenced by:  omndaddr  27456  omndadd2d  27457  omndadd2rd  27458  submomnd  27459  omndmul2  27461  omndmul3  27462  ogrpinvOLD  27464  ogrpinv0le  27465  ogrpsub  27466  ogrpaddlt  27467  orngsqr  27554  ornglmulle  27555  orngrmulle  27556
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