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Theorem omndadd2d 26171
Description: In a commutative left ordered monoid, the ordering is compatible with monoid addition. Double addition version (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
omndadd.0  |-  B  =  ( Base `  M
)
omndadd.1  |-  .<_  =  ( le `  M )
omndadd.2  |-  .+  =  ( +g  `  M )
omndadd2d.m  |-  ( ph  ->  M  e. oMnd )
omndadd2d.w  |-  ( ph  ->  W  e.  B )
omndadd2d.x  |-  ( ph  ->  X  e.  B )
omndadd2d.y  |-  ( ph  ->  Y  e.  B )
omndadd2d.z  |-  ( ph  ->  Z  e.  B )
omndadd2d.1  |-  ( ph  ->  X  .<_  Z )
omndadd2d.2  |-  ( ph  ->  Y  .<_  W )
omndadd2d.c  |-  ( ph  ->  M  e. CMnd )
Assertion
Ref Expression
omndadd2d  |-  ( ph  ->  ( X  .+  Y
)  .<_  ( Z  .+  W ) )

Proof of Theorem omndadd2d
StepHypRef Expression
1 omndadd2d.m . . 3  |-  ( ph  ->  M  e. oMnd )
2 omndtos 26168 . . 3  |-  ( M  e. oMnd  ->  M  e. Toset )
3 tospos 26119 . . 3  |-  ( M  e. Toset  ->  M  e.  Poset )
41, 2, 33syl 20 . 2  |-  ( ph  ->  M  e.  Poset )
5 omndmnd 26167 . . . . 5  |-  ( M  e. oMnd  ->  M  e.  Mnd )
61, 5syl 16 . . . 4  |-  ( ph  ->  M  e.  Mnd )
7 omndadd2d.x . . . 4  |-  ( ph  ->  X  e.  B )
8 omndadd2d.y . . . 4  |-  ( ph  ->  Y  e.  B )
9 omndadd.0 . . . . 5  |-  B  =  ( Base `  M
)
10 omndadd.2 . . . . 5  |-  .+  =  ( +g  `  M )
119, 10mndcl 15420 . . . 4  |-  ( ( M  e.  Mnd  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y
)  e.  B )
126, 7, 8, 11syl3anc 1218 . . 3  |-  ( ph  ->  ( X  .+  Y
)  e.  B )
13 omndadd2d.z . . . 4  |-  ( ph  ->  Z  e.  B )
149, 10mndcl 15420 . . . 4  |-  ( ( M  e.  Mnd  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  .+  Y
)  e.  B )
156, 13, 8, 14syl3anc 1218 . . 3  |-  ( ph  ->  ( Z  .+  Y
)  e.  B )
16 omndadd2d.w . . . 4  |-  ( ph  ->  W  e.  B )
179, 10mndcl 15420 . . . 4  |-  ( ( M  e.  Mnd  /\  Z  e.  B  /\  W  e.  B )  ->  ( Z  .+  W
)  e.  B )
186, 13, 16, 17syl3anc 1218 . . 3  |-  ( ph  ->  ( Z  .+  W
)  e.  B )
1912, 15, 183jca 1168 . 2  |-  ( ph  ->  ( ( X  .+  Y )  e.  B  /\  ( Z  .+  Y
)  e.  B  /\  ( Z  .+  W )  e.  B ) )
20 omndadd2d.1 . . 3  |-  ( ph  ->  X  .<_  Z )
21 omndadd.1 . . . 4  |-  .<_  =  ( le `  M )
229, 21, 10omndadd 26169 . . 3  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  Z  e.  B  /\  Y  e.  B )  /\  X  .<_  Z )  ->  ( X  .+  Y )  .<_  ( Z 
.+  Y ) )
231, 7, 13, 8, 20, 22syl131anc 1231 . 2  |-  ( ph  ->  ( X  .+  Y
)  .<_  ( Z  .+  Y ) )
24 omndadd2d.2 . . . 4  |-  ( ph  ->  Y  .<_  W )
259, 21, 10omndadd 26169 . . . 4  |-  ( ( M  e. oMnd  /\  ( Y  e.  B  /\  W  e.  B  /\  Z  e.  B )  /\  Y  .<_  W )  ->  ( Y  .+  Z )  .<_  ( W 
.+  Z ) )
261, 8, 16, 13, 24, 25syl131anc 1231 . . 3  |-  ( ph  ->  ( Y  .+  Z
)  .<_  ( W  .+  Z ) )
27 omndadd2d.c . . . 4  |-  ( ph  ->  M  e. CMnd )
289, 10cmncom 16293 . . . 4  |-  ( ( M  e. CMnd  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .+  Z )  =  ( Z  .+  Y
) )
2927, 8, 13, 28syl3anc 1218 . . 3  |-  ( ph  ->  ( Y  .+  Z
)  =  ( Z 
.+  Y ) )
309, 10cmncom 16293 . . . 4  |-  ( ( M  e. CMnd  /\  W  e.  B  /\  Z  e.  B )  ->  ( W  .+  Z )  =  ( Z  .+  W
) )
3127, 16, 13, 30syl3anc 1218 . . 3  |-  ( ph  ->  ( W  .+  Z
)  =  ( Z 
.+  W ) )
3226, 29, 313brtr3d 4321 . 2  |-  ( ph  ->  ( Z  .+  Y
)  .<_  ( Z  .+  W ) )
339, 21postr 15123 . . 3  |-  ( ( M  e.  Poset  /\  (
( X  .+  Y
)  e.  B  /\  ( Z  .+  Y )  e.  B  /\  ( Z  .+  W )  e.  B ) )  -> 
( ( ( X 
.+  Y )  .<_  ( Z  .+  Y )  /\  ( Z  .+  Y )  .<_  ( Z 
.+  W ) )  ->  ( X  .+  Y )  .<_  ( Z 
.+  W ) ) )
3433imp 429 . 2  |-  ( ( ( M  e.  Poset  /\  ( ( X  .+  Y )  e.  B  /\  ( Z  .+  Y
)  e.  B  /\  ( Z  .+  W )  e.  B ) )  /\  ( ( X 
.+  Y )  .<_  ( Z  .+  Y )  /\  ( Z  .+  Y )  .<_  ( Z 
.+  W ) ) )  ->  ( X  .+  Y )  .<_  ( Z 
.+  W ) )
354, 19, 23, 32, 34syl22anc 1219 1  |-  ( ph  ->  ( X  .+  Y
)  .<_  ( Z  .+  W ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   lecple 14245   Posetcpo 15110  Tosetctos 15203   Mndcmnd 15409  CMndccmn 16277  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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4421  ax-pow 4470
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-mo 2258  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-poset 15116  df-toset 15204  df-mnd 15415  df-cmn 16279  df-omnd 26162
This theorem is referenced by:  omndmul  26177  gsumle  26246
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