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Theorem omndadd2d 27522
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 27519 . . 3  |-  ( M  e. oMnd  ->  M  e. Toset )
3 tospos 27470 . . 3  |-  ( M  e. Toset  ->  M  e.  Poset )
41, 2, 33syl 20 . 2  |-  ( ph  ->  M  e.  Poset )
5 omndmnd 27518 . . . . 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 15802 . . . 4  |-  ( ( M  e.  Mnd  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y
)  e.  B )
126, 7, 8, 11syl3anc 1228 . . 3  |-  ( ph  ->  ( X  .+  Y
)  e.  B )
13 omndadd2d.z . . . 4  |-  ( ph  ->  Z  e.  B )
149, 10mndcl 15802 . . . 4  |-  ( ( M  e.  Mnd  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  .+  Y
)  e.  B )
156, 13, 8, 14syl3anc 1228 . . 3  |-  ( ph  ->  ( Z  .+  Y
)  e.  B )
16 omndadd2d.w . . . 4  |-  ( ph  ->  W  e.  B )
179, 10mndcl 15802 . . . 4  |-  ( ( M  e.  Mnd  /\  Z  e.  B  /\  W  e.  B )  ->  ( Z  .+  W
)  e.  B )
186, 13, 16, 17syl3anc 1228 . . 3  |-  ( ph  ->  ( Z  .+  W
)  e.  B )
1912, 15, 183jca 1176 . 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 27520 . . 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 1241 . 2  |-  ( ph  ->  ( X  .+  Y
)  .<_  ( Z  .+  Y ) )
24 omndadd2d.2 . . . 4  |-  ( ph  ->  Y  .<_  W )
259, 21, 10omndadd 27520 . . . 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 1241 . . 3  |-  ( ph  ->  ( Y  .+  Z
)  .<_  ( W  .+  Z ) )
27 omndadd2d.c . . . 4  |-  ( ph  ->  M  e. CMnd )
289, 10cmncom 16687 . . . 4  |-  ( ( M  e. CMnd  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .+  Z )  =  ( Z  .+  Y
) )
2927, 8, 13, 28syl3anc 1228 . . 3  |-  ( ph  ->  ( Y  .+  Z
)  =  ( Z 
.+  Y ) )
309, 10cmncom 16687 . . . 4  |-  ( ( M  e. CMnd  /\  W  e.  B  /\  Z  e.  B )  ->  ( W  .+  Z )  =  ( Z  .+  W
) )
3127, 16, 13, 30syl3anc 1228 . . 3  |-  ( ph  ->  ( W  .+  Z
)  =  ( Z 
.+  W ) )
3226, 29, 313brtr3d 4482 . 2  |-  ( ph  ->  ( Z  .+  Y
)  .<_  ( Z  .+  W ) )
339, 21postr 15457 . . 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 1229 1  |-  ( ph  ->  ( X  .+  Y
)  .<_  ( Z  .+  W ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   lecple 14579   Posetcpo 15444  Tosetctos 15537   Mndcmnd 15793  CMndccmn 16671  oMndcomnd 27511
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 4582
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298  df-poset 15450  df-toset 15538  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-cmn 16673  df-omnd 27513
This theorem is referenced by:  omndmul  27528  gsumle  27595
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