Theorem omndadd 29706

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.

Step	Hyp	Ref	Expression
1		omndadd.0	. . . . 5 |- B = ( Base ` M )
2		omndadd.2	. . . . 5 |- .+ = ( +g ` M )
3		omndadd.1	. . . . 5 |- .<_ = ( le ` M )
4	1, 2, 3	isomnd 29701	. . . 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 ) ) ) )
5	4	simp3bi 1078	. . 3 |- ( M e. oMnd -> A. a e. B A. b e. B A. c e. B ( a .<_ b -> ( a .+ c ) .<_ ( b .+ c ) ) )
6		breq1 4656	. . . . 5 |- ( a = X -> ( a .<_ b <-> X .<_ b ) )
7		oveq1 6657	. . . . . 6 |- ( a = X -> ( a .+ c ) = ( X .+ c ) )
8	7	breq1d 4663	. . . . 5 |- ( a = X -> ( ( a .+ c ) .<_ ( b .+ c ) <-> ( X .+ c ) .<_ ( b .+ c ) ) )
9	6, 8	imbi12d 334	. . . 4 |- ( a = X -> ( ( a .<_ b -> ( a .+ c ) .<_ ( b .+ c ) ) <-> ( X .<_ b -> ( X .+ c ) .<_ ( b .+ c ) ) ) )
10		breq2 4657	. . . . 5 |- ( b = Y -> ( X .<_ b <-> X .<_ Y ) )
11		oveq1 6657	. . . . . 6 |- ( b = Y -> ( b .+ c ) = ( Y .+ c ) )
12	11	breq2d 4665	. . . . 5 |- ( b = Y -> ( ( X .+ c ) .<_ ( b .+ c ) <-> ( X .+ c ) .<_ ( Y .+ c ) ) )
13	10, 12	imbi12d 334	. . . 4 |- ( b = Y -> ( ( X .<_ b -> ( X .+ c ) .<_ ( b .+ c ) ) <-> ( X .<_ Y -> ( X .+ c ) .<_ ( Y .+ c ) ) ) )
14		oveq2 6658	. . . . . 6 |- ( c = Z -> ( X .+ c ) = ( X .+ Z ) )
15		oveq2 6658	. . . . . 6 |- ( c = Z -> ( Y .+ c ) = ( Y .+ Z ) )
16	14, 15	breq12d 4666	. . . . 5 |- ( c = Z -> ( ( X .+ c ) .<_ ( Y .+ c ) <-> ( X .+ Z ) .<_ ( Y .+ Z ) ) )
17	16	imbi2d 330	. . . 4 |- ( c = Z -> ( ( X .<_ Y -> ( X .+ c ) .<_ ( Y .+ c ) ) <-> ( X .<_ Y -> ( X .+ Z ) .<_ ( Y .+ Z ) ) ) )
18	9, 13, 17	rspc3v 3325	. . 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 ) ) ) )
19	5, 18	mpan9 486	. 2 |- ( ( M e. oMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .<_ Y -> ( X .+ Z ) .<_ ( Y .+ Z ) ) )
20	19	3impia 1261	1 |- ( ( M e. oMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X .<_ Y ) -> ( X .+ Z ) .<_ ( Y .+ Z ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   lecple 15948  Tosetctos 17033   Mndcmnd 17294  oMndcomnd 29697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-omnd 29699

This theorem is referenced by:  omndaddr  29707  omndadd2d  29708  omndadd2rd  29709  submomnd  29710  omndmul2  29712  omndmul3  29713  ogrpinvOLD  29715  ogrpinv0le  29716  ogrpsub  29717  ogrpaddlt  29718  orngsqr  29804  ornglmulle  29805  orngrmulle  29806