Theorem omndadd2d 29708

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

Step	Hyp	Ref	Expression
1		omndadd2d.m	. . 3 |- ( ph -> M e. oMnd )
2		omndtos 29705	. . 3 |- ( M e. oMnd -> M e. Toset )
3		tospos 29658	. . 3 |- ( M e. Toset -> M e. Poset )
4	1, 2, 3	3syl 18	. 2 |- ( ph -> M e. Poset )
5		omndmnd 29704	. . . . 5 |- ( M e. oMnd -> M e. Mnd )
6	1, 5	syl 17	. . . 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 )
11	9, 10	mndcl 17301	. . . 4 |- ( ( M e. Mnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )
12	6, 7, 8, 11	syl3anc 1326	. . 3 |- ( ph -> ( X .+ Y ) e. B )
13		omndadd2d.z	. . . 4 |- ( ph -> Z e. B )
14	9, 10	mndcl 17301	. . . 4 |- ( ( M e. Mnd /\ Z e. B /\ Y e. B ) -> ( Z .+ Y ) e. B )
15	6, 13, 8, 14	syl3anc 1326	. . 3 |- ( ph -> ( Z .+ Y ) e. B )
16		omndadd2d.w	. . . 4 |- ( ph -> W e. B )
17	9, 10	mndcl 17301	. . . 4 |- ( ( M e. Mnd /\ Z e. B /\ W e. B ) -> ( Z .+ W ) e. B )
18	6, 13, 16, 17	syl3anc 1326	. . 3 |- ( ph -> ( Z .+ W ) e. B )
19	12, 15, 18	3jca 1242	. 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 )
22	9, 21, 10	omndadd 29706	. . 3 |- ( ( M e. oMnd /\ ( X e. B /\ Z e. B /\ Y e. B ) /\ X .<_ Z ) -> ( X .+ Y ) .<_ ( Z .+ Y ) )
23	1, 7, 13, 8, 20, 22	syl131anc 1339	. 2 |- ( ph -> ( X .+ Y ) .<_ ( Z .+ Y ) )
24		omndadd2d.2	. . . 4 |- ( ph -> Y .<_ W )
25	9, 21, 10	omndadd 29706	. . . 4 |- ( ( M e. oMnd /\ ( Y e. B /\ W e. B /\ Z e. B ) /\ Y .<_ W ) -> ( Y .+ Z ) .<_ ( W .+ Z ) )
26	1, 8, 16, 13, 24, 25	syl131anc 1339	. . 3 |- ( ph -> ( Y .+ Z ) .<_ ( W .+ Z ) )
27		omndadd2d.c	. . . 4 |- ( ph -> M e. CMnd )
28	9, 10	cmncom 18209	. . . 4 |- ( ( M e. CMnd /\ Y e. B /\ Z e. B ) -> ( Y .+ Z ) = ( Z .+ Y ) )
29	27, 8, 13, 28	syl3anc 1326	. . 3 |- ( ph -> ( Y .+ Z ) = ( Z .+ Y ) )
30	9, 10	cmncom 18209	. . . 4 |- ( ( M e. CMnd /\ W e. B /\ Z e. B ) -> ( W .+ Z ) = ( Z .+ W ) )
31	27, 16, 13, 30	syl3anc 1326	. . 3 |- ( ph -> ( W .+ Z ) = ( Z .+ W ) )
32	26, 29, 31	3brtr3d 4684	. 2 |- ( ph -> ( Z .+ Y ) .<_ ( Z .+ W ) )
33	9, 21	postr 16953	. . 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 ) ) )
34	33	imp 445	. 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 ) )
35	4, 19, 23, 32, 34	syl22anc 1327	1 |- ( ph -> ( X .+ Y ) .<_ ( Z .+ W ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   lecple 15948   Posetcpo 16940  Tosetctos 17033   Mndcmnd 17294  CMndccmn 18193  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-poset 16946  df-toset 17034  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-cmn 18195  df-omnd 29699

This theorem is referenced by:  omndmul  29714  gsumle  29779