Theorem ogrpaddlt 29718

Description: In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)

Hypotheses

Ref	Expression
ogrpaddlt.0	|- B = ( Base ` G )
ogrpaddlt.1	|- .< = ( lt ` G )
ogrpaddlt.2	|- .+ = ( +g ` G )

Assertion

Ref	Expression
ogrpaddlt	|- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X .< Y ) -> ( X .+ Z ) .< ( Y .+ Z ) )

Proof of Theorem ogrpaddlt

Step	Hyp	Ref	Expression
1		isogrp 29702	. . . . 5 |- ( G e. oGrp <-> ( G e. Grp /\ G e. oMnd ) )
2	1	simprbi 480	. . . 4 |- ( G e. oGrp -> G e. oMnd )
3	2	3ad2ant1 1082	. . 3 |- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X .< Y ) -> G e. oMnd )
4		simp2 1062	. . 3 |- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X .< Y ) -> ( X e. B /\ Y e. B /\ Z e. B ) )
5		simp1 1061	. . . 4 |- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X .< Y ) -> G e. oGrp )
6		simp21 1094	. . . 4 |- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X .< Y ) -> X e. B )
7		simp22 1095	. . . 4 |- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X .< Y ) -> Y e. B )
8		simp3 1063	. . . 4 |- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X .< Y ) -> X .< Y )
9		eqid 2622	. . . . . 6 |- ( le ` G ) = ( le ` G )
10		ogrpaddlt.1	. . . . . 6 |- .< = ( lt ` G )
11	9, 10	pltle 16961	. . . . 5 |- ( ( G e. oGrp /\ X e. B /\ Y e. B ) -> ( X .< Y -> X ( le ` G ) Y ) )
12	11	imp 445	. . . 4 |- ( ( ( G e. oGrp /\ X e. B /\ Y e. B ) /\ X .< Y ) -> X ( le ` G ) Y )
13	5, 6, 7, 8, 12	syl31anc 1329	. . 3 |- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X .< Y ) -> X ( le ` G ) Y )
14		ogrpaddlt.0	. . . 4 |- B = ( Base ` G )
15		ogrpaddlt.2	. . . 4 |- .+ = ( +g ` G )
16	14, 9, 15	omndadd 29706	. . 3 |- ( ( G e. oMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X ( le ` G ) Y ) -> ( X .+ Z ) ( le ` G ) ( Y .+ Z ) )
17	3, 4, 13, 16	syl3anc 1326	. 2 |- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X .< Y ) -> ( X .+ Z ) ( le ` G ) ( Y .+ Z ) )
18	10	pltne 16962	. . . . 5 |- ( ( G e. oGrp /\ X e. B /\ Y e. B ) -> ( X .< Y -> X =/= Y ) )
19	18	imp 445	. . . 4 |- ( ( ( G e. oGrp /\ X e. B /\ Y e. B ) /\ X .< Y ) -> X =/= Y )
20	5, 6, 7, 8, 19	syl31anc 1329	. . 3 |- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X .< Y ) -> X =/= Y )
21		ogrpgrp 29703	. . . . . 6 |- ( G e. oGrp -> G e. Grp )
22	14, 15	grprcan 17455	. . . . . . 7 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Z ) = ( Y .+ Z ) <-> X = Y ) )
23	22	biimpd 219	. . . . . 6 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Z ) = ( Y .+ Z ) -> X = Y ) )
24	21, 23	sylan 488	. . . . 5 |- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Z ) = ( Y .+ Z ) -> X = Y ) )
25	24	necon3d 2815	. . . 4 |- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X =/= Y -> ( X .+ Z ) =/= ( Y .+ Z ) ) )
26	25	3impia 1261	. . 3 |- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X =/= Y ) -> ( X .+ Z ) =/= ( Y .+ Z ) )
27	5, 4, 20, 26	syl3anc 1326	. 2 |- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X .< Y ) -> ( X .+ Z ) =/= ( Y .+ Z ) )
28		ovex 6678	. . . 4 |- ( X .+ Z ) e. _V
29		ovex 6678	. . . 4 |- ( Y .+ Z ) e. _V
30	9, 10	pltval 16960	. . . 4 |- ( ( G e. oGrp /\ ( X .+ Z ) e. _V /\ ( Y .+ Z ) e. _V ) -> ( ( X .+ Z ) .< ( Y .+ Z ) <-> ( ( X .+ Z ) ( le ` G ) ( Y .+ Z ) /\ ( X .+ Z ) =/= ( Y .+ Z ) ) ) )
31	28, 29, 30	mp3an23 1416	. . 3 |- ( G e. oGrp -> ( ( X .+ Z ) .< ( Y .+ Z ) <-> ( ( X .+ Z ) ( le ` G ) ( Y .+ Z ) /\ ( X .+ Z ) =/= ( Y .+ Z ) ) ) )
32	31	3ad2ant1 1082	. 2 |- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X .< Y ) -> ( ( X .+ Z ) .< ( Y .+ Z ) <-> ( ( X .+ Z ) ( le ` G ) ( Y .+ Z ) /\ ( X .+ Z ) =/= ( Y .+ Z ) ) ) )
33	17, 27, 32	mpbir2and 957	1 |- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X .< Y ) -> ( X .+ Z ) .< ( Y .+ Z ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   lecple 15948   ltcplt 16941   Grpcgrp 17422  oMndcomnd 29697  oGrpcogrp 29698

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-plt 16958  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-omnd 29699  df-ogrp 29700

This theorem is referenced by:  ogrpaddltbi  29719  ogrpaddltrd  29720  ogrpinv0lt  29723  isarchi3  29741  archirngz  29743  archiabllem1b  29746  archiabllem2c  29749  ofldchr  29814