Theorem ogrpaddltrbid 29721

Description: In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 4-Sep-2018.)

Hypotheses

Ref	Expression
ogrpaddlt.0	|- B = ( Base ` G )
ogrpaddlt.1	|- .< = ( lt ` G )
ogrpaddlt.2	|- .+ = ( +g ` G )
ogrpaddltrd.1	|- ( ph -> G e. V )
ogrpaddltrd.2	|- ( ph -> (oppg ` G ) e. oGrp )
ogrpaddltrd.3	|- ( ph -> X e. B )
ogrpaddltrd.4	|- ( ph -> Y e. B )
ogrpaddltrd.5	|- ( ph -> Z e. B )

Assertion

Ref	Expression
ogrpaddltrbid	|- ( ph -> ( X .< Y <-> ( Z .+ X ) .< ( Z .+ Y ) ) )

Proof of Theorem ogrpaddltrbid

Step	Hyp	Ref	Expression
1		ogrpaddlt.0	. . 3 |- B = ( Base ` G )
2		ogrpaddlt.1	. . 3 |- .< = ( lt ` G )
3		ogrpaddlt.2	. . 3 |- .+ = ( +g ` G )
4		ogrpaddltrd.1	. . . 4 |- ( ph -> G e. V )
5	4	adantr 481	. . 3 |- ( ( ph /\ X .< Y ) -> G e. V )
6		ogrpaddltrd.2	. . . 4 |- ( ph -> (oppg ` G ) e. oGrp )
7	6	adantr 481	. . 3 |- ( ( ph /\ X .< Y ) -> (oppg ` G ) e. oGrp )
8		ogrpaddltrd.3	. . . 4 |- ( ph -> X e. B )
9	8	adantr 481	. . 3 |- ( ( ph /\ X .< Y ) -> X e. B )
10		ogrpaddltrd.4	. . . 4 |- ( ph -> Y e. B )
11	10	adantr 481	. . 3 |- ( ( ph /\ X .< Y ) -> Y e. B )
12		ogrpaddltrd.5	. . . 4 |- ( ph -> Z e. B )
13	12	adantr 481	. . 3 |- ( ( ph /\ X .< Y ) -> Z e. B )
14		simpr 477	. . 3 |- ( ( ph /\ X .< Y ) -> X .< Y )
15	1, 2, 3, 5, 7, 9, 11, 13, 14	ogrpaddltrd 29720	. 2 |- ( ( ph /\ X .< Y ) -> ( Z .+ X ) .< ( Z .+ Y ) )
16	4	adantr 481	. . . 4 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> G e. V )
17	6	adantr 481	. . . 4 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> (oppg ` G ) e. oGrp )
18		ogrpgrp 29703	. . . . . . 7 |- ( (oppg ` G ) e. oGrp -> (oppg ` G ) e. Grp )
19	6, 18	syl 17	. . . . . 6 |- ( ph -> (oppg ` G ) e. Grp )
20	19	adantr 481	. . . . 5 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> (oppg ` G ) e. Grp )
21	8	adantr 481	. . . . 5 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> X e. B )
22	12	adantr 481	. . . . 5 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> Z e. B )
23		eqid 2622	. . . . . . 7 |- (oppg ` G ) = (oppg ` G )
24		eqid 2622	. . . . . . 7 |- ( +g ` (oppg ` G ) ) = ( +g ` (oppg ` G ) )
25	3, 23, 24	oppgplus 17779	. . . . . 6 |- ( X ( +g ` (oppg ` G ) ) Z ) = ( Z .+ X )
26	23, 1	oppgbas 17781	. . . . . . 7 |- B = ( Base ` (oppg ` G ) )
27	26, 24	grpcl 17430	. . . . . 6 |- ( ( (oppg ` G ) e. Grp /\ X e. B /\ Z e. B ) -> ( X ( +g ` (oppg ` G ) ) Z ) e. B )
28	25, 27	syl5eqelr 2706	. . . . 5 |- ( ( (oppg ` G ) e. Grp /\ X e. B /\ Z e. B ) -> ( Z .+ X ) e. B )
29	20, 21, 22, 28	syl3anc 1326	. . . 4 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> ( Z .+ X ) e. B )
30	10	adantr 481	. . . . 5 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> Y e. B )
31	3, 23, 24	oppgplus 17779	. . . . . 6 |- ( Y ( +g ` (oppg ` G ) ) Z ) = ( Z .+ Y )
32	26, 24	grpcl 17430	. . . . . 6 |- ( ( (oppg ` G ) e. Grp /\ Y e. B /\ Z e. B ) -> ( Y ( +g ` (oppg ` G ) ) Z ) e. B )
33	31, 32	syl5eqelr 2706	. . . . 5 |- ( ( (oppg ` G ) e. Grp /\ Y e. B /\ Z e. B ) -> ( Z .+ Y ) e. B )
34	20, 30, 22, 33	syl3anc 1326	. . . 4 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> ( Z .+ Y ) e. B )
35	23	oppggrpb 17788	. . . . . 6 |- ( G e. Grp <-> (oppg ` G ) e. Grp )
36	20, 35	sylibr 224	. . . . 5 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> G e. Grp )
37		eqid 2622	. . . . . 6 |- ( invg ` G ) = ( invg ` G )
38	1, 37	grpinvcl 17467	. . . . 5 |- ( ( G e. Grp /\ Z e. B ) -> ( ( invg ` G ) ` Z ) e. B )
39	36, 22, 38	syl2anc 693	. . . 4 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> ( ( invg ` G ) ` Z ) e. B )
40		simpr 477	. . . 4 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> ( Z .+ X ) .< ( Z .+ Y ) )
41	1, 2, 3, 16, 17, 29, 34, 39, 40	ogrpaddltrd 29720	. . 3 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> ( ( ( invg ` G ) ` Z ) .+ ( Z .+ X ) ) .< ( ( ( invg ` G ) ` Z ) .+ ( Z .+ Y ) ) )
42		eqid 2622	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
43	1, 3, 42, 37	grplinv 17468	. . . . . 6 |- ( ( G e. Grp /\ Z e. B ) -> ( ( ( invg ` G ) ` Z ) .+ Z ) = ( 0g ` G ) )
44	36, 22, 43	syl2anc 693	. . . . 5 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> ( ( ( invg ` G ) ` Z ) .+ Z ) = ( 0g ` G ) )
45	44	oveq1d 6665	. . . 4 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> ( ( ( ( invg ` G ) ` Z ) .+ Z ) .+ X ) = ( ( 0g ` G ) .+ X ) )
46	1, 3	grpass 17431	. . . . 5 |- ( ( G e. Grp /\ ( ( ( invg ` G ) ` Z ) e. B /\ Z e. B /\ X e. B ) ) -> ( ( ( ( invg ` G ) ` Z ) .+ Z ) .+ X ) = ( ( ( invg ` G ) ` Z ) .+ ( Z .+ X ) ) )
47	36, 39, 22, 21, 46	syl13anc 1328	. . . 4 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> ( ( ( ( invg ` G ) ` Z ) .+ Z ) .+ X ) = ( ( ( invg ` G ) ` Z ) .+ ( Z .+ X ) ) )
48	1, 3, 42	grplid 17452	. . . . 5 |- ( ( G e. Grp /\ X e. B ) -> ( ( 0g ` G ) .+ X ) = X )
49	36, 21, 48	syl2anc 693	. . . 4 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> ( ( 0g ` G ) .+ X ) = X )
50	45, 47, 49	3eqtr3d 2664	. . 3 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> ( ( ( invg ` G ) ` Z ) .+ ( Z .+ X ) ) = X )
51	44	oveq1d 6665	. . . 4 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> ( ( ( ( invg ` G ) ` Z ) .+ Z ) .+ Y ) = ( ( 0g ` G ) .+ Y ) )
52	1, 3	grpass 17431	. . . . 5 |- ( ( G e. Grp /\ ( ( ( invg ` G ) ` Z ) e. B /\ Z e. B /\ Y e. B ) ) -> ( ( ( ( invg ` G ) ` Z ) .+ Z ) .+ Y ) = ( ( ( invg ` G ) ` Z ) .+ ( Z .+ Y ) ) )
53	36, 39, 22, 30, 52	syl13anc 1328	. . . 4 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> ( ( ( ( invg ` G ) ` Z ) .+ Z ) .+ Y ) = ( ( ( invg ` G ) ` Z ) .+ ( Z .+ Y ) ) )
54	1, 3, 42	grplid 17452	. . . . 5 |- ( ( G e. Grp /\ Y e. B ) -> ( ( 0g ` G ) .+ Y ) = Y )
55	36, 30, 54	syl2anc 693	. . . 4 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> ( ( 0g ` G ) .+ Y ) = Y )
56	51, 53, 55	3eqtr3d 2664	. . 3 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> ( ( ( invg ` G ) ` Z ) .+ ( Z .+ Y ) ) = Y )
57	41, 50, 56	3brtr3d 4684	. 2 |- ( ( ph /\ ( Z .+ X ) .< ( Z .+ Y ) ) -> X .< Y )
58	15, 57	impbida 877	1 |- ( ph -> ( X .< Y <-> ( Z .+ X ) .< ( Z .+ Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ 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   0gc0g 16100   ltcplt 16941   Grpcgrp 17422   invgcminusg 17423  opp_gcoppg 17775  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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-dec 11494  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-ple 15961  df-0g 16102  df-plt 16958  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-oppg 17776  df-omnd 29699  df-ogrp 29700

This theorem is referenced by:  ogrpinvlt  29724