Theorem ogrpaddltbi 30719

Description: In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)

Hypotheses

Ref	Expression
ogrpaddlt.0	⊢ 𝐵 = (Base‘𝐺)
ogrpaddlt.1	⊢ < = (lt‘𝐺)
ogrpaddlt.2	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
ogrpaddltbi	⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍)))

Proof of Theorem ogrpaddltbi

Step	Hyp	Ref	Expression
1		ogrpaddlt.0	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2		ogrpaddlt.1	. . . 4 ⊢ < = (lt‘𝐺)
3		ogrpaddlt.2	. . . 4 ⊢ + = (+g‘𝐺)
4	1, 2, 3	ogrpaddlt 30718	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
5	4	3expa 1114	. 2 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
6		simpll 765	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝐺 ∈ oGrp)
7		ogrpgrp 30704	. . . . . 6 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
8	6, 7	syl 17	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝐺 ∈ Grp)
9		simplr1 1211	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋 ∈ 𝐵)
10		simplr3 1213	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑍 ∈ 𝐵)
11	1, 3	grpcl 18111	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 + 𝑍) ∈ 𝐵)
12	8, 9, 10, 11	syl3anc 1367	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) ∈ 𝐵)
13		simplr2 1212	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑌 ∈ 𝐵)
14	1, 3	grpcl 18111	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 + 𝑍) ∈ 𝐵)
15	8, 13, 10, 14	syl3anc 1367	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + 𝑍) ∈ 𝐵)
16		eqid 2821	. . . . . 6 ⊢ (invg‘𝐺) = (invg‘𝐺)
17	1, 16	grpinvcl 18151	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
18	8, 10, 17	syl2anc 586	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
19		simpr 487	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
20	1, 2, 3	ogrpaddlt 30718	. . . 4 ⊢ ((𝐺 ∈ oGrp ∧ ((𝑋 + 𝑍) ∈ 𝐵 ∧ (𝑌 + 𝑍) ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑍) ∈ 𝐵) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg‘𝐺)‘𝑍)) < ((𝑌 + 𝑍) + ((invg‘𝐺)‘𝑍)))
21	6, 12, 15, 18, 19, 20	syl131anc 1379	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg‘𝐺)‘𝑍)) < ((𝑌 + 𝑍) + ((invg‘𝐺)‘𝑍)))
22	1, 3	grpass 18112	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑍) ∈ 𝐵)) → ((𝑋 + 𝑍) + ((invg‘𝐺)‘𝑍)) = (𝑋 + (𝑍 + ((invg‘𝐺)‘𝑍))))
23	8, 9, 10, 18, 22	syl13anc 1368	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg‘𝐺)‘𝑍)) = (𝑋 + (𝑍 + ((invg‘𝐺)‘𝑍))))
24		eqid 2821	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
25	1, 3, 24, 16	grprinv 18153	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → (𝑍 + ((invg‘𝐺)‘𝑍)) = (0g‘𝐺))
26	8, 10, 25	syl2anc 586	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑍 + ((invg‘𝐺)‘𝑍)) = (0g‘𝐺))
27	26	oveq2d 7172	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + (𝑍 + ((invg‘𝐺)‘𝑍))) = (𝑋 + (0g‘𝐺)))
28	1, 3, 24	grprid 18134	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (0g‘𝐺)) = 𝑋)
29	8, 9, 28	syl2anc 586	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + (0g‘𝐺)) = 𝑋)
30	23, 27, 29	3eqtrd 2860	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg‘𝐺)‘𝑍)) = 𝑋)
31	1, 3	grpass 18112	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑍) ∈ 𝐵)) → ((𝑌 + 𝑍) + ((invg‘𝐺)‘𝑍)) = (𝑌 + (𝑍 + ((invg‘𝐺)‘𝑍))))
32	8, 13, 10, 18, 31	syl13anc 1368	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑌 + 𝑍) + ((invg‘𝐺)‘𝑍)) = (𝑌 + (𝑍 + ((invg‘𝐺)‘𝑍))))
33	26	oveq2d 7172	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + (𝑍 + ((invg‘𝐺)‘𝑍))) = (𝑌 + (0g‘𝐺)))
34	1, 3, 24	grprid 18134	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑌 + (0g‘𝐺)) = 𝑌)
35	8, 13, 34	syl2anc 586	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + (0g‘𝐺)) = 𝑌)
36	32, 33, 35	3eqtrd 2860	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑌 + 𝑍) + ((invg‘𝐺)‘𝑍)) = 𝑌)
37	21, 30, 36	3brtr3d 5097	. 2 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋 < 𝑌)
38	5, 37	impbida 799	1 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   class class class wbr 5066  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  +_gcplusg 16565  0_gc0g 16713  ltcplt 17551  Grpcgrp 18103  inv_gcminusg 18104  oGrpcogrp 30699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-riota 7114  df-ov 7159  df-0g 16715  df-plt 17568  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-omnd 30700  df-ogrp 30701

This theorem is referenced by:  ogrpinvlt  30724