Theorem ogrpaddltbi 33078

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 33077	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
5	4	3expa 1117	. 2 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
6		simpll 767	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝐺 ∈ oGrp)
7		ogrpgrp 33063	. . . . . 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 1214	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋 ∈ 𝐵)
10		simplr3 1216	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑍 ∈ 𝐵)
11	1, 3	grpcl 18972	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 + 𝑍) ∈ 𝐵)
12	8, 9, 10, 11	syl3anc 1370	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) ∈ 𝐵)
13		simplr2 1215	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑌 ∈ 𝐵)
14	1, 3	grpcl 18972	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 + 𝑍) ∈ 𝐵)
15	8, 13, 10, 14	syl3anc 1370	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + 𝑍) ∈ 𝐵)
16		eqid 2735	. . . . . 6 ⊢ (invg‘𝐺) = (invg‘𝐺)
17	1, 16	grpinvcl 19018	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
18	8, 10, 17	syl2anc 584	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
19		simpr 484	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
20	1, 2, 3	ogrpaddlt 33077	. . . 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 1382	. . 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 18973	. . . . 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 1371	. . . 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 2735	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
25	1, 3, 24, 16	grprinv 19021	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → (𝑍 + ((invg‘𝐺)‘𝑍)) = (0g‘𝐺))
26	8, 10, 25	syl2anc 584	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑍 + ((invg‘𝐺)‘𝑍)) = (0g‘𝐺))
27	26	oveq2d 7447	. . . 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 18999	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (0g‘𝐺)) = 𝑋)
29	8, 9, 28	syl2anc 584	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + (0g‘𝐺)) = 𝑋)
30	23, 27, 29	3eqtrd 2779	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg‘𝐺)‘𝑍)) = 𝑋)
31	1, 3	grpass 18973	. . . . 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 1371	. . . 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 7447	. . . 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 18999	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑌 + (0g‘𝐺)) = 𝑌)
35	8, 13, 34	syl2anc 584	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + (0g‘𝐺)) = 𝑌)
36	32, 33, 35	3eqtrd 2779	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑌 + 𝑍) + ((invg‘𝐺)‘𝑍)) = 𝑌)
37	21, 30, 36	3brtr3d 5179	. 2 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋 < 𝑌)
38	5, 37	impbida 801	1 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  +_gcplusg 17298  0_gc0g 17486  ltcplt 18366  Grpcgrp 18964  inv_gcminusg 18965  oGrpcogrp 33058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-plt 18388  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-omnd 33059  df-ogrp 33060

This theorem is referenced by:  ogrpinvlt  33083