Theorem ogrpaddltbi 30769

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 30768	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
5	4	3expa 1115	. 2 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
6		simpll 766	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝐺 ∈ oGrp)
7		ogrpgrp 30754	. . . . . 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 1212	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋 ∈ 𝐵)
10		simplr3 1214	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑍 ∈ 𝐵)
11	1, 3	grpcl 18103	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 + 𝑍) ∈ 𝐵)
12	8, 9, 10, 11	syl3anc 1368	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) ∈ 𝐵)
13		simplr2 1213	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑌 ∈ 𝐵)
14	1, 3	grpcl 18103	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 + 𝑍) ∈ 𝐵)
15	8, 13, 10, 14	syl3anc 1368	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + 𝑍) ∈ 𝐵)
16		eqid 2798	. . . . . 6 ⊢ (invg‘𝐺) = (invg‘𝐺)
17	1, 16	grpinvcl 18143	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
18	8, 10, 17	syl2anc 587	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
19		simpr 488	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
20	1, 2, 3	ogrpaddlt 30768	. . . 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 1380	. . 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 18104	. . . . 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 1369	. . . 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 2798	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
25	1, 3, 24, 16	grprinv 18145	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → (𝑍 + ((invg‘𝐺)‘𝑍)) = (0g‘𝐺))
26	8, 10, 25	syl2anc 587	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑍 + ((invg‘𝐺)‘𝑍)) = (0g‘𝐺))
27	26	oveq2d 7151	. . . 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 18126	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (0g‘𝐺)) = 𝑋)
29	8, 9, 28	syl2anc 587	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + (0g‘𝐺)) = 𝑋)
30	23, 27, 29	3eqtrd 2837	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg‘𝐺)‘𝑍)) = 𝑋)
31	1, 3	grpass 18104	. . . . 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 1369	. . . 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 7151	. . . 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 18126	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑌 + (0g‘𝐺)) = 𝑌)
35	8, 13, 34	syl2anc 587	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + (0g‘𝐺)) = 𝑌)
36	32, 33, 35	3eqtrd 2837	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑌 + 𝑍) + ((invg‘𝐺)‘𝑍)) = 𝑌)
37	21, 30, 36	3brtr3d 5061	. 2 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋 < 𝑌)
38	5, 37	impbida 800	1 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +_gcplusg 16557  0_gc0g 16705  ltcplt 17543  Grpcgrp 18095  inv_gcminusg 18096  oGrpcogrp 30749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-riota 7093  df-ov 7138  df-0g 16707  df-plt 17560  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-omnd 30750  df-ogrp 30751

This theorem is referenced by:  ogrpinvlt  30774