Theorem ogrpaddltbi 30239

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 30238	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
5	4	3expa 1148	. 2 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
6		simpll 784	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝐺 ∈ oGrp)
7		ogrpgrp 30223	. . . . . 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 1276	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋 ∈ 𝐵)
10		simplr3 1280	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑍 ∈ 𝐵)
11	1, 3	grpcl 17750	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 + 𝑍) ∈ 𝐵)
12	8, 9, 10, 11	syl3anc 1491	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) ∈ 𝐵)
13		simplr2 1278	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑌 ∈ 𝐵)
14	1, 3	grpcl 17750	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 + 𝑍) ∈ 𝐵)
15	8, 13, 10, 14	syl3anc 1491	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + 𝑍) ∈ 𝐵)
16		eqid 2803	. . . . . 6 ⊢ (invg‘𝐺) = (invg‘𝐺)
17	1, 16	grpinvcl 17787	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
18	8, 10, 17	syl2anc 580	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
19		simpr 478	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
20	1, 2, 3	ogrpaddlt 30238	. . . 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 1503	. . 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 17751	. . . . 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 1492	. . . 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 2803	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
25	1, 3, 24, 16	grprinv 17789	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → (𝑍 + ((invg‘𝐺)‘𝑍)) = (0g‘𝐺))
26	8, 10, 25	syl2anc 580	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑍 + ((invg‘𝐺)‘𝑍)) = (0g‘𝐺))
27	26	oveq2d 6898	. . . 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 17773	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (0g‘𝐺)) = 𝑋)
29	8, 9, 28	syl2anc 580	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + (0g‘𝐺)) = 𝑋)
30	23, 27, 29	3eqtrd 2841	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg‘𝐺)‘𝑍)) = 𝑋)
31	1, 3	grpass 17751	. . . . 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 1492	. . . 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 6898	. . . 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 17773	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑌 + (0g‘𝐺)) = 𝑌)
35	8, 13, 34	syl2anc 580	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + (0g‘𝐺)) = 𝑌)
36	32, 33, 35	3eqtrd 2841	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑌 + 𝑍) + ((invg‘𝐺)‘𝑍)) = 𝑌)
37	21, 30, 36	3brtr3d 4878	. 2 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋 < 𝑌)
38	5, 37	impbida 836	1 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157   class class class wbr 4847  ‘cfv 6105  (class class class)co 6882  Basecbs 16188  +_gcplusg 16271  0_gc0g 16419  ltcplt 17260  Grpcgrp 17742  inv_gcminusg 17743  oGrpcogrp 30218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-rep 4968  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ne 2976  df-ral 3098  df-rex 3099  df-reu 3100  df-rmo 3101  df-rab 3102  df-v 3391  df-sbc 3638  df-csb 3733  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-nul 4120  df-if 4282  df-sn 4373  df-pr 4375  df-op 4379  df-uni 4633  df-iun 4716  df-br 4848  df-opab 4910  df-mpt 4927  df-id 5224  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-rn 5327  df-res 5328  df-ima 5329  df-iota 6068  df-fun 6107  df-fn 6108  df-f 6109  df-f1 6110  df-fo 6111  df-f1o 6112  df-fv 6113  df-riota 6843  df-ov 6885  df-0g 16421  df-plt 17277  df-mgm 17561  df-sgrp 17603  df-mnd 17614  df-grp 17745  df-minusg 17746  df-omnd 30219  df-ogrp 30220

This theorem is referenced by:  ogrpinvlt  30244