Theorem ogrpaddltbi 31344

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 31343	. . 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 764	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝐺 ∈ oGrp)
7		ogrpgrp 31329	. . . . . 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 18585	. . . . 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 18585	. . . . 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 2738	. . . . . 6 ⊢ (invg‘𝐺) = (invg‘𝐺)
17	1, 16	grpinvcl 18627	. . . . 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 485	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
20	1, 2, 3	ogrpaddlt 31343	. . . 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 18586	. . . . 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 2738	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
25	1, 3, 24, 16	grprinv 18629	. . . . . 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 7291	. . . 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 18610	. . . . 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 2782	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg‘𝐺)‘𝑍)) = 𝑋)
31	1, 3	grpass 18586	. . . . 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 7291	. . . 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 18610	. . . . 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 2782	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑌 + 𝑍) + ((invg‘𝐺)‘𝑍)) = 𝑌)
37	21, 30, 36	3brtr3d 5105	. 2 ⊢ (((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋 < 𝑌)
38	5, 37	impbida 798	1 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  0_gc0g 17150  ltcplt 18026  Grpcgrp 18577  inv_gcminusg 18578  oGrpcogrp 31324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-riota 7232  df-ov 7278  df-0g 17152  df-plt 18048  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-minusg 18581  df-omnd 31325  df-ogrp 31326

This theorem is referenced by:  ogrpinvlt  31349