Theorem ogrpinv0lt 32845

Description: In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)

Hypotheses

Ref	Expression
ogrpinvlt.0	⊢ 𝐵 = (Base‘𝐺)
ogrpinvlt.1	⊢ < = (lt‘𝐺)
ogrpinvlt.2	⊢ 𝐼 = (invg‘𝐺)
ogrpinv0lt.3	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
ogrpinv0lt	⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ (𝐼‘𝑋) < 0 ))

Proof of Theorem ogrpinv0lt

Step	Hyp	Ref	Expression
1		simpll 765	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → 𝐺 ∈ oGrp)
2		ogrpgrp 32826	. . . . . 6 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
3	1, 2	syl 17	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → 𝐺 ∈ Grp)
4		ogrpinvlt.0	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
5		ogrpinv0lt.3	. . . . . 6 ⊢ 0 = (0g‘𝐺)
6	4, 5	grpidcl 18924	. . . . 5 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
7	3, 6	syl 17	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → 0 ∈ 𝐵)
8		simplr 767	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → 𝑋 ∈ 𝐵)
9		ogrpinvlt.2	. . . . . 6 ⊢ 𝐼 = (invg‘𝐺)
10	4, 9	grpinvcl 18946	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵)
11	3, 8, 10	syl2anc 582	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → (𝐼‘𝑋) ∈ 𝐵)
12		simpr 483	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → 0 < 𝑋)
13		ogrpinvlt.1	. . . . 5 ⊢ < = (lt‘𝐺)
14		eqid 2725	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
15	4, 13, 14	ogrpaddlt 32840	. . . 4 ⊢ ((𝐺 ∈ oGrp ∧ ( 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ (𝐼‘𝑋) ∈ 𝐵) ∧ 0 < 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) < (𝑋(+g‘𝐺)(𝐼‘𝑋)))
16	1, 7, 8, 11, 12, 15	syl131anc 1380	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) < (𝑋(+g‘𝐺)(𝐼‘𝑋)))
17	4, 14, 5	grplid 18926	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝐼‘𝑋) ∈ 𝐵) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
18	3, 11, 17	syl2anc 582	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
19	4, 14, 5, 9	grprinv 18949	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
20	3, 8, 19	syl2anc 582	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
21	16, 18, 20	3brtr3d 5174	. 2 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → (𝐼‘𝑋) < 0 )
22		simpll 765	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → 𝐺 ∈ oGrp)
23	22, 2	syl 17	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → 𝐺 ∈ Grp)
24		simplr 767	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → 𝑋 ∈ 𝐵)
25	23, 24, 10	syl2anc 582	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → (𝐼‘𝑋) ∈ 𝐵)
26	22, 2, 6	3syl 18	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → 0 ∈ 𝐵)
27		simpr 483	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → (𝐼‘𝑋) < 0 )
28	4, 13, 14	ogrpaddlt 32840	. . . 4 ⊢ ((𝐺 ∈ oGrp ∧ ((𝐼‘𝑋) ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) < ( 0 (+g‘𝐺)𝑋))
29	22, 25, 26, 24, 27, 28	syl131anc 1380	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) < ( 0 (+g‘𝐺)𝑋))
30	4, 14, 5, 9	grplinv 18948	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) = 0 )
31	23, 24, 30	syl2anc 582	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) = 0 )
32	4, 14, 5	grplid 18926	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑋) = 𝑋)
33	23, 24, 32	syl2anc 582	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → ( 0 (+g‘𝐺)𝑋) = 𝑋)
34	29, 31, 33	3brtr3d 5174	. 2 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → 0 < 𝑋)
35	21, 34	impbida 799	1 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ (𝐼‘𝑋) < 0 ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   class class class wbr 5143  ‘cfv 6542  (class class class)co 7415  Basecbs 17177  +_gcplusg 17230  0_gc0g 17418  ltcplt 18297  Grpcgrp 18892  inv_gcminusg 18893  oGrpcogrp 32821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-riota 7371  df-ov 7418  df-0g 17420  df-plt 18319  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-grp 18895  df-minusg 18896  df-omnd 32822  df-ogrp 32823

This theorem is referenced by:  archirngz  32940