Theorem ogrpinv0lt 20244

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 779	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → 𝐺 ∈ oGrp)
2		ogrpgrp 20226	. . . . . 6 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
3	1, 2	syl 18	. . . . 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 19063	. . . . 5 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
7	3, 6	syl 18	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → 0 ∈ 𝐵)
8		simplr 781	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → 𝑋 ∈ 𝐵)
9		ogrpinvlt.2	. . . . . 6 ⊢ 𝐼 = (invg‘𝐺)
10	4, 9	grpinvcl 19085	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵)
11	3, 8, 10	syl2anc 596	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → (𝐼‘𝑋) ∈ 𝐵)
12		simpr 490	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → 0 < 𝑋)
13		ogrpinvlt.1	. . . . 5 ⊢ < = (lt‘𝐺)
14		eqid 2766	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
15	4, 13, 14	ogrpaddlt 20239	. . . 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 1410	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) < (𝑋(+g‘𝐺)(𝐼‘𝑋)))
17	4, 14, 5	grplid 19065	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝐼‘𝑋) ∈ 𝐵) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
18	3, 11, 17	syl2anc 596	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
19	4, 14, 5, 9	grprinv 19088	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
20	3, 8, 19	syl2anc 596	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
21	16, 18, 20	3brtr3d 5147	. 2 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → (𝐼‘𝑋) < 0 )
22		simpll 779	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → 𝐺 ∈ oGrp)
23	22, 2	syl 18	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → 𝐺 ∈ Grp)
24		simplr 781	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → 𝑋 ∈ 𝐵)
25	23, 24, 10	syl2anc 596	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → (𝐼‘𝑋) ∈ 𝐵)
26	22, 2, 6	3syl 19	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → 0 ∈ 𝐵)
27		simpr 490	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → (𝐼‘𝑋) < 0 )
28	4, 13, 14	ogrpaddlt 20239	. . . 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 1410	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) < ( 0 (+g‘𝐺)𝑋))
30	4, 14, 5, 9	grplinv 19087	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) = 0 )
31	23, 24, 30	syl2anc 596	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) = 0 )
32	4, 14, 5	grplid 19065	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑋) = 𝑋)
33	23, 24, 32	syl2anc 596	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → ( 0 (+g‘𝐺)𝑋) = 𝑋)
34	29, 31, 33	3brtr3d 5147	. 2 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → 0 < 𝑋)
35	21, 34	impbida 813	1 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ (𝐼‘𝑋) < 0 ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  +_gcplusg 17335  0_gc0g 17517  ltcplt 18389  Grpcgrp 19031  inv_gcminusg 19032  oGrpcogrp 20221

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-riota 7380  df-ov 7426  df-0g 17519  df-plt 18409  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-grp 19034  df-minusg 19035  df-omnd 20222  df-ogrp 20223

This theorem is used by:  archirngz  33540