Theorem ogrpinv0lt 33027

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 766	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → 𝐺 ∈ oGrp)
2		ogrpgrp 33008	. . . . . 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 18935	. . . . 5 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
7	3, 6	syl 17	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → 0 ∈ 𝐵)
8		simplr 768	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → 𝑋 ∈ 𝐵)
9		ogrpinvlt.2	. . . . . 6 ⊢ 𝐼 = (invg‘𝐺)
10	4, 9	grpinvcl 18957	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵)
11	3, 8, 10	syl2anc 584	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → (𝐼‘𝑋) ∈ 𝐵)
12		simpr 484	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → 0 < 𝑋)
13		ogrpinvlt.1	. . . . 5 ⊢ < = (lt‘𝐺)
14		eqid 2734	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
15	4, 13, 14	ogrpaddlt 33022	. . . 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 1384	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) < (𝑋(+g‘𝐺)(𝐼‘𝑋)))
17	4, 14, 5	grplid 18937	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝐼‘𝑋) ∈ 𝐵) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
18	3, 11, 17	syl2anc 584	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
19	4, 14, 5, 9	grprinv 18960	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
20	3, 8, 19	syl2anc 584	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
21	16, 18, 20	3brtr3d 5148	. 2 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → (𝐼‘𝑋) < 0 )
22		simpll 766	. . . 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 768	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → 𝑋 ∈ 𝐵)
25	23, 24, 10	syl2anc 584	. . . 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 484	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → (𝐼‘𝑋) < 0 )
28	4, 13, 14	ogrpaddlt 33022	. . . 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 1384	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) < ( 0 (+g‘𝐺)𝑋))
30	4, 14, 5, 9	grplinv 18959	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) = 0 )
31	23, 24, 30	syl2anc 584	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) = 0 )
32	4, 14, 5	grplid 18937	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑋) = 𝑋)
33	23, 24, 32	syl2anc 584	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → ( 0 (+g‘𝐺)𝑋) = 𝑋)
34	29, 31, 33	3brtr3d 5148	. 2 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) < 0 ) → 0 < 𝑋)
35	21, 34	impbida 800	1 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ (𝐼‘𝑋) < 0 ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107   class class class wbr 5117  ‘cfv 6528  (class class class)co 7400  Basecbs 17215  +_gcplusg 17258  0_gc0g 17440  ltcplt 18307  Grpcgrp 18903  inv_gcminusg 18904  oGrpcogrp 33003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3357  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-fv 6536  df-riota 7357  df-ov 7403  df-0g 17442  df-plt 18327  df-mgm 18605  df-sgrp 18684  df-mnd 18700  df-grp 18906  df-minusg 18907  df-omnd 33004  df-ogrp 33005

This theorem is referenced by:  archirngz  33124