Theorem ogrpinv0lt 20362

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 20344	. . . . . 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 19181	. . . . 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 19203	. . . . 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 2760	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
15	4, 13, 14	ogrpaddlt 20357	. . . 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 19183	. . . 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 19206	. . . 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 5135	. 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 20357	. . . 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 19205	. . . 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 19183	. . . 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 5135	. 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 1569   ∈ wcel 2144   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Basecbs 17392  +_gcplusg 17433  0_gc0g 17615  ltcplt 18487  Grpcgrp 19149  inv_gcminusg 19150  oGrpcogrp 20339

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-riota 7377  df-ov 7423  df-0g 17617  df-plt 18507  df-mgm 18821  df-sgrp 18913  df-mnd 18929  df-grp 19152  df-minusg 19153  df-omnd 20340  df-ogrp 20341

This theorem is used by:  archirngz  33796