Metamath Proof Explorer

Theorem ogrpinvlt

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

Ref Expression
Hypotheses ogrpinvlt.0 ⊢ 𝐵 = ( Base ‘ 𝐺 )
ogrpinvlt.1 ⊢ < = ( lt ‘ 𝐺 )
ogrpinvlt.2 ⊢ 𝐼 = ( invg ‘ 𝐺 )
Assertion ogrpinvlt ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 ↔ ( 𝐼 ‘ 𝑌 ) < ( 𝐼 ‘ 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 ogrpinvlt.0 ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 ogrpinvlt.1 ⊢ < = ( lt ‘ 𝐺 )
3 ogrpinvlt.2 ⊢ 𝐼 = ( invg ‘ 𝐺 )
4 simp1l ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐺 ∈ oGrp )
5 simp2 ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
6 simp3 ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 )
7 ogrpgrp ⊢ ( 𝐺 ∈ oGrp → 𝐺 ∈ Grp )
8 4 7 syl ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐺 ∈ Grp )
9 1 3 grpinvcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑌 ) ∈ 𝐵 )
10 8 6 9 syl2anc ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑌 ) ∈ 𝐵 )
11 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
12 1 2 11 ogrpaddltbi ⊢ ( ( 𝐺 ∈ oGrp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( 𝐼 ‘ 𝑌 ) ∈ 𝐵 ) ) → ( 𝑋 < 𝑌 ↔ ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) < ( 𝑌 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) ) )
13 4 5 6 10 12 syl13anc ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 ↔ ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) < ( 𝑌 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) ) )
14 eqid ⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
15 1 11 14 3 grprinv ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( 𝑌 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) = ( 0g ‘ 𝐺 ) )
16 8 6 15 syl2anc ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑌 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) = ( 0g ‘ 𝐺 ) )
17 16 breq2d ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) < ( 𝑌 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) ↔ ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) < ( 0g ‘ 𝐺 ) ) )
18 simp1r ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( oppg ‘ 𝐺 ) ∈ oGrp )
19 1 11 grpcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ ( 𝐼 ‘ 𝑌 ) ∈ 𝐵 ) → ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) ∈ 𝐵 )
20 8 5 10 19 syl3anc ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) ∈ 𝐵 )
21 1 14 grpidcl ⊢ ( 𝐺 ∈ Grp → ( 0g ‘ 𝐺 ) ∈ 𝐵 )
22 4 7 21 3syl ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 0g ‘ 𝐺 ) ∈ 𝐵 )
23 1 3 grpinvcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 )
24 8 5 23 syl2anc ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 )
25 1 2 11 4 18 20 22 24 ogrpaddltrbid ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) < ( 0g ‘ 𝐺 ) ↔ ( ( 𝐼 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) ) < ( ( 𝐼 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ) )
26 13 17 25 3bitrd ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 ↔ ( ( 𝐼 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) ) < ( ( 𝐼 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ) )
27 1 11 14 3 grplinv ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑋 ) ( +g ‘ 𝐺 ) 𝑋 ) = ( 0g ‘ 𝐺 ) )
28 8 5 27 syl2anc ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑋 ) ( +g ‘ 𝐺 ) 𝑋 ) = ( 0g ‘ 𝐺 ) )
29 28 oveq1d ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( 𝐼 ‘ 𝑋 ) ( +g ‘ 𝐺 ) 𝑋 ) ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) = ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) )
30 1 11 grpass ⊢ ( ( 𝐺 ∈ Grp ∧ ( ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ ( 𝐼 ‘ 𝑌 ) ∈ 𝐵 ) ) → ( ( ( 𝐼 ‘ 𝑋 ) ( +g ‘ 𝐺 ) 𝑋 ) ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) ) )
31 8 24 5 10 30 syl13anc ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( 𝐼 ‘ 𝑋 ) ( +g ‘ 𝐺 ) 𝑋 ) ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) ) )
32 1 11 14 grplid ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝐼 ‘ 𝑌 ) ∈ 𝐵 ) → ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) = ( 𝐼 ‘ 𝑌 ) )
33 8 10 32 syl2anc ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) = ( 𝐼 ‘ 𝑌 ) )
34 29 31 33 3eqtr3d ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) ) = ( 𝐼 ‘ 𝑌 ) )
35 1 11 14 grprid ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) = ( 𝐼 ‘ 𝑋 ) )
36 8 24 35 syl2anc ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) = ( 𝐼 ‘ 𝑋 ) )
37 34 36 breq12d ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( 𝐼 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐼 ‘ 𝑌 ) ) ) < ( ( 𝐼 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ↔ ( 𝐼 ‘ 𝑌 ) < ( 𝐼 ‘ 𝑋 ) ) )
38 26 37 bitrd ⊢ ( ( ( 𝐺 ∈ oGrp ∧ ( oppg ‘ 𝐺 ) ∈ oGrp ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 ↔ ( 𝐼 ‘ 𝑌 ) < ( 𝐼 ‘ 𝑋 ) ) )