| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ngpgrp |
⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ Grp ) |
| 2 |
1
|
adantr |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) → 𝐺 ∈ Grp ) |
| 3 |
|
ngpms |
⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp ) |
| 4 |
3
|
adantr |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) → 𝐺 ∈ MetSp ) |
| 5 |
|
mstps |
⊢ ( 𝐺 ∈ MetSp → 𝐺 ∈ TopSp ) |
| 6 |
4 5
|
syl |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) → 𝐺 ∈ TopSp ) |
| 7 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
| 8 |
|
eqid |
⊢ ( -g ‘ 𝐺 ) = ( -g ‘ 𝐺 ) |
| 9 |
7 8
|
grpsubf |
⊢ ( 𝐺 ∈ Grp → ( -g ‘ 𝐺 ) : ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ⟶ ( Base ‘ 𝐺 ) ) |
| 10 |
2 9
|
syl |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) → ( -g ‘ 𝐺 ) : ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ⟶ ( Base ‘ 𝐺 ) ) |
| 11 |
|
rphalfcl |
⊢ ( 𝑧 ∈ ℝ+ → ( 𝑧 / 2 ) ∈ ℝ+ ) |
| 12 |
|
simplll |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ) |
| 13 |
12 4
|
syl |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → 𝐺 ∈ MetSp ) |
| 14 |
|
simpllr |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) |
| 15 |
14
|
simpld |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → 𝑥 ∈ ( Base ‘ 𝐺 ) ) |
| 16 |
|
simprl |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → 𝑢 ∈ ( Base ‘ 𝐺 ) ) |
| 17 |
|
eqid |
⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 ) |
| 18 |
7 17
|
mscl |
⊢ ( ( 𝐺 ∈ MetSp ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑢 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) ∈ ℝ ) |
| 19 |
13 15 16 18
|
syl3anc |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) ∈ ℝ ) |
| 20 |
14
|
simprd |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → 𝑦 ∈ ( Base ‘ 𝐺 ) ) |
| 21 |
|
simprr |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → 𝑣 ∈ ( Base ‘ 𝐺 ) ) |
| 22 |
7 17
|
mscl |
⊢ ( ( 𝐺 ∈ MetSp ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ∈ ℝ ) |
| 23 |
13 20 21 22
|
syl3anc |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ∈ ℝ ) |
| 24 |
|
rpre |
⊢ ( 𝑧 ∈ ℝ+ → 𝑧 ∈ ℝ ) |
| 25 |
24
|
ad2antlr |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → 𝑧 ∈ ℝ ) |
| 26 |
|
lt2halves |
⊢ ( ( ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) ∈ ℝ ∧ ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ∈ ℝ ∧ 𝑧 ∈ ℝ ) → ( ( ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) < ( 𝑧 / 2 ) ∧ ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) < ( 𝑧 / 2 ) ) → ( ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) + ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) |
| 27 |
19 23 25 26
|
syl3anc |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) < ( 𝑧 / 2 ) ∧ ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) < ( 𝑧 / 2 ) ) → ( ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) + ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) |
| 28 |
12 2
|
syl |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → 𝐺 ∈ Grp ) |
| 29 |
7 8
|
grpsubcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ∈ ( Base ‘ 𝐺 ) ) |
| 30 |
28 15 20 29
|
syl3anc |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ∈ ( Base ‘ 𝐺 ) ) |
| 31 |
7 8
|
grpsubcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ∈ ( Base ‘ 𝐺 ) ) |
| 32 |
28 16 21 31
|
syl3anc |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ∈ ( Base ‘ 𝐺 ) ) |
| 33 |
7 8
|
grpsubcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑢 ( -g ‘ 𝐺 ) 𝑦 ) ∈ ( Base ‘ 𝐺 ) ) |
| 34 |
28 16 20 33
|
syl3anc |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑢 ( -g ‘ 𝐺 ) 𝑦 ) ∈ ( Base ‘ 𝐺 ) ) |
| 35 |
7 17
|
mstri |
⊢ ( ( 𝐺 ∈ MetSp ∧ ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑢 ( -g ‘ 𝐺 ) 𝑦 ) ∈ ( Base ‘ 𝐺 ) ) ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) ≤ ( ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑦 ) ) + ( ( 𝑢 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) ) ) |
| 36 |
13 30 32 34 35
|
syl13anc |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) ≤ ( ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑦 ) ) + ( ( 𝑢 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) ) ) |
| 37 |
12
|
simpld |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → 𝐺 ∈ NrmGrp ) |
| 38 |
7 8 17
|
ngpsubcan |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑦 ) ) = ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) ) |
| 39 |
37 15 16 20 38
|
syl13anc |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑦 ) ) = ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) ) |
| 40 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
| 41 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
| 42 |
7 40 41 8
|
grpsubval |
⊢ ( ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑢 ( -g ‘ 𝐺 ) 𝑦 ) = ( 𝑢 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ) ) |
| 43 |
16 20 42
|
syl2anc |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑢 ( -g ‘ 𝐺 ) 𝑦 ) = ( 𝑢 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ) ) |
| 44 |
7 40 41 8
|
grpsubval |
⊢ ( ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) = ( 𝑢 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑣 ) ) ) |
| 45 |
44
|
adantl |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) = ( 𝑢 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑣 ) ) ) |
| 46 |
43 45
|
oveq12d |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( 𝑢 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) = ( ( 𝑢 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ) ( dist ‘ 𝐺 ) ( 𝑢 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑣 ) ) ) ) |
| 47 |
7 41
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ∈ ( Base ‘ 𝐺 ) ) |
| 48 |
28 20 47
|
syl2anc |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ∈ ( Base ‘ 𝐺 ) ) |
| 49 |
7 41
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑣 ) ∈ ( Base ‘ 𝐺 ) ) |
| 50 |
28 21 49
|
syl2anc |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑣 ) ∈ ( Base ‘ 𝐺 ) ) |
| 51 |
7 40 17
|
ngplcan |
⊢ ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ∈ ( Base ‘ 𝐺 ) ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑣 ) ∈ ( Base ‘ 𝐺 ) ∧ 𝑢 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( 𝑢 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ) ( dist ‘ 𝐺 ) ( 𝑢 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑣 ) ) ) = ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( dist ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑣 ) ) ) |
| 52 |
12 48 50 16 51
|
syl13anc |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( 𝑢 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ) ( dist ‘ 𝐺 ) ( 𝑢 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑣 ) ) ) = ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( dist ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑣 ) ) ) |
| 53 |
7 41 17
|
ngpinvds |
⊢ ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑦 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( dist ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑣 ) ) = ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ) |
| 54 |
12 20 21 53
|
syl12anc |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( dist ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑣 ) ) = ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ) |
| 55 |
46 52 54
|
3eqtrd |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( 𝑢 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) = ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ) |
| 56 |
39 55
|
oveq12d |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑦 ) ) + ( ( 𝑢 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) ) = ( ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) + ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ) ) |
| 57 |
36 56
|
breqtrd |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) ≤ ( ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) + ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ) ) |
| 58 |
7 17
|
mscl |
⊢ ( ( 𝐺 ∈ MetSp ∧ ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ∈ ( Base ‘ 𝐺 ) ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) ∈ ℝ ) |
| 59 |
13 30 32 58
|
syl3anc |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) ∈ ℝ ) |
| 60 |
19 23
|
readdcld |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) + ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ) ∈ ℝ ) |
| 61 |
|
lelttr |
⊢ ( ( ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) ∈ ℝ ∧ ( ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) + ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ) ∈ ℝ ∧ 𝑧 ∈ ℝ ) → ( ( ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) ≤ ( ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) + ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ) ∧ ( ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) + ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) |
| 62 |
59 60 25 61
|
syl3anc |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) ≤ ( ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) + ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ) ∧ ( ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) + ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) |
| 63 |
57 62
|
mpand |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) + ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ) < 𝑧 → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) |
| 64 |
27 63
|
syld |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) < ( 𝑧 / 2 ) ∧ ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) < ( 𝑧 / 2 ) ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) |
| 65 |
15 16
|
ovresd |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) = ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) ) |
| 66 |
65
|
breq1d |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < ( 𝑧 / 2 ) ↔ ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) < ( 𝑧 / 2 ) ) ) |
| 67 |
20 21
|
ovresd |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) = ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) ) |
| 68 |
67
|
breq1d |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < ( 𝑧 / 2 ) ↔ ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) < ( 𝑧 / 2 ) ) ) |
| 69 |
66 68
|
anbi12d |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < ( 𝑧 / 2 ) ∧ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < ( 𝑧 / 2 ) ) ↔ ( ( 𝑥 ( dist ‘ 𝐺 ) 𝑢 ) < ( 𝑧 / 2 ) ∧ ( 𝑦 ( dist ‘ 𝐺 ) 𝑣 ) < ( 𝑧 / 2 ) ) ) ) |
| 70 |
30 32
|
ovresd |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) = ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) ) |
| 71 |
70
|
breq1d |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ↔ ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( dist ‘ 𝐺 ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) |
| 72 |
64 69 71
|
3imtr4d |
⊢ ( ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) ∧ ( 𝑢 ∈ ( Base ‘ 𝐺 ) ∧ 𝑣 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < ( 𝑧 / 2 ) ∧ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < ( 𝑧 / 2 ) ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) |
| 73 |
72
|
ralrimivva |
⊢ ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) → ∀ 𝑢 ∈ ( Base ‘ 𝐺 ) ∀ 𝑣 ∈ ( Base ‘ 𝐺 ) ( ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < ( 𝑧 / 2 ) ∧ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < ( 𝑧 / 2 ) ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) |
| 74 |
|
breq2 |
⊢ ( 𝑟 = ( 𝑧 / 2 ) → ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < 𝑟 ↔ ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < ( 𝑧 / 2 ) ) ) |
| 75 |
|
breq2 |
⊢ ( 𝑟 = ( 𝑧 / 2 ) → ( ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < 𝑟 ↔ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < ( 𝑧 / 2 ) ) ) |
| 76 |
74 75
|
anbi12d |
⊢ ( 𝑟 = ( 𝑧 / 2 ) → ( ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < 𝑟 ∧ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < 𝑟 ) ↔ ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < ( 𝑧 / 2 ) ∧ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < ( 𝑧 / 2 ) ) ) ) |
| 77 |
76
|
imbi1d |
⊢ ( 𝑟 = ( 𝑧 / 2 ) → ( ( ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < 𝑟 ∧ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < 𝑟 ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ↔ ( ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < ( 𝑧 / 2 ) ∧ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < ( 𝑧 / 2 ) ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) ) |
| 78 |
77
|
2ralbidv |
⊢ ( 𝑟 = ( 𝑧 / 2 ) → ( ∀ 𝑢 ∈ ( Base ‘ 𝐺 ) ∀ 𝑣 ∈ ( Base ‘ 𝐺 ) ( ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < 𝑟 ∧ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < 𝑟 ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ↔ ∀ 𝑢 ∈ ( Base ‘ 𝐺 ) ∀ 𝑣 ∈ ( Base ‘ 𝐺 ) ( ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < ( 𝑧 / 2 ) ∧ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < ( 𝑧 / 2 ) ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) ) |
| 79 |
78
|
rspcev |
⊢ ( ( ( 𝑧 / 2 ) ∈ ℝ+ ∧ ∀ 𝑢 ∈ ( Base ‘ 𝐺 ) ∀ 𝑣 ∈ ( Base ‘ 𝐺 ) ( ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < ( 𝑧 / 2 ) ∧ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < ( 𝑧 / 2 ) ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) → ∃ 𝑟 ∈ ℝ+ ∀ 𝑢 ∈ ( Base ‘ 𝐺 ) ∀ 𝑣 ∈ ( Base ‘ 𝐺 ) ( ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < 𝑟 ∧ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < 𝑟 ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) |
| 80 |
11 73 79
|
syl2an2 |
⊢ ( ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) ∧ 𝑧 ∈ ℝ+ ) → ∃ 𝑟 ∈ ℝ+ ∀ 𝑢 ∈ ( Base ‘ 𝐺 ) ∀ 𝑣 ∈ ( Base ‘ 𝐺 ) ( ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < 𝑟 ∧ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < 𝑟 ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) |
| 81 |
80
|
ralrimiva |
⊢ ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) → ∀ 𝑧 ∈ ℝ+ ∃ 𝑟 ∈ ℝ+ ∀ 𝑢 ∈ ( Base ‘ 𝐺 ) ∀ 𝑣 ∈ ( Base ‘ 𝐺 ) ( ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < 𝑟 ∧ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < 𝑟 ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) |
| 82 |
81
|
ralrimivva |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) → ∀ 𝑥 ∈ ( Base ‘ 𝐺 ) ∀ 𝑦 ∈ ( Base ‘ 𝐺 ) ∀ 𝑧 ∈ ℝ+ ∃ 𝑟 ∈ ℝ+ ∀ 𝑢 ∈ ( Base ‘ 𝐺 ) ∀ 𝑣 ∈ ( Base ‘ 𝐺 ) ( ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < 𝑟 ∧ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < 𝑟 ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) |
| 83 |
|
msxms |
⊢ ( 𝐺 ∈ MetSp → 𝐺 ∈ ∞MetSp ) |
| 84 |
|
eqid |
⊢ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) = ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) |
| 85 |
7 84
|
xmsxmet |
⊢ ( 𝐺 ∈ ∞MetSp → ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝐺 ) ) ) |
| 86 |
4 83 85
|
3syl |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) → ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝐺 ) ) ) |
| 87 |
|
eqid |
⊢ ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) = ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) |
| 88 |
87 87 87
|
txmetcn |
⊢ ( ( ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝐺 ) ) ∧ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝐺 ) ) ∧ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝐺 ) ) ) → ( ( -g ‘ 𝐺 ) ∈ ( ( ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) ×t ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) ) Cn ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) ) ↔ ( ( -g ‘ 𝐺 ) : ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ⟶ ( Base ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐺 ) ∀ 𝑦 ∈ ( Base ‘ 𝐺 ) ∀ 𝑧 ∈ ℝ+ ∃ 𝑟 ∈ ℝ+ ∀ 𝑢 ∈ ( Base ‘ 𝐺 ) ∀ 𝑣 ∈ ( Base ‘ 𝐺 ) ( ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < 𝑟 ∧ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < 𝑟 ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) ) ) |
| 89 |
86 86 86 88
|
syl3anc |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) → ( ( -g ‘ 𝐺 ) ∈ ( ( ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) ×t ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) ) Cn ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) ) ↔ ( ( -g ‘ 𝐺 ) : ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ⟶ ( Base ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐺 ) ∀ 𝑦 ∈ ( Base ‘ 𝐺 ) ∀ 𝑧 ∈ ℝ+ ∃ 𝑟 ∈ ℝ+ ∀ 𝑢 ∈ ( Base ‘ 𝐺 ) ∀ 𝑣 ∈ ( Base ‘ 𝐺 ) ( ( ( 𝑥 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑢 ) < 𝑟 ∧ ( 𝑦 ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) 𝑣 ) < 𝑟 ) → ( ( 𝑥 ( -g ‘ 𝐺 ) 𝑦 ) ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ( 𝑢 ( -g ‘ 𝐺 ) 𝑣 ) ) < 𝑧 ) ) ) ) |
| 90 |
10 82 89
|
mpbir2and |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) → ( -g ‘ 𝐺 ) ∈ ( ( ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) ×t ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) ) Cn ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) ) ) |
| 91 |
|
eqid |
⊢ ( TopOpen ‘ 𝐺 ) = ( TopOpen ‘ 𝐺 ) |
| 92 |
91 7 84
|
mstopn |
⊢ ( 𝐺 ∈ MetSp → ( TopOpen ‘ 𝐺 ) = ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) ) |
| 93 |
4 92
|
syl |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) → ( TopOpen ‘ 𝐺 ) = ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) ) |
| 94 |
93 93
|
oveq12d |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) → ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) = ( ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) ×t ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) ) ) |
| 95 |
94 93
|
oveq12d |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) → ( ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) Cn ( TopOpen ‘ 𝐺 ) ) = ( ( ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) ×t ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) ) Cn ( MetOpen ‘ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ) ) ) |
| 96 |
90 95
|
eleqtrrd |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) → ( -g ‘ 𝐺 ) ∈ ( ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) Cn ( TopOpen ‘ 𝐺 ) ) ) |
| 97 |
91 8
|
istgp2 |
⊢ ( 𝐺 ∈ TopGrp ↔ ( 𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ( -g ‘ 𝐺 ) ∈ ( ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) Cn ( TopOpen ‘ 𝐺 ) ) ) ) |
| 98 |
2 6 96 97
|
syl3anbrc |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) → 𝐺 ∈ TopGrp ) |