Step | Hyp | Ref
| Expression |
1 | | ngpgrp 23766 |
. . 3
⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) |
2 | 1 | adantr 481 |
. 2
⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ Grp) |
3 | | ngpms 23767 |
. . . 4
⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) |
4 | 3 | adantr 481 |
. . 3
⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ MetSp) |
5 | | mstps 23619 |
. . 3
⊢ (𝐺 ∈ MetSp → 𝐺 ∈ TopSp) |
6 | 4, 5 | syl 17 |
. 2
⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ TopSp) |
7 | | eqid 2740 |
. . . . . 6
⊢
(Base‘𝐺) =
(Base‘𝐺) |
8 | | eqid 2740 |
. . . . . 6
⊢
(-g‘𝐺) = (-g‘𝐺) |
9 | 7, 8 | grpsubf 18665 |
. . . . 5
⊢ (𝐺 ∈ Grp →
(-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺)) |
10 | 2, 9 | syl 17 |
. . . 4
⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) →
(-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺)) |
11 | | rphalfcl 12768 |
. . . . . . 7
⊢ (𝑧 ∈ ℝ+
→ (𝑧 / 2) ∈
ℝ+) |
12 | | simplll 772 |
. . . . . . . . . . . . 13
⊢
(((((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 17 |
. . . . . . . . . . . 12
⊢
(((((𝐺 ∈ NrmGrp
∧ 𝐺 ∈ Abel) ∧
(𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝐺 ∈ MetSp) |
14 | | simpllr 773 |
. . . . . . . . . . . . 13
⊢
(((((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 495 |
. . . . . . . . . . . 12
⊢
(((((𝐺 ∈ NrmGrp
∧ 𝐺 ∈ Abel) ∧
(𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑥 ∈ (Base‘𝐺)) |
16 | | simprl 768 |
. . . . . . . . . . . 12
⊢
(((((𝐺 ∈ NrmGrp
∧ 𝐺 ∈ Abel) ∧
(𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑢 ∈ (Base‘𝐺)) |
17 | | eqid 2740 |
. . . . . . . . . . . . 13
⊢
(dist‘𝐺) =
(dist‘𝐺) |
18 | 7, 17 | mscl 23625 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ MetSp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → (𝑥(dist‘𝐺)𝑢) ∈ ℝ) |
19 | 13, 15, 16, 18 | syl3anc 1370 |
. . . . . . . . . . 11
⊢
(((((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 496 |
. . . . . . . . . . . 12
⊢
(((((𝐺 ∈ NrmGrp
∧ 𝐺 ∈ Abel) ∧
(𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑦 ∈ (Base‘𝐺)) |
21 | | simprr 770 |
. . . . . . . . . . . 12
⊢
(((((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 23625 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ MetSp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑦(dist‘𝐺)𝑣) ∈ ℝ) |
23 | 13, 20, 21, 22 | syl3anc 1370 |
. . . . . . . . . . 11
⊢
(((((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 12749 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ℝ+
→ 𝑧 ∈
ℝ) |
25 | 24 | ad2antlr 724 |
. . . . . . . . . . 11
⊢
(((((𝐺 ∈ NrmGrp
∧ 𝐺 ∈ Abel) ∧
(𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑧 ∈ ℝ) |
26 | | lt2halves 12219 |
. . . . . . . . . . 11
⊢ (((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 1370 |
. . . . . . . . . 10
⊢
(((((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 17 |
. . . . . . . . . . . . . 14
⊢
(((((𝐺 ∈ NrmGrp
∧ 𝐺 ∈ Abel) ∧
(𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝐺 ∈ Grp) |
29 | 7, 8 | grpsubcl 18666 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(-g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
30 | 28, 15, 20, 29 | syl3anc 1370 |
. . . . . . . . . . . . 13
⊢
(((((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 18666 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑢(-g‘𝐺)𝑣) ∈ (Base‘𝐺)) |
32 | 28, 16, 21, 31 | syl3anc 1370 |
. . . . . . . . . . . . 13
⊢
(((((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 18666 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑢(-g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
34 | 28, 16, 20, 33 | syl3anc 1370 |
. . . . . . . . . . . . 13
⊢
(((((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 23633 |
. . . . . . . . . . . . 13
⊢ ((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 1371 |
. . . . . . . . . . . 12
⊢
(((((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 495 |
. . . . . . . . . . . . . 14
⊢
(((((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 23781 |
. . . . . . . . . . . . . 14
⊢ ((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 1371 |
. . . . . . . . . . . . 13
⊢
(((((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 2740 |
. . . . . . . . . . . . . . . . 17
⊢
(+g‘𝐺) = (+g‘𝐺) |
41 | | eqid 2740 |
. . . . . . . . . . . . . . . . 17
⊢
(invg‘𝐺) = (invg‘𝐺) |
42 | 7, 40, 41, 8 | grpsubval 18636 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑢 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑢(-g‘𝐺)𝑦) = (𝑢(+g‘𝐺)((invg‘𝐺)‘𝑦))) |
43 | 16, 20, 42 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢
(((((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 18636 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑢(-g‘𝐺)𝑣) = (𝑢(+g‘𝐺)((invg‘𝐺)‘𝑣))) |
45 | 44 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢
(((((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 7290 |
. . . . . . . . . . . . . 14
⊢
(((((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 18638 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺)) →
((invg‘𝐺)‘𝑦) ∈ (Base‘𝐺)) |
48 | 28, 20, 47 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢
(((((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 18638 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺 ∈ Grp ∧ 𝑣 ∈ (Base‘𝐺)) →
((invg‘𝐺)‘𝑣) ∈ (Base‘𝐺)) |
50 | 28, 21, 49 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢
(((((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 23778 |
. . . . . . . . . . . . . . 15
⊢ (((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 1371 |
. . . . . . . . . . . . . 14
⊢
(((((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 23780 |
. . . . . . . . . . . . . . 15
⊢ (((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 834 |
. . . . . . . . . . . . . 14
⊢
(((((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 2784 |
. . . . . . . . . . . . 13
⊢
(((((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 7290 |
. . . . . . . . . . . 12
⊢
(((((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 5105 |
. . . . . . . . . . 11
⊢
(((((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 23625 |
. . . . . . . . . . . . 13
⊢ ((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 1370 |
. . . . . . . . . . . 12
⊢
(((((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 11015 |
. . . . . . . . . . . 12
⊢
(((((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 11076 |
. . . . . . . . . . . 12
⊢ ((((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 1370 |
. . . . . . . . . . 11
⊢
(((((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 692 |
. . . . . . . . . 10
⊢
(((((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 47 |
. . . . . . . . 9
⊢
(((((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 7434 |
. . . . . . . . . . 11
⊢
(((((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 5089 |
. . . . . . . . . 10
⊢
(((((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 7434 |
. . . . . . . . . . 11
⊢
(((((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 5089 |
. . . . . . . . . 10
⊢
(((((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 631 |
. . . . . . . . 9
⊢
(((((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 7434 |
. . . . . . . . . 10
⊢
(((((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 5089 |
. . . . . . . . 9
⊢
(((((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 294 |
. . . . . . . 8
⊢
(((((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 3117 |
. . . . . . 7
⊢ ((((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 5083 |
. . . . . . . . . . 11
⊢ (𝑟 = (𝑧 / 2) → ((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < 𝑟 ↔ (𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < (𝑧 / 2))) |
75 | | breq2 5083 |
. . . . . . . . . . 11
⊢ (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 631 |
. . . . . . . . . 10
⊢ (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 342 |
. . . . . . . . 9
⊢ (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 3125 |
. . . . . . . 8
⊢ (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 3561 |
. . . . . . 7
⊢ (((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 683 |
. . . . . 6
⊢ ((((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 3110 |
. . . . 5
⊢ (((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 3117 |
. . . 4
⊢ ((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 23618 |
. . . . . 6
⊢ (𝐺 ∈ MetSp → 𝐺 ∈
∞MetSp) |
84 | | eqid 2740 |
. . . . . . 7
⊢
((dist‘𝐺)
↾ ((Base‘𝐺)
× (Base‘𝐺))) =
((dist‘𝐺) ↾
((Base‘𝐺) ×
(Base‘𝐺))) |
85 | 7, 84 | xmsxmet 23620 |
. . . . . 6
⊢ (𝐺 ∈ ∞MetSp →
((dist‘𝐺) ↾
((Base‘𝐺) ×
(Base‘𝐺))) ∈
(∞Met‘(Base‘𝐺))) |
86 | 4, 83, 85 | 3syl 18 |
. . . . 5
⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) →
((dist‘𝐺) ↾
((Base‘𝐺) ×
(Base‘𝐺))) ∈
(∞Met‘(Base‘𝐺))) |
87 | | eqid 2740 |
. . . . . 6
⊢
(MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))) = (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))) |
88 | 87, 87, 87 | txmetcn 23715 |
. . . . 5
⊢
((((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 1370 |
. . . 4
⊢ ((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 710 |
. . 3
⊢ ((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 2740 |
. . . . . . 7
⊢
(TopOpen‘𝐺) =
(TopOpen‘𝐺) |
92 | 91, 7, 84 | mstopn 23616 |
. . . . . 6
⊢ (𝐺 ∈ MetSp →
(TopOpen‘𝐺) =
(MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))) |
93 | 4, 92 | syl 17 |
. . . . 5
⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) →
(TopOpen‘𝐺) =
(MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))) |
94 | 93, 93 | oveq12d 7290 |
. . . 4
⊢ ((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 7290 |
. . 3
⊢ ((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 2844 |
. 2
⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) →
(-g‘𝐺)
∈ (((TopOpen‘𝐺)
×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))) |
97 | 91, 8 | istgp2 23253 |
. 2
⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧
(-g‘𝐺)
∈ (((TopOpen‘𝐺)
×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺)))) |
98 | 2, 6, 96, 97 | syl3anbrc 1342 |
1
⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ TopGrp) |