Step |
Hyp |
Ref |
Expression |
1 |
|
gexcl.1 |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
2 |
|
gexcl.2 |
⊢ 𝐸 = ( gEx ‘ 𝐺 ) |
3 |
|
gexid.3 |
⊢ · = ( .g ‘ 𝐺 ) |
4 |
|
gexid.4 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
5 |
|
oveq1 |
⊢ ( 𝑦 = 𝑁 → ( 𝑦 · 𝑥 ) = ( 𝑁 · 𝑥 ) ) |
6 |
5
|
eqeq1d |
⊢ ( 𝑦 = 𝑁 → ( ( 𝑦 · 𝑥 ) = 0 ↔ ( 𝑁 · 𝑥 ) = 0 ) ) |
7 |
6
|
ralbidv |
⊢ ( 𝑦 = 𝑁 → ( ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 ↔ ∀ 𝑥 ∈ 𝑋 ( 𝑁 · 𝑥 ) = 0 ) ) |
8 |
7
|
elrab |
⊢ ( 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ↔ ( 𝑁 ∈ ℕ ∧ ∀ 𝑥 ∈ 𝑋 ( 𝑁 · 𝑥 ) = 0 ) ) |
9 |
|
eqid |
⊢ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } = { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } |
10 |
1 3 4 2 9
|
gexval |
⊢ ( 𝐺 ∈ 𝑉 → 𝐸 = if ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } = ∅ , 0 , inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ) ) |
11 |
|
ne0i |
⊢ ( 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } → { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ≠ ∅ ) |
12 |
|
ifnefalse |
⊢ ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ≠ ∅ → if ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } = ∅ , 0 , inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ) = inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ) |
13 |
11 12
|
syl |
⊢ ( 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } → if ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } = ∅ , 0 , inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ) = inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ) |
14 |
10 13
|
sylan9eq |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → 𝐸 = inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ) |
15 |
|
ssrab2 |
⊢ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ⊆ ℕ |
16 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
17 |
15 16
|
sseqtri |
⊢ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ⊆ ( ℤ≥ ‘ 1 ) |
18 |
11
|
adantl |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ≠ ∅ ) |
19 |
|
infssuzcl |
⊢ ( ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ⊆ ( ℤ≥ ‘ 1 ) ∧ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ≠ ∅ ) → inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ) |
20 |
17 18 19
|
sylancr |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ) |
21 |
15 20
|
sselid |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ ℕ ) |
22 |
|
infssuzle |
⊢ ( ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ⊆ ( ℤ≥ ‘ 1 ) ∧ 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ≤ 𝑁 ) |
23 |
17 22
|
mpan |
⊢ ( 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } → inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ≤ 𝑁 ) |
24 |
23
|
adantl |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ≤ 𝑁 ) |
25 |
|
elrabi |
⊢ ( 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } → 𝑁 ∈ ℕ ) |
26 |
25
|
nnzd |
⊢ ( 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } → 𝑁 ∈ ℤ ) |
27 |
|
fznn |
⊢ ( 𝑁 ∈ ℤ → ( inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ ( 1 ... 𝑁 ) ↔ ( inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ ℕ ∧ inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ≤ 𝑁 ) ) ) |
28 |
26 27
|
syl |
⊢ ( 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } → ( inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ ( 1 ... 𝑁 ) ↔ ( inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ ℕ ∧ inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ≤ 𝑁 ) ) ) |
29 |
28
|
adantl |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → ( inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ ( 1 ... 𝑁 ) ↔ ( inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ ℕ ∧ inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ≤ 𝑁 ) ) ) |
30 |
21 24 29
|
mpbir2and |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → inf ( { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } , ℝ , < ) ∈ ( 1 ... 𝑁 ) ) |
31 |
14 30
|
eqeltrd |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑁 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ) → 𝐸 ∈ ( 1 ... 𝑁 ) ) |
32 |
8 31
|
sylan2br |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ ( 𝑁 ∈ ℕ ∧ ∀ 𝑥 ∈ 𝑋 ( 𝑁 · 𝑥 ) = 0 ) ) → 𝐸 ∈ ( 1 ... 𝑁 ) ) |
33 |
32
|
3impb |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ ∀ 𝑥 ∈ 𝑋 ( 𝑁 · 𝑥 ) = 0 ) → 𝐸 ∈ ( 1 ... 𝑁 ) ) |