Step |
Hyp |
Ref |
Expression |
1 |
|
gexval.1 |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
2 |
|
gexval.2 |
⊢ · = ( .g ‘ 𝐺 ) |
3 |
|
gexval.3 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
4 |
|
gexval.4 |
⊢ 𝐸 = ( gEx ‘ 𝐺 ) |
5 |
|
gexval.i |
⊢ 𝐼 = { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } |
6 |
|
df-gex |
⊢ gEx = ( 𝑔 ∈ V ↦ ⦋ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( 𝑦 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) } / 𝑖 ⦌ if ( 𝑖 = ∅ , 0 , inf ( 𝑖 , ℝ , < ) ) ) |
7 |
|
nnex |
⊢ ℕ ∈ V |
8 |
7
|
rabex |
⊢ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( 𝑦 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) } ∈ V |
9 |
8
|
a1i |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) → { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( 𝑦 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) } ∈ V ) |
10 |
|
simpr |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) → 𝑔 = 𝐺 ) |
11 |
10
|
fveq2d |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) → ( Base ‘ 𝑔 ) = ( Base ‘ 𝐺 ) ) |
12 |
11 1
|
eqtr4di |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) → ( Base ‘ 𝑔 ) = 𝑋 ) |
13 |
10
|
fveq2d |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) → ( .g ‘ 𝑔 ) = ( .g ‘ 𝐺 ) ) |
14 |
13 2
|
eqtr4di |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) → ( .g ‘ 𝑔 ) = · ) |
15 |
14
|
oveqd |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) → ( 𝑦 ( .g ‘ 𝑔 ) 𝑥 ) = ( 𝑦 · 𝑥 ) ) |
16 |
10
|
fveq2d |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) → ( 0g ‘ 𝑔 ) = ( 0g ‘ 𝐺 ) ) |
17 |
16 3
|
eqtr4di |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) → ( 0g ‘ 𝑔 ) = 0 ) |
18 |
15 17
|
eqeq12d |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) → ( ( 𝑦 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) ↔ ( 𝑦 · 𝑥 ) = 0 ) ) |
19 |
12 18
|
raleqbidv |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) → ( ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( 𝑦 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) ↔ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 ) ) |
20 |
19
|
rabbidv |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) → { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( 𝑦 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) } = { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 · 𝑥 ) = 0 } ) |
21 |
20 5
|
eqtr4di |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) → { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( 𝑦 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) } = 𝐼 ) |
22 |
21
|
eqeq2d |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) → ( 𝑖 = { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( 𝑦 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) } ↔ 𝑖 = 𝐼 ) ) |
23 |
22
|
biimpa |
⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) ∧ 𝑖 = { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( 𝑦 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) } ) → 𝑖 = 𝐼 ) |
24 |
23
|
eqeq1d |
⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) ∧ 𝑖 = { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( 𝑦 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) } ) → ( 𝑖 = ∅ ↔ 𝐼 = ∅ ) ) |
25 |
23
|
infeq1d |
⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) ∧ 𝑖 = { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( 𝑦 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) } ) → inf ( 𝑖 , ℝ , < ) = inf ( 𝐼 , ℝ , < ) ) |
26 |
24 25
|
ifbieq2d |
⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) ∧ 𝑖 = { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( 𝑦 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) } ) → if ( 𝑖 = ∅ , 0 , inf ( 𝑖 , ℝ , < ) ) = if ( 𝐼 = ∅ , 0 , inf ( 𝐼 , ℝ , < ) ) ) |
27 |
9 26
|
csbied |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺 ) → ⦋ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( 𝑦 ( .g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) } / 𝑖 ⦌ if ( 𝑖 = ∅ , 0 , inf ( 𝑖 , ℝ , < ) ) = if ( 𝐼 = ∅ , 0 , inf ( 𝐼 , ℝ , < ) ) ) |
28 |
|
elex |
⊢ ( 𝐺 ∈ 𝑉 → 𝐺 ∈ V ) |
29 |
|
c0ex |
⊢ 0 ∈ V |
30 |
|
ltso |
⊢ < Or ℝ |
31 |
30
|
infex |
⊢ inf ( 𝐼 , ℝ , < ) ∈ V |
32 |
29 31
|
ifex |
⊢ if ( 𝐼 = ∅ , 0 , inf ( 𝐼 , ℝ , < ) ) ∈ V |
33 |
32
|
a1i |
⊢ ( 𝐺 ∈ 𝑉 → if ( 𝐼 = ∅ , 0 , inf ( 𝐼 , ℝ , < ) ) ∈ V ) |
34 |
6 27 28 33
|
fvmptd2 |
⊢ ( 𝐺 ∈ 𝑉 → ( gEx ‘ 𝐺 ) = if ( 𝐼 = ∅ , 0 , inf ( 𝐼 , ℝ , < ) ) ) |
35 |
4 34
|
syl5eq |
⊢ ( 𝐺 ∈ 𝑉 → 𝐸 = if ( 𝐼 = ∅ , 0 , inf ( 𝐼 , ℝ , < ) ) ) |