Step |
Hyp |
Ref |
Expression |
1 |
|
isphg.1 |
⊢ 𝑋 = ran 𝐺 |
2 |
|
df-ph |
⊢ CPreHilOLD = ( NrmCVec ∩ { 〈 〈 𝑔 , 𝑠 〉 , 𝑛 〉 ∣ ∀ 𝑥 ∈ ran 𝑔 ∀ 𝑦 ∈ ran 𝑔 ( ( ( 𝑛 ‘ ( 𝑥 𝑔 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝑔 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) } ) |
3 |
2
|
elin2 |
⊢ ( 〈 〈 𝐺 , 𝑆 〉 , 𝑁 〉 ∈ CPreHilOLD ↔ ( 〈 〈 𝐺 , 𝑆 〉 , 𝑁 〉 ∈ NrmCVec ∧ 〈 〈 𝐺 , 𝑆 〉 , 𝑁 〉 ∈ { 〈 〈 𝑔 , 𝑠 〉 , 𝑛 〉 ∣ ∀ 𝑥 ∈ ran 𝑔 ∀ 𝑦 ∈ ran 𝑔 ( ( ( 𝑛 ‘ ( 𝑥 𝑔 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝑔 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) } ) ) |
4 |
|
rneq |
⊢ ( 𝑔 = 𝐺 → ran 𝑔 = ran 𝐺 ) |
5 |
4 1
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ran 𝑔 = 𝑋 ) |
6 |
|
oveq |
⊢ ( 𝑔 = 𝐺 → ( 𝑥 𝑔 𝑦 ) = ( 𝑥 𝐺 𝑦 ) ) |
7 |
6
|
fveq2d |
⊢ ( 𝑔 = 𝐺 → ( 𝑛 ‘ ( 𝑥 𝑔 𝑦 ) ) = ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) ) |
8 |
7
|
oveq1d |
⊢ ( 𝑔 = 𝐺 → ( ( 𝑛 ‘ ( 𝑥 𝑔 𝑦 ) ) ↑ 2 ) = ( ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) ) |
9 |
|
oveq |
⊢ ( 𝑔 = 𝐺 → ( 𝑥 𝑔 ( - 1 𝑠 𝑦 ) ) = ( 𝑥 𝐺 ( - 1 𝑠 𝑦 ) ) ) |
10 |
9
|
fveq2d |
⊢ ( 𝑔 = 𝐺 → ( 𝑛 ‘ ( 𝑥 𝑔 ( - 1 𝑠 𝑦 ) ) ) = ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑠 𝑦 ) ) ) ) |
11 |
10
|
oveq1d |
⊢ ( 𝑔 = 𝐺 → ( ( 𝑛 ‘ ( 𝑥 𝑔 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) = ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) ) |
12 |
8 11
|
oveq12d |
⊢ ( 𝑔 = 𝐺 → ( ( ( 𝑛 ‘ ( 𝑥 𝑔 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝑔 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) ) = ( ( ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) ) ) |
13 |
12
|
eqeq1d |
⊢ ( 𝑔 = 𝐺 → ( ( ( ( 𝑛 ‘ ( 𝑥 𝑔 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝑔 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) ↔ ( ( ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) ) ) |
14 |
5 13
|
raleqbidv |
⊢ ( 𝑔 = 𝐺 → ( ∀ 𝑦 ∈ ran 𝑔 ( ( ( 𝑛 ‘ ( 𝑥 𝑔 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝑔 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) ↔ ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) ) ) |
15 |
5 14
|
raleqbidv |
⊢ ( 𝑔 = 𝐺 → ( ∀ 𝑥 ∈ ran 𝑔 ∀ 𝑦 ∈ ran 𝑔 ( ( ( 𝑛 ‘ ( 𝑥 𝑔 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝑔 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) ↔ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) ) ) |
16 |
|
oveq |
⊢ ( 𝑠 = 𝑆 → ( - 1 𝑠 𝑦 ) = ( - 1 𝑆 𝑦 ) ) |
17 |
16
|
oveq2d |
⊢ ( 𝑠 = 𝑆 → ( 𝑥 𝐺 ( - 1 𝑠 𝑦 ) ) = ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) |
18 |
17
|
fveq2d |
⊢ ( 𝑠 = 𝑆 → ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑠 𝑦 ) ) ) = ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ) |
19 |
18
|
oveq1d |
⊢ ( 𝑠 = 𝑆 → ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) = ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ↑ 2 ) ) |
20 |
19
|
oveq2d |
⊢ ( 𝑠 = 𝑆 → ( ( ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) ) = ( ( ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ↑ 2 ) ) ) |
21 |
20
|
eqeq1d |
⊢ ( 𝑠 = 𝑆 → ( ( ( ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) ↔ ( ( ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) ) ) |
22 |
21
|
2ralbidv |
⊢ ( 𝑠 = 𝑆 → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) ↔ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) ) ) |
23 |
|
fveq1 |
⊢ ( 𝑛 = 𝑁 → ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ) |
24 |
23
|
oveq1d |
⊢ ( 𝑛 = 𝑁 → ( ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) = ( ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) ) |
25 |
|
fveq1 |
⊢ ( 𝑛 = 𝑁 → ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) = ( 𝑁 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ) |
26 |
25
|
oveq1d |
⊢ ( 𝑛 = 𝑁 → ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ↑ 2 ) = ( ( 𝑁 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ↑ 2 ) ) |
27 |
24 26
|
oveq12d |
⊢ ( 𝑛 = 𝑁 → ( ( ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ↑ 2 ) ) = ( ( ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑁 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ↑ 2 ) ) ) |
28 |
|
fveq1 |
⊢ ( 𝑛 = 𝑁 → ( 𝑛 ‘ 𝑥 ) = ( 𝑁 ‘ 𝑥 ) ) |
29 |
28
|
oveq1d |
⊢ ( 𝑛 = 𝑁 → ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) = ( ( 𝑁 ‘ 𝑥 ) ↑ 2 ) ) |
30 |
|
fveq1 |
⊢ ( 𝑛 = 𝑁 → ( 𝑛 ‘ 𝑦 ) = ( 𝑁 ‘ 𝑦 ) ) |
31 |
30
|
oveq1d |
⊢ ( 𝑛 = 𝑁 → ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) = ( ( 𝑁 ‘ 𝑦 ) ↑ 2 ) ) |
32 |
29 31
|
oveq12d |
⊢ ( 𝑛 = 𝑁 → ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) = ( ( ( 𝑁 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑁 ‘ 𝑦 ) ↑ 2 ) ) ) |
33 |
32
|
oveq2d |
⊢ ( 𝑛 = 𝑁 → ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) = ( 2 · ( ( ( 𝑁 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑁 ‘ 𝑦 ) ↑ 2 ) ) ) ) |
34 |
27 33
|
eqeq12d |
⊢ ( 𝑛 = 𝑁 → ( ( ( ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) ↔ ( ( ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑁 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑁 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑁 ‘ 𝑦 ) ↑ 2 ) ) ) ) ) |
35 |
34
|
2ralbidv |
⊢ ( 𝑛 = 𝑁 → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑛 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) ↔ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑁 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑁 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑁 ‘ 𝑦 ) ↑ 2 ) ) ) ) ) |
36 |
15 22 35
|
eloprabg |
⊢ ( ( 𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶 ) → ( 〈 〈 𝐺 , 𝑆 〉 , 𝑁 〉 ∈ { 〈 〈 𝑔 , 𝑠 〉 , 𝑛 〉 ∣ ∀ 𝑥 ∈ ran 𝑔 ∀ 𝑦 ∈ ran 𝑔 ( ( ( 𝑛 ‘ ( 𝑥 𝑔 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝑔 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) } ↔ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑁 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑁 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑁 ‘ 𝑦 ) ↑ 2 ) ) ) ) ) |
37 |
36
|
anbi2d |
⊢ ( ( 𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶 ) → ( ( 〈 〈 𝐺 , 𝑆 〉 , 𝑁 〉 ∈ NrmCVec ∧ 〈 〈 𝐺 , 𝑆 〉 , 𝑁 〉 ∈ { 〈 〈 𝑔 , 𝑠 〉 , 𝑛 〉 ∣ ∀ 𝑥 ∈ ran 𝑔 ∀ 𝑦 ∈ ran 𝑔 ( ( ( 𝑛 ‘ ( 𝑥 𝑔 𝑦 ) ) ↑ 2 ) + ( ( 𝑛 ‘ ( 𝑥 𝑔 ( - 1 𝑠 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑛 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑛 ‘ 𝑦 ) ↑ 2 ) ) ) } ) ↔ ( 〈 〈 𝐺 , 𝑆 〉 , 𝑁 〉 ∈ NrmCVec ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑁 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑁 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑁 ‘ 𝑦 ) ↑ 2 ) ) ) ) ) ) |
38 |
3 37
|
syl5bb |
⊢ ( ( 𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶 ) → ( 〈 〈 𝐺 , 𝑆 〉 , 𝑁 〉 ∈ CPreHilOLD ↔ ( 〈 〈 𝐺 , 𝑆 〉 , 𝑁 〉 ∈ NrmCVec ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ↑ 2 ) + ( ( 𝑁 ‘ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑁 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝑁 ‘ 𝑦 ) ↑ 2 ) ) ) ) ) ) |