| Step |
Hyp |
Ref |
Expression |
| 1 |
|
veronesevrow.1 |
⊢ ( 𝜑 → 𝑃 ∈ ( ℝ ↑m ( 1 ... 3 ) ) ) |
| 2 |
|
ovex |
⊢ ( ( ( if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , 0 ) + if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , 0 ) ) + if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , 0 ) ) + ( ( if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , 0 ) + if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , 0 ) ) + if ( 𝑘 = 6 , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) , 0 ) ) ) ∈ V |
| 3 |
|
eqid |
⊢ ( 𝑘 ∈ ( 1 ... 6 ) ↦ ( ( ( if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , 0 ) + if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , 0 ) ) + if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , 0 ) ) + ( ( if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , 0 ) + if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , 0 ) ) + if ( 𝑘 = 6 , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) , 0 ) ) ) ) = ( 𝑘 ∈ ( 1 ... 6 ) ↦ ( ( ( if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , 0 ) + if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , 0 ) ) + if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , 0 ) ) + ( ( if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , 0 ) + if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , 0 ) ) + if ( 𝑘 = 6 , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) , 0 ) ) ) ) |
| 4 |
2 3
|
fnmpti |
⊢ ( 𝑘 ∈ ( 1 ... 6 ) ↦ ( ( ( if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , 0 ) + if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , 0 ) ) + if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , 0 ) ) + ( ( if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , 0 ) + if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , 0 ) ) + if ( 𝑘 = 6 , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) , 0 ) ) ) ) Fn ( 1 ... 6 ) |
| 5 |
1
|
veronesevald |
⊢ ( 𝜑 → ( veronese ‘ 𝑃 ) = ( 𝑘 ∈ ( 1 ... 6 ) ↦ ( ( ( if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , 0 ) + if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , 0 ) ) + if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , 0 ) ) + ( ( if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , 0 ) + if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , 0 ) ) + if ( 𝑘 = 6 , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) , 0 ) ) ) ) ) |
| 6 |
5
|
fneq1d |
⊢ ( 𝜑 → ( ( veronese ‘ 𝑃 ) Fn ( 1 ... 6 ) ↔ ( 𝑘 ∈ ( 1 ... 6 ) ↦ ( ( ( if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , 0 ) + if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , 0 ) ) + if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , 0 ) ) + ( ( if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , 0 ) + if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , 0 ) ) + if ( 𝑘 = 6 , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) , 0 ) ) ) ) Fn ( 1 ... 6 ) ) ) |
| 7 |
4 6
|
mpbiri |
⊢ ( 𝜑 → ( veronese ‘ 𝑃 ) Fn ( 1 ... 6 ) ) |
| 8 |
|
ovex |
⊢ ( ( 𝑃 ‘ 1 ) ↑ 2 ) ∈ V |
| 9 |
|
ovex |
⊢ ( ( 𝑃 ‘ 2 ) ↑ 2 ) ∈ V |
| 10 |
|
ovex |
⊢ ( ( 𝑃 ‘ 3 ) ↑ 2 ) ∈ V |
| 11 |
|
ovex |
⊢ ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) ∈ V |
| 12 |
|
ovex |
⊢ ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) ∈ V |
| 13 |
|
ovex |
⊢ ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ∈ V |
| 14 |
12 13
|
ifex |
⊢ if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ∈ V |
| 15 |
11 14
|
ifex |
⊢ if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ∈ V |
| 16 |
10 15
|
ifex |
⊢ if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ∈ V |
| 17 |
9 16
|
ifex |
⊢ if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ∈ V |
| 18 |
8 17
|
ifex |
⊢ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ∈ V |
| 19 |
|
eqid |
⊢ ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) = ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) |
| 20 |
18 19
|
fnmpti |
⊢ ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) Fn ( 1 ... 6 ) |
| 21 |
20
|
a1i |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) Fn ( 1 ... 6 ) ) |
| 22 |
1
|
veronesev1lem |
⊢ ( 𝜑 → ( ( veronese ‘ 𝑃 ) ‘ 1 ) = ( ( 𝑃 ‘ 1 ) ↑ 2 ) ) |
| 23 |
22
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 1 ) → ( ( veronese ‘ 𝑃 ) ‘ 1 ) = ( ( 𝑃 ‘ 1 ) ↑ 2 ) ) |
| 24 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 1 ) → 𝑥 = 1 ) |
| 25 |
24
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 1 ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( veronese ‘ 𝑃 ) ‘ 1 ) ) |
| 26 |
24
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 1 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 1 ) ) |
| 27 |
|
1nn |
⊢ 1 ∈ ℕ |
| 28 |
|
6nn |
⊢ 6 ∈ ℕ |
| 29 |
|
1re |
⊢ 1 ∈ ℝ |
| 30 |
|
6re |
⊢ 6 ∈ ℝ |
| 31 |
|
1lt6 |
⊢ 1 < 6 |
| 32 |
29 30 31
|
ltleii |
⊢ 1 ≤ 6 |
| 33 |
|
elfz1b |
⊢ ( 1 ∈ ( 1 ... 6 ) ↔ ( 1 ∈ ℕ ∧ 6 ∈ ℕ ∧ 1 ≤ 6 ) ) |
| 34 |
27 28 32 33
|
mpbir3an |
⊢ 1 ∈ ( 1 ... 6 ) |
| 35 |
|
iftrue |
⊢ ( 𝑘 = 1 → if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) = ( ( 𝑃 ‘ 1 ) ↑ 2 ) ) |
| 36 |
35 19 18
|
fvmpt3i |
⊢ ( 1 ∈ ( 1 ... 6 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 1 ) = ( ( 𝑃 ‘ 1 ) ↑ 2 ) ) |
| 37 |
34 36
|
ax-mp |
⊢ ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 1 ) = ( ( 𝑃 ‘ 1 ) ↑ 2 ) |
| 38 |
26 37
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 1 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) = ( ( 𝑃 ‘ 1 ) ↑ 2 ) ) |
| 39 |
23 25 38
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 1 ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) ) |
| 40 |
1
|
veronesev2lem |
⊢ ( 𝜑 → ( ( veronese ‘ 𝑃 ) ‘ 2 ) = ( ( 𝑃 ‘ 2 ) ↑ 2 ) ) |
| 41 |
40
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 2 ) → ( ( veronese ‘ 𝑃 ) ‘ 2 ) = ( ( 𝑃 ‘ 2 ) ↑ 2 ) ) |
| 42 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 2 ) → 𝑥 = 2 ) |
| 43 |
42
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 2 ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( veronese ‘ 𝑃 ) ‘ 2 ) ) |
| 44 |
42
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 2 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 2 ) ) |
| 45 |
|
2nn |
⊢ 2 ∈ ℕ |
| 46 |
|
2re |
⊢ 2 ∈ ℝ |
| 47 |
|
2lt6 |
⊢ 2 < 6 |
| 48 |
46 30 47
|
ltleii |
⊢ 2 ≤ 6 |
| 49 |
|
elfz1b |
⊢ ( 2 ∈ ( 1 ... 6 ) ↔ ( 2 ∈ ℕ ∧ 6 ∈ ℕ ∧ 2 ≤ 6 ) ) |
| 50 |
45 28 48 49
|
mpbir3an |
⊢ 2 ∈ ( 1 ... 6 ) |
| 51 |
|
1ne2 |
⊢ 1 ≠ 2 |
| 52 |
51
|
necomi |
⊢ 2 ≠ 1 |
| 53 |
|
neeq1 |
⊢ ( 𝑘 = 2 → ( 𝑘 ≠ 1 ↔ 2 ≠ 1 ) ) |
| 54 |
52 53
|
mpbiri |
⊢ ( 𝑘 = 2 → 𝑘 ≠ 1 ) |
| 55 |
54
|
neneqd |
⊢ ( 𝑘 = 2 → ¬ 𝑘 = 1 ) |
| 56 |
55
|
iffalsed |
⊢ ( 𝑘 = 2 → if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) = if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) |
| 57 |
|
iftrue |
⊢ ( 𝑘 = 2 → if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) = ( ( 𝑃 ‘ 2 ) ↑ 2 ) ) |
| 58 |
56 57
|
eqtrd |
⊢ ( 𝑘 = 2 → if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) = ( ( 𝑃 ‘ 2 ) ↑ 2 ) ) |
| 59 |
58 19 18
|
fvmpt3i |
⊢ ( 2 ∈ ( 1 ... 6 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 2 ) = ( ( 𝑃 ‘ 2 ) ↑ 2 ) ) |
| 60 |
50 59
|
ax-mp |
⊢ ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 2 ) = ( ( 𝑃 ‘ 2 ) ↑ 2 ) |
| 61 |
44 60
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 2 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) = ( ( 𝑃 ‘ 2 ) ↑ 2 ) ) |
| 62 |
41 43 61
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 2 ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) ) |
| 63 |
39 62
|
jaodan |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ ( 𝑥 = 1 ∨ 𝑥 = 2 ) ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) ) |
| 64 |
1
|
veronesev3lem |
⊢ ( 𝜑 → ( ( veronese ‘ 𝑃 ) ‘ 3 ) = ( ( 𝑃 ‘ 3 ) ↑ 2 ) ) |
| 65 |
64
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 3 ) → ( ( veronese ‘ 𝑃 ) ‘ 3 ) = ( ( 𝑃 ‘ 3 ) ↑ 2 ) ) |
| 66 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 3 ) → 𝑥 = 3 ) |
| 67 |
66
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 3 ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( veronese ‘ 𝑃 ) ‘ 3 ) ) |
| 68 |
66
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 3 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 3 ) ) |
| 69 |
|
3nn |
⊢ 3 ∈ ℕ |
| 70 |
|
3re |
⊢ 3 ∈ ℝ |
| 71 |
|
3lt6 |
⊢ 3 < 6 |
| 72 |
70 30 71
|
ltleii |
⊢ 3 ≤ 6 |
| 73 |
|
elfz1b |
⊢ ( 3 ∈ ( 1 ... 6 ) ↔ ( 3 ∈ ℕ ∧ 6 ∈ ℕ ∧ 3 ≤ 6 ) ) |
| 74 |
69 28 72 73
|
mpbir3an |
⊢ 3 ∈ ( 1 ... 6 ) |
| 75 |
|
1ne3 |
⊢ 1 ≠ 3 |
| 76 |
75
|
necomi |
⊢ 3 ≠ 1 |
| 77 |
|
neeq1 |
⊢ ( 𝑘 = 3 → ( 𝑘 ≠ 1 ↔ 3 ≠ 1 ) ) |
| 78 |
76 77
|
mpbiri |
⊢ ( 𝑘 = 3 → 𝑘 ≠ 1 ) |
| 79 |
78
|
neneqd |
⊢ ( 𝑘 = 3 → ¬ 𝑘 = 1 ) |
| 80 |
79
|
iffalsed |
⊢ ( 𝑘 = 3 → if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) = if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) |
| 81 |
|
2ne3 |
⊢ 2 ≠ 3 |
| 82 |
81
|
necomi |
⊢ 3 ≠ 2 |
| 83 |
|
neeq1 |
⊢ ( 𝑘 = 3 → ( 𝑘 ≠ 2 ↔ 3 ≠ 2 ) ) |
| 84 |
82 83
|
mpbiri |
⊢ ( 𝑘 = 3 → 𝑘 ≠ 2 ) |
| 85 |
84
|
neneqd |
⊢ ( 𝑘 = 3 → ¬ 𝑘 = 2 ) |
| 86 |
85
|
iffalsed |
⊢ ( 𝑘 = 3 → if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) = if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) |
| 87 |
|
iftrue |
⊢ ( 𝑘 = 3 → if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) = ( ( 𝑃 ‘ 3 ) ↑ 2 ) ) |
| 88 |
80 86 87
|
3eqtrd |
⊢ ( 𝑘 = 3 → if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) = ( ( 𝑃 ‘ 3 ) ↑ 2 ) ) |
| 89 |
88 19 18
|
fvmpt3i |
⊢ ( 3 ∈ ( 1 ... 6 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 3 ) = ( ( 𝑃 ‘ 3 ) ↑ 2 ) ) |
| 90 |
74 89
|
ax-mp |
⊢ ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 3 ) = ( ( 𝑃 ‘ 3 ) ↑ 2 ) |
| 91 |
68 90
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 3 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) = ( ( 𝑃 ‘ 3 ) ↑ 2 ) ) |
| 92 |
65 67 91
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 3 ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) ) |
| 93 |
63 92
|
jaodan |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ ( ( 𝑥 = 1 ∨ 𝑥 = 2 ) ∨ 𝑥 = 3 ) ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) ) |
| 94 |
1
|
veronesev4lem |
⊢ ( 𝜑 → ( ( veronese ‘ 𝑃 ) ‘ 4 ) = ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) ) |
| 95 |
94
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 4 ) → ( ( veronese ‘ 𝑃 ) ‘ 4 ) = ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) ) |
| 96 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 4 ) → 𝑥 = 4 ) |
| 97 |
96
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 4 ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( veronese ‘ 𝑃 ) ‘ 4 ) ) |
| 98 |
96
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 4 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 4 ) ) |
| 99 |
|
4nn |
⊢ 4 ∈ ℕ |
| 100 |
|
4re |
⊢ 4 ∈ ℝ |
| 101 |
|
4lt6 |
⊢ 4 < 6 |
| 102 |
100 30 101
|
ltleii |
⊢ 4 ≤ 6 |
| 103 |
|
elfz1b |
⊢ ( 4 ∈ ( 1 ... 6 ) ↔ ( 4 ∈ ℕ ∧ 6 ∈ ℕ ∧ 4 ≤ 6 ) ) |
| 104 |
99 28 102 103
|
mpbir3an |
⊢ 4 ∈ ( 1 ... 6 ) |
| 105 |
|
1lt4 |
⊢ 1 < 4 |
| 106 |
29 105
|
gtneii |
⊢ 4 ≠ 1 |
| 107 |
|
neeq1 |
⊢ ( 𝑘 = 4 → ( 𝑘 ≠ 1 ↔ 4 ≠ 1 ) ) |
| 108 |
106 107
|
mpbiri |
⊢ ( 𝑘 = 4 → 𝑘 ≠ 1 ) |
| 109 |
108
|
neneqd |
⊢ ( 𝑘 = 4 → ¬ 𝑘 = 1 ) |
| 110 |
109
|
iffalsed |
⊢ ( 𝑘 = 4 → if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) = if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) |
| 111 |
|
2lt4 |
⊢ 2 < 4 |
| 112 |
46 111
|
gtneii |
⊢ 4 ≠ 2 |
| 113 |
|
neeq1 |
⊢ ( 𝑘 = 4 → ( 𝑘 ≠ 2 ↔ 4 ≠ 2 ) ) |
| 114 |
112 113
|
mpbiri |
⊢ ( 𝑘 = 4 → 𝑘 ≠ 2 ) |
| 115 |
114
|
neneqd |
⊢ ( 𝑘 = 4 → ¬ 𝑘 = 2 ) |
| 116 |
115
|
iffalsed |
⊢ ( 𝑘 = 4 → if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) = if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) |
| 117 |
110 116
|
eqtrd |
⊢ ( 𝑘 = 4 → if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) = if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) |
| 118 |
|
3lt4 |
⊢ 3 < 4 |
| 119 |
70 118
|
gtneii |
⊢ 4 ≠ 3 |
| 120 |
|
neeq1 |
⊢ ( 𝑘 = 4 → ( 𝑘 ≠ 3 ↔ 4 ≠ 3 ) ) |
| 121 |
119 120
|
mpbiri |
⊢ ( 𝑘 = 4 → 𝑘 ≠ 3 ) |
| 122 |
121
|
neneqd |
⊢ ( 𝑘 = 4 → ¬ 𝑘 = 3 ) |
| 123 |
122
|
iffalsed |
⊢ ( 𝑘 = 4 → if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) = if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) |
| 124 |
|
iftrue |
⊢ ( 𝑘 = 4 → if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) = ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) ) |
| 125 |
117 123 124
|
3eqtrd |
⊢ ( 𝑘 = 4 → if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) = ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) ) |
| 126 |
125 19 18
|
fvmpt3i |
⊢ ( 4 ∈ ( 1 ... 6 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 4 ) = ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) ) |
| 127 |
104 126
|
ax-mp |
⊢ ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 4 ) = ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) |
| 128 |
98 127
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 4 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) = ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) ) |
| 129 |
95 97 128
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 4 ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) ) |
| 130 |
93 129
|
jaodan |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ ( ( ( 𝑥 = 1 ∨ 𝑥 = 2 ) ∨ 𝑥 = 3 ) ∨ 𝑥 = 4 ) ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) ) |
| 131 |
1
|
veronesev5lem |
⊢ ( 𝜑 → ( ( veronese ‘ 𝑃 ) ‘ 5 ) = ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) ) |
| 132 |
131
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 5 ) → ( ( veronese ‘ 𝑃 ) ‘ 5 ) = ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) ) |
| 133 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 5 ) → 𝑥 = 5 ) |
| 134 |
133
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 5 ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( veronese ‘ 𝑃 ) ‘ 5 ) ) |
| 135 |
133
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 5 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 5 ) ) |
| 136 |
|
5nn |
⊢ 5 ∈ ℕ |
| 137 |
|
5re |
⊢ 5 ∈ ℝ |
| 138 |
|
5lt6 |
⊢ 5 < 6 |
| 139 |
137 30 138
|
ltleii |
⊢ 5 ≤ 6 |
| 140 |
|
elfz1b |
⊢ ( 5 ∈ ( 1 ... 6 ) ↔ ( 5 ∈ ℕ ∧ 6 ∈ ℕ ∧ 5 ≤ 6 ) ) |
| 141 |
136 28 139 140
|
mpbir3an |
⊢ 5 ∈ ( 1 ... 6 ) |
| 142 |
|
1lt5 |
⊢ 1 < 5 |
| 143 |
29 142
|
gtneii |
⊢ 5 ≠ 1 |
| 144 |
|
neeq1 |
⊢ ( 𝑘 = 5 → ( 𝑘 ≠ 1 ↔ 5 ≠ 1 ) ) |
| 145 |
143 144
|
mpbiri |
⊢ ( 𝑘 = 5 → 𝑘 ≠ 1 ) |
| 146 |
145
|
neneqd |
⊢ ( 𝑘 = 5 → ¬ 𝑘 = 1 ) |
| 147 |
146
|
iffalsed |
⊢ ( 𝑘 = 5 → if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) = if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) |
| 148 |
|
2lt5 |
⊢ 2 < 5 |
| 149 |
46 148
|
gtneii |
⊢ 5 ≠ 2 |
| 150 |
|
neeq1 |
⊢ ( 𝑘 = 5 → ( 𝑘 ≠ 2 ↔ 5 ≠ 2 ) ) |
| 151 |
149 150
|
mpbiri |
⊢ ( 𝑘 = 5 → 𝑘 ≠ 2 ) |
| 152 |
151
|
neneqd |
⊢ ( 𝑘 = 5 → ¬ 𝑘 = 2 ) |
| 153 |
152
|
iffalsed |
⊢ ( 𝑘 = 5 → if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) = if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) |
| 154 |
|
3lt5 |
⊢ 3 < 5 |
| 155 |
70 154
|
gtneii |
⊢ 5 ≠ 3 |
| 156 |
|
neeq1 |
⊢ ( 𝑘 = 5 → ( 𝑘 ≠ 3 ↔ 5 ≠ 3 ) ) |
| 157 |
155 156
|
mpbiri |
⊢ ( 𝑘 = 5 → 𝑘 ≠ 3 ) |
| 158 |
157
|
neneqd |
⊢ ( 𝑘 = 5 → ¬ 𝑘 = 3 ) |
| 159 |
158
|
iffalsed |
⊢ ( 𝑘 = 5 → if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) = if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) |
| 160 |
147 153 159
|
3eqtrd |
⊢ ( 𝑘 = 5 → if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) = if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) |
| 161 |
|
4lt5 |
⊢ 4 < 5 |
| 162 |
100 161
|
gtneii |
⊢ 5 ≠ 4 |
| 163 |
|
neeq1 |
⊢ ( 𝑘 = 5 → ( 𝑘 ≠ 4 ↔ 5 ≠ 4 ) ) |
| 164 |
162 163
|
mpbiri |
⊢ ( 𝑘 = 5 → 𝑘 ≠ 4 ) |
| 165 |
164
|
neneqd |
⊢ ( 𝑘 = 5 → ¬ 𝑘 = 4 ) |
| 166 |
165
|
iffalsed |
⊢ ( 𝑘 = 5 → if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) = if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) |
| 167 |
|
iftrue |
⊢ ( 𝑘 = 5 → if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) = ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) ) |
| 168 |
160 166 167
|
3eqtrd |
⊢ ( 𝑘 = 5 → if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) = ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) ) |
| 169 |
168 19 18
|
fvmpt3i |
⊢ ( 5 ∈ ( 1 ... 6 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 5 ) = ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) ) |
| 170 |
141 169
|
ax-mp |
⊢ ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 5 ) = ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) |
| 171 |
135 170
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 5 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) = ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) ) |
| 172 |
132 134 171
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 5 ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) ) |
| 173 |
130 172
|
jaodan |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ ( ( ( ( 𝑥 = 1 ∨ 𝑥 = 2 ) ∨ 𝑥 = 3 ) ∨ 𝑥 = 4 ) ∨ 𝑥 = 5 ) ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) ) |
| 174 |
1
|
veronesev6lem |
⊢ ( 𝜑 → ( ( veronese ‘ 𝑃 ) ‘ 6 ) = ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) |
| 175 |
174
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 6 ) → ( ( veronese ‘ 𝑃 ) ‘ 6 ) = ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) |
| 176 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 6 ) → 𝑥 = 6 ) |
| 177 |
176
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 6 ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( veronese ‘ 𝑃 ) ‘ 6 ) ) |
| 178 |
176
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 6 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 6 ) ) |
| 179 |
30
|
leidi |
⊢ 6 ≤ 6 |
| 180 |
|
elfz1b |
⊢ ( 6 ∈ ( 1 ... 6 ) ↔ ( 6 ∈ ℕ ∧ 6 ∈ ℕ ∧ 6 ≤ 6 ) ) |
| 181 |
28 28 179 180
|
mpbir3an |
⊢ 6 ∈ ( 1 ... 6 ) |
| 182 |
29 31
|
gtneii |
⊢ 6 ≠ 1 |
| 183 |
|
neeq1 |
⊢ ( 𝑘 = 6 → ( 𝑘 ≠ 1 ↔ 6 ≠ 1 ) ) |
| 184 |
182 183
|
mpbiri |
⊢ ( 𝑘 = 6 → 𝑘 ≠ 1 ) |
| 185 |
184
|
neneqd |
⊢ ( 𝑘 = 6 → ¬ 𝑘 = 1 ) |
| 186 |
185
|
iffalsed |
⊢ ( 𝑘 = 6 → if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) = if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) |
| 187 |
46 47
|
gtneii |
⊢ 6 ≠ 2 |
| 188 |
|
neeq1 |
⊢ ( 𝑘 = 6 → ( 𝑘 ≠ 2 ↔ 6 ≠ 2 ) ) |
| 189 |
187 188
|
mpbiri |
⊢ ( 𝑘 = 6 → 𝑘 ≠ 2 ) |
| 190 |
189
|
neneqd |
⊢ ( 𝑘 = 6 → ¬ 𝑘 = 2 ) |
| 191 |
190
|
iffalsed |
⊢ ( 𝑘 = 6 → if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) = if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) |
| 192 |
70 71
|
gtneii |
⊢ 6 ≠ 3 |
| 193 |
|
neeq1 |
⊢ ( 𝑘 = 6 → ( 𝑘 ≠ 3 ↔ 6 ≠ 3 ) ) |
| 194 |
192 193
|
mpbiri |
⊢ ( 𝑘 = 6 → 𝑘 ≠ 3 ) |
| 195 |
194
|
neneqd |
⊢ ( 𝑘 = 6 → ¬ 𝑘 = 3 ) |
| 196 |
195
|
iffalsed |
⊢ ( 𝑘 = 6 → if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) = if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) |
| 197 |
186 191 196
|
3eqtrd |
⊢ ( 𝑘 = 6 → if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) = if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) |
| 198 |
100 101
|
gtneii |
⊢ 6 ≠ 4 |
| 199 |
|
neeq1 |
⊢ ( 𝑘 = 6 → ( 𝑘 ≠ 4 ↔ 6 ≠ 4 ) ) |
| 200 |
198 199
|
mpbiri |
⊢ ( 𝑘 = 6 → 𝑘 ≠ 4 ) |
| 201 |
200
|
neneqd |
⊢ ( 𝑘 = 6 → ¬ 𝑘 = 4 ) |
| 202 |
201
|
iffalsed |
⊢ ( 𝑘 = 6 → if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) = if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) |
| 203 |
137 138
|
gtneii |
⊢ 6 ≠ 5 |
| 204 |
|
neeq1 |
⊢ ( 𝑘 = 6 → ( 𝑘 ≠ 5 ↔ 6 ≠ 5 ) ) |
| 205 |
203 204
|
mpbiri |
⊢ ( 𝑘 = 6 → 𝑘 ≠ 5 ) |
| 206 |
205
|
neneqd |
⊢ ( 𝑘 = 6 → ¬ 𝑘 = 5 ) |
| 207 |
206
|
iffalsed |
⊢ ( 𝑘 = 6 → if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) = ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) |
| 208 |
197 202 207
|
3eqtrd |
⊢ ( 𝑘 = 6 → if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) = ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) |
| 209 |
208 19 18
|
fvmpt3i |
⊢ ( 6 ∈ ( 1 ... 6 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 6 ) = ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) |
| 210 |
181 209
|
ax-mp |
⊢ ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 6 ) = ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) |
| 211 |
178 210
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 6 ) → ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) = ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) |
| 212 |
175 177 211
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) ∧ 𝑥 = 6 ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) ) |
| 213 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) → 𝑥 ∈ ( 1 ... 6 ) ) |
| 214 |
|
elnnuz |
⊢ ( 5 ∈ ℕ ↔ 5 ∈ ( ℤ≥ ‘ 1 ) ) |
| 215 |
136 214
|
mpbi |
⊢ 5 ∈ ( ℤ≥ ‘ 1 ) |
| 216 |
|
elfzp1 |
⊢ ( 5 ∈ ( ℤ≥ ‘ 1 ) → ( 𝑥 ∈ ( 1 ... ( 5 + 1 ) ) ↔ ( 𝑥 ∈ ( 1 ... 5 ) ∨ 𝑥 = ( 5 + 1 ) ) ) ) |
| 217 |
215 216
|
ax-mp |
⊢ ( 𝑥 ∈ ( 1 ... ( 5 + 1 ) ) ↔ ( 𝑥 ∈ ( 1 ... 5 ) ∨ 𝑥 = ( 5 + 1 ) ) ) |
| 218 |
|
5p1e6 |
⊢ ( 5 + 1 ) = 6 |
| 219 |
218
|
oveq2i |
⊢ ( 1 ... ( 5 + 1 ) ) = ( 1 ... 6 ) |
| 220 |
219
|
eleq2i |
⊢ ( 𝑥 ∈ ( 1 ... ( 5 + 1 ) ) ↔ 𝑥 ∈ ( 1 ... 6 ) ) |
| 221 |
218
|
eqeq2i |
⊢ ( 𝑥 = ( 5 + 1 ) ↔ 𝑥 = 6 ) |
| 222 |
221
|
orbi2i |
⊢ ( ( 𝑥 ∈ ( 1 ... 5 ) ∨ 𝑥 = ( 5 + 1 ) ) ↔ ( 𝑥 ∈ ( 1 ... 5 ) ∨ 𝑥 = 6 ) ) |
| 223 |
217 220 222
|
3bitr3i |
⊢ ( 𝑥 ∈ ( 1 ... 6 ) ↔ ( 𝑥 ∈ ( 1 ... 5 ) ∨ 𝑥 = 6 ) ) |
| 224 |
|
elnnuz |
⊢ ( 4 ∈ ℕ ↔ 4 ∈ ( ℤ≥ ‘ 1 ) ) |
| 225 |
99 224
|
mpbi |
⊢ 4 ∈ ( ℤ≥ ‘ 1 ) |
| 226 |
|
elfzp1 |
⊢ ( 4 ∈ ( ℤ≥ ‘ 1 ) → ( 𝑥 ∈ ( 1 ... ( 4 + 1 ) ) ↔ ( 𝑥 ∈ ( 1 ... 4 ) ∨ 𝑥 = ( 4 + 1 ) ) ) ) |
| 227 |
225 226
|
ax-mp |
⊢ ( 𝑥 ∈ ( 1 ... ( 4 + 1 ) ) ↔ ( 𝑥 ∈ ( 1 ... 4 ) ∨ 𝑥 = ( 4 + 1 ) ) ) |
| 228 |
|
4p1e5 |
⊢ ( 4 + 1 ) = 5 |
| 229 |
228
|
oveq2i |
⊢ ( 1 ... ( 4 + 1 ) ) = ( 1 ... 5 ) |
| 230 |
229
|
eleq2i |
⊢ ( 𝑥 ∈ ( 1 ... ( 4 + 1 ) ) ↔ 𝑥 ∈ ( 1 ... 5 ) ) |
| 231 |
228
|
eqeq2i |
⊢ ( 𝑥 = ( 4 + 1 ) ↔ 𝑥 = 5 ) |
| 232 |
231
|
orbi2i |
⊢ ( ( 𝑥 ∈ ( 1 ... 4 ) ∨ 𝑥 = ( 4 + 1 ) ) ↔ ( 𝑥 ∈ ( 1 ... 4 ) ∨ 𝑥 = 5 ) ) |
| 233 |
227 230 232
|
3bitr3i |
⊢ ( 𝑥 ∈ ( 1 ... 5 ) ↔ ( 𝑥 ∈ ( 1 ... 4 ) ∨ 𝑥 = 5 ) ) |
| 234 |
|
elnnuz |
⊢ ( 3 ∈ ℕ ↔ 3 ∈ ( ℤ≥ ‘ 1 ) ) |
| 235 |
69 234
|
mpbi |
⊢ 3 ∈ ( ℤ≥ ‘ 1 ) |
| 236 |
|
elfzp1 |
⊢ ( 3 ∈ ( ℤ≥ ‘ 1 ) → ( 𝑥 ∈ ( 1 ... ( 3 + 1 ) ) ↔ ( 𝑥 ∈ ( 1 ... 3 ) ∨ 𝑥 = ( 3 + 1 ) ) ) ) |
| 237 |
235 236
|
ax-mp |
⊢ ( 𝑥 ∈ ( 1 ... ( 3 + 1 ) ) ↔ ( 𝑥 ∈ ( 1 ... 3 ) ∨ 𝑥 = ( 3 + 1 ) ) ) |
| 238 |
|
3p1e4 |
⊢ ( 3 + 1 ) = 4 |
| 239 |
238
|
oveq2i |
⊢ ( 1 ... ( 3 + 1 ) ) = ( 1 ... 4 ) |
| 240 |
239
|
eleq2i |
⊢ ( 𝑥 ∈ ( 1 ... ( 3 + 1 ) ) ↔ 𝑥 ∈ ( 1 ... 4 ) ) |
| 241 |
238
|
eqeq2i |
⊢ ( 𝑥 = ( 3 + 1 ) ↔ 𝑥 = 4 ) |
| 242 |
241
|
orbi2i |
⊢ ( ( 𝑥 ∈ ( 1 ... 3 ) ∨ 𝑥 = ( 3 + 1 ) ) ↔ ( 𝑥 ∈ ( 1 ... 3 ) ∨ 𝑥 = 4 ) ) |
| 243 |
237 240 242
|
3bitr3i |
⊢ ( 𝑥 ∈ ( 1 ... 4 ) ↔ ( 𝑥 ∈ ( 1 ... 3 ) ∨ 𝑥 = 4 ) ) |
| 244 |
|
2eluzge1 |
⊢ 2 ∈ ( ℤ≥ ‘ 1 ) |
| 245 |
|
elfzp1 |
⊢ ( 2 ∈ ( ℤ≥ ‘ 1 ) → ( 𝑥 ∈ ( 1 ... ( 2 + 1 ) ) ↔ ( 𝑥 ∈ ( 1 ... 2 ) ∨ 𝑥 = ( 2 + 1 ) ) ) ) |
| 246 |
244 245
|
ax-mp |
⊢ ( 𝑥 ∈ ( 1 ... ( 2 + 1 ) ) ↔ ( 𝑥 ∈ ( 1 ... 2 ) ∨ 𝑥 = ( 2 + 1 ) ) ) |
| 247 |
|
2p1e3 |
⊢ ( 2 + 1 ) = 3 |
| 248 |
247
|
oveq2i |
⊢ ( 1 ... ( 2 + 1 ) ) = ( 1 ... 3 ) |
| 249 |
248
|
eleq2i |
⊢ ( 𝑥 ∈ ( 1 ... ( 2 + 1 ) ) ↔ 𝑥 ∈ ( 1 ... 3 ) ) |
| 250 |
247
|
eqeq2i |
⊢ ( 𝑥 = ( 2 + 1 ) ↔ 𝑥 = 3 ) |
| 251 |
250
|
orbi2i |
⊢ ( ( 𝑥 ∈ ( 1 ... 2 ) ∨ 𝑥 = ( 2 + 1 ) ) ↔ ( 𝑥 ∈ ( 1 ... 2 ) ∨ 𝑥 = 3 ) ) |
| 252 |
246 249 251
|
3bitr3i |
⊢ ( 𝑥 ∈ ( 1 ... 3 ) ↔ ( 𝑥 ∈ ( 1 ... 2 ) ∨ 𝑥 = 3 ) ) |
| 253 |
|
elnnuz |
⊢ ( 1 ∈ ℕ ↔ 1 ∈ ( ℤ≥ ‘ 1 ) ) |
| 254 |
27 253
|
mpbi |
⊢ 1 ∈ ( ℤ≥ ‘ 1 ) |
| 255 |
|
elfzp1 |
⊢ ( 1 ∈ ( ℤ≥ ‘ 1 ) → ( 𝑥 ∈ ( 1 ... ( 1 + 1 ) ) ↔ ( 𝑥 ∈ ( 1 ... 1 ) ∨ 𝑥 = ( 1 + 1 ) ) ) ) |
| 256 |
254 255
|
ax-mp |
⊢ ( 𝑥 ∈ ( 1 ... ( 1 + 1 ) ) ↔ ( 𝑥 ∈ ( 1 ... 1 ) ∨ 𝑥 = ( 1 + 1 ) ) ) |
| 257 |
|
1p1e2 |
⊢ ( 1 + 1 ) = 2 |
| 258 |
257
|
oveq2i |
⊢ ( 1 ... ( 1 + 1 ) ) = ( 1 ... 2 ) |
| 259 |
258
|
eleq2i |
⊢ ( 𝑥 ∈ ( 1 ... ( 1 + 1 ) ) ↔ 𝑥 ∈ ( 1 ... 2 ) ) |
| 260 |
257
|
eqeq2i |
⊢ ( 𝑥 = ( 1 + 1 ) ↔ 𝑥 = 2 ) |
| 261 |
260
|
orbi2i |
⊢ ( ( 𝑥 ∈ ( 1 ... 1 ) ∨ 𝑥 = ( 1 + 1 ) ) ↔ ( 𝑥 ∈ ( 1 ... 1 ) ∨ 𝑥 = 2 ) ) |
| 262 |
256 259 261
|
3bitr3i |
⊢ ( 𝑥 ∈ ( 1 ... 2 ) ↔ ( 𝑥 ∈ ( 1 ... 1 ) ∨ 𝑥 = 2 ) ) |
| 263 |
|
elfz1eq |
⊢ ( 𝑥 ∈ ( 1 ... 1 ) → 𝑥 = 1 ) |
| 264 |
263
|
orim1i |
⊢ ( ( 𝑥 ∈ ( 1 ... 1 ) ∨ 𝑥 = 2 ) → ( 𝑥 = 1 ∨ 𝑥 = 2 ) ) |
| 265 |
262 264
|
sylbi |
⊢ ( 𝑥 ∈ ( 1 ... 2 ) → ( 𝑥 = 1 ∨ 𝑥 = 2 ) ) |
| 266 |
265
|
orim1i |
⊢ ( ( 𝑥 ∈ ( 1 ... 2 ) ∨ 𝑥 = 3 ) → ( ( 𝑥 = 1 ∨ 𝑥 = 2 ) ∨ 𝑥 = 3 ) ) |
| 267 |
252 266
|
sylbi |
⊢ ( 𝑥 ∈ ( 1 ... 3 ) → ( ( 𝑥 = 1 ∨ 𝑥 = 2 ) ∨ 𝑥 = 3 ) ) |
| 268 |
267
|
orim1i |
⊢ ( ( 𝑥 ∈ ( 1 ... 3 ) ∨ 𝑥 = 4 ) → ( ( ( 𝑥 = 1 ∨ 𝑥 = 2 ) ∨ 𝑥 = 3 ) ∨ 𝑥 = 4 ) ) |
| 269 |
243 268
|
sylbi |
⊢ ( 𝑥 ∈ ( 1 ... 4 ) → ( ( ( 𝑥 = 1 ∨ 𝑥 = 2 ) ∨ 𝑥 = 3 ) ∨ 𝑥 = 4 ) ) |
| 270 |
269
|
orim1i |
⊢ ( ( 𝑥 ∈ ( 1 ... 4 ) ∨ 𝑥 = 5 ) → ( ( ( ( 𝑥 = 1 ∨ 𝑥 = 2 ) ∨ 𝑥 = 3 ) ∨ 𝑥 = 4 ) ∨ 𝑥 = 5 ) ) |
| 271 |
233 270
|
sylbi |
⊢ ( 𝑥 ∈ ( 1 ... 5 ) → ( ( ( ( 𝑥 = 1 ∨ 𝑥 = 2 ) ∨ 𝑥 = 3 ) ∨ 𝑥 = 4 ) ∨ 𝑥 = 5 ) ) |
| 272 |
271
|
orim1i |
⊢ ( ( 𝑥 ∈ ( 1 ... 5 ) ∨ 𝑥 = 6 ) → ( ( ( ( ( 𝑥 = 1 ∨ 𝑥 = 2 ) ∨ 𝑥 = 3 ) ∨ 𝑥 = 4 ) ∨ 𝑥 = 5 ) ∨ 𝑥 = 6 ) ) |
| 273 |
223 272
|
sylbi |
⊢ ( 𝑥 ∈ ( 1 ... 6 ) → ( ( ( ( ( 𝑥 = 1 ∨ 𝑥 = 2 ) ∨ 𝑥 = 3 ) ∨ 𝑥 = 4 ) ∨ 𝑥 = 5 ) ∨ 𝑥 = 6 ) ) |
| 274 |
213 273
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) → ( ( ( ( ( 𝑥 = 1 ∨ 𝑥 = 2 ) ∨ 𝑥 = 3 ) ∨ 𝑥 = 4 ) ∨ 𝑥 = 5 ) ∨ 𝑥 = 6 ) ) |
| 275 |
173 212 274
|
mpjaodan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 6 ) ) → ( ( veronese ‘ 𝑃 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ‘ 𝑥 ) ) |
| 276 |
7 21 275
|
eqfnfvd |
⊢ ( 𝜑 → ( veronese ‘ 𝑃 ) = ( 𝑘 ∈ ( 1 ... 6 ) ↦ if ( 𝑘 = 1 , ( ( 𝑃 ‘ 1 ) ↑ 2 ) , if ( 𝑘 = 2 , ( ( 𝑃 ‘ 2 ) ↑ 2 ) , if ( 𝑘 = 3 , ( ( 𝑃 ‘ 3 ) ↑ 2 ) , if ( 𝑘 = 4 , ( ( 𝑃 ‘ 1 ) · ( 𝑃 ‘ 2 ) ) , if ( 𝑘 = 5 , ( ( 𝑃 ‘ 2 ) · ( 𝑃 ‘ 3 ) ) , ( ( 𝑃 ‘ 3 ) · ( 𝑃 ‘ 1 ) ) ) ) ) ) ) ) ) |