Metamath Proof Explorer

Theorem veronesev5lem

Description: Lemma for veronesevrowd . Value of the fifth coordinate of the Veronese map at a point. (Contributed by Jiamin Zhao, 17-Aug-2026)

Ref Expression
Hypothesis veronesevrow.1 
|- ( ph -> P e. ( RR ^m ( 1 ... 3 ) ) )
Assertion veronesev5lem 
|- ( ph -> ( ( veronese ` P ) ` 5 ) = ( ( P ` 2 ) x. ( P ` 3 ) ) )

Proof

Step Hyp Ref Expression
1 veronesevrow.1 
 |-  ( ph -> P e. ( RR ^m ( 1 ... 3 ) ) )
2 1 veronesevald 
 |-  ( ph -> ( veronese ` P ) = ( x e. ( 1 ... 6 ) |-> ( ( ( if ( x = 1 , ( ( P ` 1 ) ^ 2 ) , 0 ) + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) + ( ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) ) )
3 1re 
 |-  1 e. RR
4 1lt5 
 |-  1 < 5
5 3 4 gtneii 
 |-  5 =/= 1
6 neeq1 
 |-  ( x = 5 -> ( x =/= 1 <-> 5 =/= 1 ) )
7 5 6 mpbiri 
 |-  ( x = 5 -> x =/= 1 )
8 7 neneqd 
 |-  ( x = 5 -> -. x = 1 )
9 8 iffalsed 
 |-  ( x = 5 -> if ( x = 1 , ( ( P ` 1 ) ^ 2 ) , 0 ) = 0 )
10 9 oveq1d 
 |-  ( x = 5 -> ( if ( x = 1 , ( ( P ` 1 ) ^ 2 ) , 0 ) + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) = ( 0 + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) )
11 10 oveq1d 
 |-  ( x = 5 -> ( ( if ( x = 1 , ( ( P ` 1 ) ^ 2 ) , 0 ) + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) = ( ( 0 + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) )
12 iftrue 
 |-  ( x = 5 -> if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) = ( ( P ` 2 ) x. ( P ` 3 ) ) )
13 12 oveq2d 
 |-  ( x = 5 -> ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) ) = ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + ( ( P ` 2 ) x. ( P ` 3 ) ) ) )
14 13 oveq1d 
 |-  ( x = 5 -> ( ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) = ( ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) )
15 11 14 oveq12d 
 |-  ( x = 5 -> ( ( ( if ( x = 1 , ( ( P ` 1 ) ^ 2 ) , 0 ) + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) + ( ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) = ( ( ( 0 + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) + ( ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) )
16 2re 
 |-  2 e. RR
17 2lt5 
 |-  2 < 5
18 16 17 gtneii 
 |-  5 =/= 2
19 neeq1 
 |-  ( x = 5 -> ( x =/= 2 <-> 5 =/= 2 ) )
20 18 19 mpbiri 
 |-  ( x = 5 -> x =/= 2 )
21 20 neneqd 
 |-  ( x = 5 -> -. x = 2 )
22 21 iffalsed 
 |-  ( x = 5 -> if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) = 0 )
23 22 oveq2d 
 |-  ( x = 5 -> ( 0 + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) = ( 0 + 0 ) )
24 23 oveq1d 
 |-  ( x = 5 -> ( ( 0 + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) = ( ( 0 + 0 ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) )
25 24 oveq1d 
 |-  ( x = 5 -> ( ( ( 0 + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) + ( ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) = ( ( ( 0 + 0 ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) + ( ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) )
26 3re 
 |-  3 e. RR
27 3lt5 
 |-  3 < 5
28 26 27 gtneii 
 |-  5 =/= 3
29 neeq1 
 |-  ( x = 5 -> ( x =/= 3 <-> 5 =/= 3 ) )
30 28 29 mpbiri 
 |-  ( x = 5 -> x =/= 3 )
31 30 neneqd 
 |-  ( x = 5 -> -. x = 3 )
32 31 iffalsed 
 |-  ( x = 5 -> if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) = 0 )
33 32 oveq2d 
 |-  ( x = 5 -> ( ( 0 + 0 ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) = ( ( 0 + 0 ) + 0 ) )
34 33 oveq1d 
 |-  ( x = 5 -> ( ( ( 0 + 0 ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) + ( ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) = ( ( ( 0 + 0 ) + 0 ) + ( ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) )
35 15 25 34 3eqtrd 
 |-  ( x = 5 -> ( ( ( if ( x = 1 , ( ( P ` 1 ) ^ 2 ) , 0 ) + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) + ( ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) = ( ( ( 0 + 0 ) + 0 ) + ( ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) )
36 4re 
 |-  4 e. RR
37 4lt5 
 |-  4 < 5
38 36 37 gtneii 
 |-  5 =/= 4
39 neeq1 
 |-  ( x = 5 -> ( x =/= 4 <-> 5 =/= 4 ) )
40 38 39 mpbiri 
 |-  ( x = 5 -> x =/= 4 )
41 40 neneqd 
 |-  ( x = 5 -> -. x = 4 )
42 41 iffalsed 
 |-  ( x = 5 -> if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) = 0 )
43 42 oveq1d 
 |-  ( x = 5 -> ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + ( ( P ` 2 ) x. ( P ` 3 ) ) ) = ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) )
44 43 oveq1d 
 |-  ( x = 5 -> ( ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) = ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) )
45 44 oveq2d 
 |-  ( x = 5 -> ( ( ( 0 + 0 ) + 0 ) + ( ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) = ( ( ( 0 + 0 ) + 0 ) + ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) )
46 5re 
 |-  5 e. RR
47 5lt6 
 |-  5 < 6
48 46 47 ltneii 
 |-  5 =/= 6
49 neeq1 
 |-  ( x = 5 -> ( x =/= 6 <-> 5 =/= 6 ) )
50 48 49 mpbiri 
 |-  ( x = 5 -> x =/= 6 )
51 50 neneqd 
 |-  ( x = 5 -> -. x = 6 )
52 51 iffalsed 
 |-  ( x = 5 -> if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) = 0 )
53 52 oveq2d 
 |-  ( x = 5 -> ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) = ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + 0 ) )
54 53 oveq2d 
 |-  ( x = 5 -> ( ( ( 0 + 0 ) + 0 ) + ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) = ( ( ( 0 + 0 ) + 0 ) + ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + 0 ) ) )
55 35 45 54 3eqtrd 
 |-  ( x = 5 -> ( ( ( if ( x = 1 , ( ( P ` 1 ) ^ 2 ) , 0 ) + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) + ( ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) = ( ( ( 0 + 0 ) + 0 ) + ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + 0 ) ) )
56 55 adantl 
 |-  ( ( ph /\ x = 5 ) -> ( ( ( if ( x = 1 , ( ( P ` 1 ) ^ 2 ) , 0 ) + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) + ( ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) = ( ( ( 0 + 0 ) + 0 ) + ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + 0 ) ) )
57 0red 
 |-  ( ( ph /\ x = 5 ) -> 0 e. RR )
58 57 57 readdcld 
 |-  ( ( ph /\ x = 5 ) -> ( 0 + 0 ) e. RR )
59 58 recnd 
 |-  ( ( ph /\ x = 5 ) -> ( 0 + 0 ) e. CC )
60 59 addridd 
 |-  ( ( ph /\ x = 5 ) -> ( ( 0 + 0 ) + 0 ) = ( 0 + 0 ) )
61 60 oveq1d 
 |-  ( ( ph /\ x = 5 ) -> ( ( ( 0 + 0 ) + 0 ) + ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + 0 ) ) = ( ( 0 + 0 ) + ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + 0 ) ) )
62 00id 
 |-  ( 0 + 0 ) = 0
63 62 a1i 
 |-  ( ( ph /\ x = 5 ) -> ( 0 + 0 ) = 0 )
64 63 oveq1d 
 |-  ( ( ph /\ x = 5 ) -> ( ( 0 + 0 ) + ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + 0 ) ) = ( 0 + ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + 0 ) ) )
65 61 64 eqtrd 
 |-  ( ( ph /\ x = 5 ) -> ( ( ( 0 + 0 ) + 0 ) + ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + 0 ) ) = ( 0 + ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + 0 ) ) )
66 1 rr3fv2cld 
 |-  ( ph -> ( P ` 2 ) e. RR )
67 66 adantr 
 |-  ( ( ph /\ x = 5 ) -> ( P ` 2 ) e. RR )
68 1 rr3fv3cld 
 |-  ( ph -> ( P ` 3 ) e. RR )
69 68 adantr 
 |-  ( ( ph /\ x = 5 ) -> ( P ` 3 ) e. RR )
70 67 69 remulcld 
 |-  ( ( ph /\ x = 5 ) -> ( ( P ` 2 ) x. ( P ` 3 ) ) e. RR )
71 57 70 readdcld 
 |-  ( ( ph /\ x = 5 ) -> ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) e. RR )
72 71 recnd 
 |-  ( ( ph /\ x = 5 ) -> ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) e. CC )
73 72 addridd 
 |-  ( ( ph /\ x = 5 ) -> ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + 0 ) = ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) )
74 73 oveq2d 
 |-  ( ( ph /\ x = 5 ) -> ( 0 + ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + 0 ) ) = ( 0 + ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) ) )
75 70 recnd 
 |-  ( ( ph /\ x = 5 ) -> ( ( P ` 2 ) x. ( P ` 3 ) ) e. CC )
76 75 addlidd 
 |-  ( ( ph /\ x = 5 ) -> ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) = ( ( P ` 2 ) x. ( P ` 3 ) ) )
77 76 oveq2d 
 |-  ( ( ph /\ x = 5 ) -> ( 0 + ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) ) = ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) )
78 65 74 77 3eqtrd 
 |-  ( ( ph /\ x = 5 ) -> ( ( ( 0 + 0 ) + 0 ) + ( ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) + 0 ) ) = ( 0 + ( ( P ` 2 ) x. ( P ` 3 ) ) ) )
79 56 78 76 3eqtrd 
 |-  ( ( ph /\ x = 5 ) -> ( ( ( if ( x = 1 , ( ( P ` 1 ) ^ 2 ) , 0 ) + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) + ( ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) = ( ( P ` 2 ) x. ( P ` 3 ) ) )
80 1zzd 
 |-  ( ph -> 1 e. ZZ )
81 6nn 
 |-  6 e. NN
82 81 nnzi 
 |-  6 e. ZZ
83 82 a1i 
 |-  ( ph -> 6 e. ZZ )
84 5nn 
 |-  5 e. NN
85 84 nnzi 
 |-  5 e. ZZ
86 85 a1i 
 |-  ( ph -> 5 e. ZZ )
87 3 46 4 ltleii 
 |-  1 <_ 5
88 87 a1i 
 |-  ( ph -> 1 <_ 5 )
89 6re 
 |-  6 e. RR
90 46 89 47 ltleii 
 |-  5 <_ 6
91 90 a1i 
 |-  ( ph -> 5 <_ 6 )
92 80 83 86 88 91 elfzd 
 |-  ( ph -> 5 e. ( 1 ... 6 ) )
93 66 68 remulcld 
 |-  ( ph -> ( ( P ` 2 ) x. ( P ` 3 ) ) e. RR )
94 2 79 92 93 fvmptd 
 |-  ( ph -> ( ( veronese ` P ) ` 5 ) = ( ( P ` 2 ) x. ( P ` 3 ) ) )