Metamath Proof Explorer

Theorem veronesev1lem

Description: Lemma for veronesevrowd . Value of the first coordinate of the Veronese map at a point. (Contributed by Jiamin Zhao, 15-Aug-2026)

Ref Expression
Hypothesis veronesevrow.1 
|- ( ph -> P e. ( RR ^m ( 1 ... 3 ) ) )
Assertion veronesev1lem 
|- ( ph -> ( ( veronese ` P ) ` 1 ) = ( ( P ` 1 ) ^ 2 ) )

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 iftrue 
 |-  ( x = 1 -> if ( x = 1 , ( ( P ` 1 ) ^ 2 ) , 0 ) = ( ( P ` 1 ) ^ 2 ) )
4 3 oveq1d 
 |-  ( x = 1 -> ( if ( x = 1 , ( ( P ` 1 ) ^ 2 ) , 0 ) + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) = ( ( ( P ` 1 ) ^ 2 ) + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) )
5 4 oveq1d 
 |-  ( x = 1 -> ( ( if ( x = 1 , ( ( P ` 1 ) ^ 2 ) , 0 ) + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) = ( ( ( ( P ` 1 ) ^ 2 ) + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) )
6 5 oveq1d 
 |-  ( x = 1 -> ( ( ( 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 ` 1 ) ^ 2 ) + 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 ) ) ) )
7 1ne2 
 |-  1 =/= 2
8 neeq1 
 |-  ( x = 1 -> ( x =/= 2 <-> 1 =/= 2 ) )
9 7 8 mpbiri 
 |-  ( x = 1 -> x =/= 2 )
10 9 neneqd 
 |-  ( x = 1 -> -. x = 2 )
11 10 iffalsed 
 |-  ( x = 1 -> if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) = 0 )
12 11 oveq2d 
 |-  ( x = 1 -> ( ( ( P ` 1 ) ^ 2 ) + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) = ( ( ( P ` 1 ) ^ 2 ) + 0 ) )
13 12 oveq1d 
 |-  ( x = 1 -> ( ( ( ( P ` 1 ) ^ 2 ) + if ( x = 2 , ( ( P ` 2 ) ^ 2 ) , 0 ) ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) = ( ( ( ( P ` 1 ) ^ 2 ) + 0 ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) )
14 13 oveq1d 
 |-  ( x = 1 -> ( ( ( ( ( P ` 1 ) ^ 2 ) + 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 ` 1 ) ^ 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 ) ) ) )
15 6 14 eqtrd 
 |-  ( x = 1 -> ( ( ( 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 ` 1 ) ^ 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 ) ) ) )
16 1ne3 
 |-  1 =/= 3
17 neeq1 
 |-  ( x = 1 -> ( x =/= 3 <-> 1 =/= 3 ) )
18 16 17 mpbiri 
 |-  ( x = 1 -> x =/= 3 )
19 18 neneqd 
 |-  ( x = 1 -> -. x = 3 )
20 19 iffalsed 
 |-  ( x = 1 -> if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) = 0 )
21 20 oveq2d 
 |-  ( x = 1 -> ( ( ( ( P ` 1 ) ^ 2 ) + 0 ) + if ( x = 3 , ( ( P ` 3 ) ^ 2 ) , 0 ) ) = ( ( ( ( P ` 1 ) ^ 2 ) + 0 ) + 0 ) )
22 21 oveq1d 
 |-  ( x = 1 -> ( ( ( ( ( P ` 1 ) ^ 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 ` 1 ) ^ 2 ) + 0 ) + 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 ) ) ) )
23 1re 
 |-  1 e. RR
24 1lt4 
 |-  1 < 4
25 23 24 ltneii 
 |-  1 =/= 4
26 neeq1 
 |-  ( x = 1 -> ( x =/= 4 <-> 1 =/= 4 ) )
27 25 26 mpbiri 
 |-  ( x = 1 -> x =/= 4 )
28 27 neneqd 
 |-  ( x = 1 -> -. x = 4 )
29 28 iffalsed 
 |-  ( x = 1 -> if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) = 0 )
30 29 oveq1d 
 |-  ( x = 1 -> ( if ( x = 4 , ( ( P ` 1 ) x. ( P ` 2 ) ) , 0 ) + if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) ) = ( 0 + if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) ) )
31 30 oveq1d 
 |-  ( x = 1 -> ( ( 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 = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) )
32 31 oveq2d 
 |-  ( x = 1 -> ( ( ( ( ( P ` 1 ) ^ 2 ) + 0 ) + 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 ` 1 ) ^ 2 ) + 0 ) + 0 ) + ( ( 0 + if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) )
33 15 22 32 3eqtrd 
 |-  ( x = 1 -> ( ( ( 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 ` 1 ) ^ 2 ) + 0 ) + 0 ) + ( ( 0 + if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) )
34 1lt5 
 |-  1 < 5
35 23 34 ltneii 
 |-  1 =/= 5
36 neeq1 
 |-  ( x = 1 -> ( x =/= 5 <-> 1 =/= 5 ) )
37 35 36 mpbiri 
 |-  ( x = 1 -> x =/= 5 )
38 37 neneqd 
 |-  ( x = 1 -> -. x = 5 )
39 38 iffalsed 
 |-  ( x = 1 -> if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) = 0 )
40 39 oveq2d 
 |-  ( x = 1 -> ( 0 + if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) ) = ( 0 + 0 ) )
41 40 oveq1d 
 |-  ( x = 1 -> ( ( 0 + if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) = ( ( 0 + 0 ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) )
42 41 oveq2d 
 |-  ( x = 1 -> ( ( ( ( ( P ` 1 ) ^ 2 ) + 0 ) + 0 ) + ( ( 0 + if ( x = 5 , ( ( P ` 2 ) x. ( P ` 3 ) ) , 0 ) ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) = ( ( ( ( ( P ` 1 ) ^ 2 ) + 0 ) + 0 ) + ( ( 0 + 0 ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) )
43 1lt6 
 |-  1 < 6
44 23 43 ltneii 
 |-  1 =/= 6
45 neeq1 
 |-  ( x = 1 -> ( x =/= 6 <-> 1 =/= 6 ) )
46 44 45 mpbiri 
 |-  ( x = 1 -> x =/= 6 )
47 46 neneqd 
 |-  ( x = 1 -> -. x = 6 )
48 47 iffalsed 
 |-  ( x = 1 -> if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) = 0 )
49 48 oveq2d 
 |-  ( x = 1 -> ( ( 0 + 0 ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) = ( ( 0 + 0 ) + 0 ) )
50 49 oveq2d 
 |-  ( x = 1 -> ( ( ( ( ( P ` 1 ) ^ 2 ) + 0 ) + 0 ) + ( ( 0 + 0 ) + if ( x = 6 , ( ( P ` 3 ) x. ( P ` 1 ) ) , 0 ) ) ) = ( ( ( ( ( P ` 1 ) ^ 2 ) + 0 ) + 0 ) + ( ( 0 + 0 ) + 0 ) ) )
51 33 42 50 3eqtrd 
 |-  ( x = 1 -> ( ( ( 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 ` 1 ) ^ 2 ) + 0 ) + 0 ) + ( ( 0 + 0 ) + 0 ) ) )
52 51 adantl 
 |-  ( ( ph /\ x = 1 ) -> ( ( ( 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 ` 1 ) ^ 2 ) + 0 ) + 0 ) + ( ( 0 + 0 ) + 0 ) ) )
53 1 rr3fv1cld 
 |-  ( ph -> ( P ` 1 ) e. RR )
54 53 adantr 
 |-  ( ( ph /\ x = 1 ) -> ( P ` 1 ) e. RR )
55 54 resqcld 
 |-  ( ( ph /\ x = 1 ) -> ( ( P ` 1 ) ^ 2 ) e. RR )
56 0red 
 |-  ( ( ph /\ x = 1 ) -> 0 e. RR )
57 55 56 readdcld 
 |-  ( ( ph /\ x = 1 ) -> ( ( ( P ` 1 ) ^ 2 ) + 0 ) e. RR )
58 57 recnd 
 |-  ( ( ph /\ x = 1 ) -> ( ( ( P ` 1 ) ^ 2 ) + 0 ) e. CC )
59 58 addridd 
 |-  ( ( ph /\ x = 1 ) -> ( ( ( ( P ` 1 ) ^ 2 ) + 0 ) + 0 ) = ( ( ( P ` 1 ) ^ 2 ) + 0 ) )
60 59 oveq1d 
 |-  ( ( ph /\ x = 1 ) -> ( ( ( ( ( P ` 1 ) ^ 2 ) + 0 ) + 0 ) + ( ( 0 + 0 ) + 0 ) ) = ( ( ( ( P ` 1 ) ^ 2 ) + 0 ) + ( ( 0 + 0 ) + 0 ) ) )
61 55 recnd 
 |-  ( ( ph /\ x = 1 ) -> ( ( P ` 1 ) ^ 2 ) e. CC )
62 61 addridd 
 |-  ( ( ph /\ x = 1 ) -> ( ( ( P ` 1 ) ^ 2 ) + 0 ) = ( ( P ` 1 ) ^ 2 ) )
63 62 oveq1d 
 |-  ( ( ph /\ x = 1 ) -> ( ( ( ( P ` 1 ) ^ 2 ) + 0 ) + ( ( 0 + 0 ) + 0 ) ) = ( ( ( P ` 1 ) ^ 2 ) + ( ( 0 + 0 ) + 0 ) ) )
64 60 63 eqtrd 
 |-  ( ( ph /\ x = 1 ) -> ( ( ( ( ( P ` 1 ) ^ 2 ) + 0 ) + 0 ) + ( ( 0 + 0 ) + 0 ) ) = ( ( ( P ` 1 ) ^ 2 ) + ( ( 0 + 0 ) + 0 ) ) )
65 56 56 readdcld 
 |-  ( ( ph /\ x = 1 ) -> ( 0 + 0 ) e. RR )
66 65 recnd 
 |-  ( ( ph /\ x = 1 ) -> ( 0 + 0 ) e. CC )
67 66 addridd 
 |-  ( ( ph /\ x = 1 ) -> ( ( 0 + 0 ) + 0 ) = ( 0 + 0 ) )
68 67 oveq2d 
 |-  ( ( ph /\ x = 1 ) -> ( ( ( P ` 1 ) ^ 2 ) + ( ( 0 + 0 ) + 0 ) ) = ( ( ( P ` 1 ) ^ 2 ) + ( 0 + 0 ) ) )
69 00id 
 |-  ( 0 + 0 ) = 0
70 69 a1i 
 |-  ( ( ph /\ x = 1 ) -> ( 0 + 0 ) = 0 )
71 70 oveq2d 
 |-  ( ( ph /\ x = 1 ) -> ( ( ( P ` 1 ) ^ 2 ) + ( 0 + 0 ) ) = ( ( ( P ` 1 ) ^ 2 ) + 0 ) )
72 64 68 71 3eqtrd 
 |-  ( ( ph /\ x = 1 ) -> ( ( ( ( ( P ` 1 ) ^ 2 ) + 0 ) + 0 ) + ( ( 0 + 0 ) + 0 ) ) = ( ( ( P ` 1 ) ^ 2 ) + 0 ) )
73 52 72 62 3eqtrd 
 |-  ( ( ph /\ x = 1 ) -> ( ( ( 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 ` 1 ) ^ 2 ) )
74 1zzd 
 |-  ( ph -> 1 e. ZZ )
75 6nn 
 |-  6 e. NN
76 75 nnzi 
 |-  6 e. ZZ
77 76 a1i 
 |-  ( ph -> 6 e. ZZ )
78 1le1 
 |-  1 <_ 1
79 78 a1i 
 |-  ( ph -> 1 <_ 1 )
80 6re 
 |-  6 e. RR
81 23 80 43 ltleii 
 |-  1 <_ 6
82 81 a1i 
 |-  ( ph -> 1 <_ 6 )
83 74 77 74 79 82 elfzd 
 |-  ( ph -> 1 e. ( 1 ... 6 ) )
84 53 resqcld 
 |-  ( ph -> ( ( P ` 1 ) ^ 2 ) e. RR )
85 2 73 83 84 fvmptd 
 |-  ( ph -> ( ( veronese ` P ) ` 1 ) = ( ( P ` 1 ) ^ 2 ) )