Metamath Proof Explorer


Theorem nn0sumshdiglem2

Description: Lemma 2 for nn0sumshdig . (Contributed by AV, 7-Jun-2020)

Ref Expression
Assertion nn0sumshdiglem2 L a 0 # b a = L a = k 0 ..^ L k digit 2 a 2 k

Proof

Step Hyp Ref Expression
1 eqeq2 x = 1 # b a = x # b a = 1
2 oveq2 x = 1 0 ..^ x = 0 ..^ 1
3 fzo01 0 ..^ 1 = 0
4 2 3 eqtrdi x = 1 0 ..^ x = 0
5 4 sumeq1d x = 1 k 0 ..^ x k digit 2 a 2 k = k 0 k digit 2 a 2 k
6 5 eqeq2d x = 1 a = k 0 ..^ x k digit 2 a 2 k a = k 0 k digit 2 a 2 k
7 1 6 imbi12d x = 1 # b a = x a = k 0 ..^ x k digit 2 a 2 k # b a = 1 a = k 0 k digit 2 a 2 k
8 7 ralbidv x = 1 a 0 # b a = x a = k 0 ..^ x k digit 2 a 2 k a 0 # b a = 1 a = k 0 k digit 2 a 2 k
9 eqeq2 x = y # b a = x # b a = y
10 oveq2 x = y 0 ..^ x = 0 ..^ y
11 10 sumeq1d x = y k 0 ..^ x k digit 2 a 2 k = k 0 ..^ y k digit 2 a 2 k
12 11 eqeq2d x = y a = k 0 ..^ x k digit 2 a 2 k a = k 0 ..^ y k digit 2 a 2 k
13 9 12 imbi12d x = y # b a = x a = k 0 ..^ x k digit 2 a 2 k # b a = y a = k 0 ..^ y k digit 2 a 2 k
14 13 ralbidv x = y a 0 # b a = x a = k 0 ..^ x k digit 2 a 2 k a 0 # b a = y a = k 0 ..^ y k digit 2 a 2 k
15 eqeq2 x = y + 1 # b a = x # b a = y + 1
16 oveq2 x = y + 1 0 ..^ x = 0 ..^ y + 1
17 16 sumeq1d x = y + 1 k 0 ..^ x k digit 2 a 2 k = k 0 ..^ y + 1 k digit 2 a 2 k
18 17 eqeq2d x = y + 1 a = k 0 ..^ x k digit 2 a 2 k a = k 0 ..^ y + 1 k digit 2 a 2 k
19 15 18 imbi12d x = y + 1 # b a = x a = k 0 ..^ x k digit 2 a 2 k # b a = y + 1 a = k 0 ..^ y + 1 k digit 2 a 2 k
20 19 ralbidv x = y + 1 a 0 # b a = x a = k 0 ..^ x k digit 2 a 2 k a 0 # b a = y + 1 a = k 0 ..^ y + 1 k digit 2 a 2 k
21 eqeq2 x = L # b a = x # b a = L
22 oveq2 x = L 0 ..^ x = 0 ..^ L
23 22 sumeq1d x = L k 0 ..^ x k digit 2 a 2 k = k 0 ..^ L k digit 2 a 2 k
24 23 eqeq2d x = L a = k 0 ..^ x k digit 2 a 2 k a = k 0 ..^ L k digit 2 a 2 k
25 21 24 imbi12d x = L # b a = x a = k 0 ..^ x k digit 2 a 2 k # b a = L a = k 0 ..^ L k digit 2 a 2 k
26 25 ralbidv x = L a 0 # b a = x a = k 0 ..^ x k digit 2 a 2 k a 0 # b a = L a = k 0 ..^ L k digit 2 a 2 k
27 0cnd a 0 0
28 2nn 2
29 28 a1i a 0 2
30 0zd a 0 0
31 nn0rp0 a 0 a 0 +∞
32 digvalnn0 2 0 a 0 +∞ 0 digit 2 a 0
33 29 30 31 32 syl3anc a 0 0 digit 2 a 0
34 33 nn0cnd a 0 0 digit 2 a
35 1cnd a 0 1
36 34 35 mulcld a 0 0 digit 2 a 1
37 27 36 jca a 0 0 0 digit 2 a 1
38 37 adantr a 0 # b a = 1 0 0 digit 2 a 1
39 oveq1 k = 0 k digit 2 a = 0 digit 2 a
40 oveq2 k = 0 2 k = 2 0
41 2cn 2
42 exp0 2 2 0 = 1
43 41 42 ax-mp 2 0 = 1
44 40 43 eqtrdi k = 0 2 k = 1
45 39 44 oveq12d k = 0 k digit 2 a 2 k = 0 digit 2 a 1
46 45 sumsn 0 0 digit 2 a 1 k 0 k digit 2 a 2 k = 0 digit 2 a 1
47 38 46 syl a 0 # b a = 1 k 0 k digit 2 a 2 k = 0 digit 2 a 1
48 34 adantr a 0 # b a = 1 0 digit 2 a
49 48 mulid1d a 0 # b a = 1 0 digit 2 a 1 = 0 digit 2 a
50 blen1b a 0 # b a = 1 a = 0 a = 1
51 50 biimpa a 0 # b a = 1 a = 0 a = 1
52 vex a V
53 52 elpr a 0 1 a = 0 a = 1
54 51 53 sylibr a 0 # b a = 1 a 0 1
55 0dig2pr01 a 0 1 0 digit 2 a = a
56 54 55 syl a 0 # b a = 1 0 digit 2 a = a
57 47 49 56 3eqtrrd a 0 # b a = 1 a = k 0 k digit 2 a 2 k
58 57 ex a 0 # b a = 1 a = k 0 k digit 2 a 2 k
59 58 rgen a 0 # b a = 1 a = k 0 k digit 2 a 2 k
60 nn0sumshdiglem1 y a 0 # b a = y a = k 0 ..^ y k digit 2 a 2 k a 0 # b a = y + 1 a = k 0 ..^ y + 1 k digit 2 a 2 k
61 8 14 20 26 59 60 nnind L a 0 # b a = L a = k 0 ..^ L k digit 2 a 2 k