Step |
Hyp |
Ref |
Expression |
1 |
|
peano2nn0 |
⊢ ( 𝐾 ∈ ℕ0 → ( 𝐾 + 1 ) ∈ ℕ0 ) |
2 |
|
vdwapval |
⊢ ( ( ( 𝐾 + 1 ) ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝑥 ∈ ( 𝐴 ( AP ‘ ( 𝐾 + 1 ) ) 𝐷 ) ↔ ∃ 𝑛 ∈ ( 0 ... ( ( 𝐾 + 1 ) − 1 ) ) 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) |
3 |
1 2
|
syl3an1 |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝑥 ∈ ( 𝐴 ( AP ‘ ( 𝐾 + 1 ) ) 𝐷 ) ↔ ∃ 𝑛 ∈ ( 0 ... ( ( 𝐾 + 1 ) − 1 ) ) 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) |
4 |
|
simp1 |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → 𝐾 ∈ ℕ0 ) |
5 |
4
|
nn0cnd |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → 𝐾 ∈ ℂ ) |
6 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
7 |
|
pncan |
⊢ ( ( 𝐾 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝐾 + 1 ) − 1 ) = 𝐾 ) |
8 |
5 6 7
|
sylancl |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( ( 𝐾 + 1 ) − 1 ) = 𝐾 ) |
9 |
8
|
oveq2d |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 0 ... ( ( 𝐾 + 1 ) − 1 ) ) = ( 0 ... 𝐾 ) ) |
10 |
9
|
eleq2d |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝑛 ∈ ( 0 ... ( ( 𝐾 + 1 ) − 1 ) ) ↔ 𝑛 ∈ ( 0 ... 𝐾 ) ) ) |
11 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
12 |
4 11
|
eleqtrdi |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → 𝐾 ∈ ( ℤ≥ ‘ 0 ) ) |
13 |
|
elfzp12 |
⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 0 ) → ( 𝑛 ∈ ( 0 ... 𝐾 ) ↔ ( 𝑛 = 0 ∨ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) ) ) |
14 |
12 13
|
syl |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝑛 ∈ ( 0 ... 𝐾 ) ↔ ( 𝑛 = 0 ∨ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) ) ) |
15 |
10 14
|
bitrd |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝑛 ∈ ( 0 ... ( ( 𝐾 + 1 ) − 1 ) ) ↔ ( 𝑛 = 0 ∨ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) ) ) |
16 |
15
|
anbi1d |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( ( 𝑛 ∈ ( 0 ... ( ( 𝐾 + 1 ) − 1 ) ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ↔ ( ( 𝑛 = 0 ∨ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) ) |
17 |
|
andir |
⊢ ( ( ( 𝑛 = 0 ∨ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ↔ ( ( 𝑛 = 0 ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ∨ ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) ) |
18 |
16 17
|
bitrdi |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( ( 𝑛 ∈ ( 0 ... ( ( 𝐾 + 1 ) − 1 ) ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ↔ ( ( 𝑛 = 0 ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ∨ ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) ) ) |
19 |
18
|
exbidv |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( ∃ 𝑛 ( 𝑛 ∈ ( 0 ... ( ( 𝐾 + 1 ) − 1 ) ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ↔ ∃ 𝑛 ( ( 𝑛 = 0 ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ∨ ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) ) ) |
20 |
|
df-rex |
⊢ ( ∃ 𝑛 ∈ ( 0 ... ( ( 𝐾 + 1 ) − 1 ) ) 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ↔ ∃ 𝑛 ( 𝑛 ∈ ( 0 ... ( ( 𝐾 + 1 ) − 1 ) ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) |
21 |
|
19.43 |
⊢ ( ∃ 𝑛 ( ( 𝑛 = 0 ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ∨ ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) ↔ ( ∃ 𝑛 ( 𝑛 = 0 ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ∨ ∃ 𝑛 ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) ) |
22 |
21
|
bicomi |
⊢ ( ( ∃ 𝑛 ( 𝑛 = 0 ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ∨ ∃ 𝑛 ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) ↔ ∃ 𝑛 ( ( 𝑛 = 0 ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ∨ ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) ) |
23 |
19 20 22
|
3bitr4g |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( ∃ 𝑛 ∈ ( 0 ... ( ( 𝐾 + 1 ) − 1 ) ) 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ↔ ( ∃ 𝑛 ( 𝑛 = 0 ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ∨ ∃ 𝑛 ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) ) ) |
24 |
3 23
|
bitrd |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝑥 ∈ ( 𝐴 ( AP ‘ ( 𝐾 + 1 ) ) 𝐷 ) ↔ ( ∃ 𝑛 ( 𝑛 = 0 ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ∨ ∃ 𝑛 ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) ) ) |
25 |
|
nncn |
⊢ ( 𝐷 ∈ ℕ → 𝐷 ∈ ℂ ) |
26 |
25
|
3ad2ant3 |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → 𝐷 ∈ ℂ ) |
27 |
26
|
mul02d |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 0 · 𝐷 ) = 0 ) |
28 |
27
|
oveq2d |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝐴 + ( 0 · 𝐷 ) ) = ( 𝐴 + 0 ) ) |
29 |
|
nncn |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℂ ) |
30 |
29
|
3ad2ant2 |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → 𝐴 ∈ ℂ ) |
31 |
30
|
addid1d |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝐴 + 0 ) = 𝐴 ) |
32 |
28 31
|
eqtrd |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝐴 + ( 0 · 𝐷 ) ) = 𝐴 ) |
33 |
32
|
eqeq2d |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝑥 = ( 𝐴 + ( 0 · 𝐷 ) ) ↔ 𝑥 = 𝐴 ) ) |
34 |
|
c0ex |
⊢ 0 ∈ V |
35 |
|
oveq1 |
⊢ ( 𝑛 = 0 → ( 𝑛 · 𝐷 ) = ( 0 · 𝐷 ) ) |
36 |
35
|
oveq2d |
⊢ ( 𝑛 = 0 → ( 𝐴 + ( 𝑛 · 𝐷 ) ) = ( 𝐴 + ( 0 · 𝐷 ) ) ) |
37 |
36
|
eqeq2d |
⊢ ( 𝑛 = 0 → ( 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ↔ 𝑥 = ( 𝐴 + ( 0 · 𝐷 ) ) ) ) |
38 |
34 37
|
ceqsexv |
⊢ ( ∃ 𝑛 ( 𝑛 = 0 ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ↔ 𝑥 = ( 𝐴 + ( 0 · 𝐷 ) ) ) |
39 |
|
velsn |
⊢ ( 𝑥 ∈ { 𝐴 } ↔ 𝑥 = 𝐴 ) |
40 |
33 38 39
|
3bitr4g |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( ∃ 𝑛 ( 𝑛 = 0 ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ↔ 𝑥 ∈ { 𝐴 } ) ) |
41 |
|
simpr |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) |
42 |
|
0p1e1 |
⊢ ( 0 + 1 ) = 1 |
43 |
42
|
oveq1i |
⊢ ( ( 0 + 1 ) ... 𝐾 ) = ( 1 ... 𝐾 ) |
44 |
41 43
|
eleqtrdi |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → 𝑛 ∈ ( 1 ... 𝐾 ) ) |
45 |
|
1zzd |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → 1 ∈ ℤ ) |
46 |
4
|
adantr |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → 𝐾 ∈ ℕ0 ) |
47 |
46
|
nn0zd |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → 𝐾 ∈ ℤ ) |
48 |
|
elfzelz |
⊢ ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) → 𝑛 ∈ ℤ ) |
49 |
48
|
adantl |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → 𝑛 ∈ ℤ ) |
50 |
|
fzsubel |
⊢ ( ( ( 1 ∈ ℤ ∧ 𝐾 ∈ ℤ ) ∧ ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( 𝑛 ∈ ( 1 ... 𝐾 ) ↔ ( 𝑛 − 1 ) ∈ ( ( 1 − 1 ) ... ( 𝐾 − 1 ) ) ) ) |
51 |
45 47 49 45 50
|
syl22anc |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( 𝑛 ∈ ( 1 ... 𝐾 ) ↔ ( 𝑛 − 1 ) ∈ ( ( 1 − 1 ) ... ( 𝐾 − 1 ) ) ) ) |
52 |
44 51
|
mpbid |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( 𝑛 − 1 ) ∈ ( ( 1 − 1 ) ... ( 𝐾 − 1 ) ) ) |
53 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
54 |
53
|
oveq1i |
⊢ ( ( 1 − 1 ) ... ( 𝐾 − 1 ) ) = ( 0 ... ( 𝐾 − 1 ) ) |
55 |
52 54
|
eleqtrdi |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( 𝑛 − 1 ) ∈ ( 0 ... ( 𝐾 − 1 ) ) ) |
56 |
49
|
zcnd |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → 𝑛 ∈ ℂ ) |
57 |
|
1cnd |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → 1 ∈ ℂ ) |
58 |
26
|
adantr |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → 𝐷 ∈ ℂ ) |
59 |
56 57 58
|
subdird |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( ( 𝑛 − 1 ) · 𝐷 ) = ( ( 𝑛 · 𝐷 ) − ( 1 · 𝐷 ) ) ) |
60 |
58
|
mulid2d |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( 1 · 𝐷 ) = 𝐷 ) |
61 |
60
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( ( 𝑛 · 𝐷 ) − ( 1 · 𝐷 ) ) = ( ( 𝑛 · 𝐷 ) − 𝐷 ) ) |
62 |
59 61
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( ( 𝑛 − 1 ) · 𝐷 ) = ( ( 𝑛 · 𝐷 ) − 𝐷 ) ) |
63 |
62
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( 𝐷 + ( ( 𝑛 − 1 ) · 𝐷 ) ) = ( 𝐷 + ( ( 𝑛 · 𝐷 ) − 𝐷 ) ) ) |
64 |
56 58
|
mulcld |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( 𝑛 · 𝐷 ) ∈ ℂ ) |
65 |
58 64
|
pncan3d |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( 𝐷 + ( ( 𝑛 · 𝐷 ) − 𝐷 ) ) = ( 𝑛 · 𝐷 ) ) |
66 |
63 65
|
eqtr2d |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( 𝑛 · 𝐷 ) = ( 𝐷 + ( ( 𝑛 − 1 ) · 𝐷 ) ) ) |
67 |
66
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( 𝐴 + ( 𝑛 · 𝐷 ) ) = ( 𝐴 + ( 𝐷 + ( ( 𝑛 − 1 ) · 𝐷 ) ) ) ) |
68 |
30
|
adantr |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → 𝐴 ∈ ℂ ) |
69 |
|
subcl |
⊢ ( ( 𝑛 ∈ ℂ ∧ 1 ∈ ℂ ) → ( 𝑛 − 1 ) ∈ ℂ ) |
70 |
56 6 69
|
sylancl |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( 𝑛 − 1 ) ∈ ℂ ) |
71 |
70 58
|
mulcld |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( ( 𝑛 − 1 ) · 𝐷 ) ∈ ℂ ) |
72 |
68 58 71
|
addassd |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( ( 𝐴 + 𝐷 ) + ( ( 𝑛 − 1 ) · 𝐷 ) ) = ( 𝐴 + ( 𝐷 + ( ( 𝑛 − 1 ) · 𝐷 ) ) ) ) |
73 |
67 72
|
eqtr4d |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( 𝐴 + ( 𝑛 · 𝐷 ) ) = ( ( 𝐴 + 𝐷 ) + ( ( 𝑛 − 1 ) · 𝐷 ) ) ) |
74 |
|
oveq1 |
⊢ ( 𝑚 = ( 𝑛 − 1 ) → ( 𝑚 · 𝐷 ) = ( ( 𝑛 − 1 ) · 𝐷 ) ) |
75 |
74
|
oveq2d |
⊢ ( 𝑚 = ( 𝑛 − 1 ) → ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) = ( ( 𝐴 + 𝐷 ) + ( ( 𝑛 − 1 ) · 𝐷 ) ) ) |
76 |
75
|
rspceeqv |
⊢ ( ( ( 𝑛 − 1 ) ∈ ( 0 ... ( 𝐾 − 1 ) ) ∧ ( 𝐴 + ( 𝑛 · 𝐷 ) ) = ( ( 𝐴 + 𝐷 ) + ( ( 𝑛 − 1 ) · 𝐷 ) ) ) → ∃ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ( 𝐴 + ( 𝑛 · 𝐷 ) ) = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) ) |
77 |
55 73 76
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ∃ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ( 𝐴 + ( 𝑛 · 𝐷 ) ) = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) ) |
78 |
|
eqeq1 |
⊢ ( 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) → ( 𝑥 = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) ↔ ( 𝐴 + ( 𝑛 · 𝐷 ) ) = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) ) ) |
79 |
78
|
rexbidv |
⊢ ( 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) → ( ∃ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) 𝑥 = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) ↔ ∃ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ( 𝐴 + ( 𝑛 · 𝐷 ) ) = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) ) ) |
80 |
77 79
|
syl5ibrcom |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ) → ( 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) → ∃ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) 𝑥 = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) ) ) |
81 |
80
|
expimpd |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) → ∃ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) 𝑥 = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) ) ) |
82 |
81
|
exlimdv |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( ∃ 𝑛 ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) → ∃ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) 𝑥 = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) ) ) |
83 |
|
simpr |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) |
84 |
|
0zd |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → 0 ∈ ℤ ) |
85 |
4
|
adantr |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → 𝐾 ∈ ℕ0 ) |
86 |
85
|
nn0zd |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → 𝐾 ∈ ℤ ) |
87 |
|
peano2zm |
⊢ ( 𝐾 ∈ ℤ → ( 𝐾 − 1 ) ∈ ℤ ) |
88 |
86 87
|
syl |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( 𝐾 − 1 ) ∈ ℤ ) |
89 |
|
elfzelz |
⊢ ( 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) → 𝑚 ∈ ℤ ) |
90 |
89
|
adantl |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → 𝑚 ∈ ℤ ) |
91 |
|
1zzd |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → 1 ∈ ℤ ) |
92 |
|
fzaddel |
⊢ ( ( ( 0 ∈ ℤ ∧ ( 𝐾 − 1 ) ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ↔ ( 𝑚 + 1 ) ∈ ( ( 0 + 1 ) ... ( ( 𝐾 − 1 ) + 1 ) ) ) ) |
93 |
84 88 90 91 92
|
syl22anc |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ↔ ( 𝑚 + 1 ) ∈ ( ( 0 + 1 ) ... ( ( 𝐾 − 1 ) + 1 ) ) ) ) |
94 |
83 93
|
mpbid |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( 𝑚 + 1 ) ∈ ( ( 0 + 1 ) ... ( ( 𝐾 − 1 ) + 1 ) ) ) |
95 |
85
|
nn0cnd |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → 𝐾 ∈ ℂ ) |
96 |
|
npcan |
⊢ ( ( 𝐾 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝐾 − 1 ) + 1 ) = 𝐾 ) |
97 |
95 6 96
|
sylancl |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( ( 𝐾 − 1 ) + 1 ) = 𝐾 ) |
98 |
97
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( ( 0 + 1 ) ... ( ( 𝐾 − 1 ) + 1 ) ) = ( ( 0 + 1 ) ... 𝐾 ) ) |
99 |
94 98
|
eleqtrd |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( 𝑚 + 1 ) ∈ ( ( 0 + 1 ) ... 𝐾 ) ) |
100 |
30
|
adantr |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → 𝐴 ∈ ℂ ) |
101 |
26
|
adantr |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → 𝐷 ∈ ℂ ) |
102 |
90
|
zcnd |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → 𝑚 ∈ ℂ ) |
103 |
102 101
|
mulcld |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( 𝑚 · 𝐷 ) ∈ ℂ ) |
104 |
100 101 103
|
addassd |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) = ( 𝐴 + ( 𝐷 + ( 𝑚 · 𝐷 ) ) ) ) |
105 |
|
1cnd |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → 1 ∈ ℂ ) |
106 |
102 105 101
|
adddird |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( ( 𝑚 + 1 ) · 𝐷 ) = ( ( 𝑚 · 𝐷 ) + ( 1 · 𝐷 ) ) ) |
107 |
101 103
|
addcomd |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( 𝐷 + ( 𝑚 · 𝐷 ) ) = ( ( 𝑚 · 𝐷 ) + 𝐷 ) ) |
108 |
101
|
mulid2d |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( 1 · 𝐷 ) = 𝐷 ) |
109 |
108
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( ( 𝑚 · 𝐷 ) + ( 1 · 𝐷 ) ) = ( ( 𝑚 · 𝐷 ) + 𝐷 ) ) |
110 |
107 109
|
eqtr4d |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( 𝐷 + ( 𝑚 · 𝐷 ) ) = ( ( 𝑚 · 𝐷 ) + ( 1 · 𝐷 ) ) ) |
111 |
106 110
|
eqtr4d |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( ( 𝑚 + 1 ) · 𝐷 ) = ( 𝐷 + ( 𝑚 · 𝐷 ) ) ) |
112 |
111
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( 𝐴 + ( ( 𝑚 + 1 ) · 𝐷 ) ) = ( 𝐴 + ( 𝐷 + ( 𝑚 · 𝐷 ) ) ) ) |
113 |
104 112
|
eqtr4d |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) = ( 𝐴 + ( ( 𝑚 + 1 ) · 𝐷 ) ) ) |
114 |
|
ovex |
⊢ ( 𝑚 + 1 ) ∈ V |
115 |
|
eleq1 |
⊢ ( 𝑛 = ( 𝑚 + 1 ) → ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ↔ ( 𝑚 + 1 ) ∈ ( ( 0 + 1 ) ... 𝐾 ) ) ) |
116 |
|
oveq1 |
⊢ ( 𝑛 = ( 𝑚 + 1 ) → ( 𝑛 · 𝐷 ) = ( ( 𝑚 + 1 ) · 𝐷 ) ) |
117 |
116
|
oveq2d |
⊢ ( 𝑛 = ( 𝑚 + 1 ) → ( 𝐴 + ( 𝑛 · 𝐷 ) ) = ( 𝐴 + ( ( 𝑚 + 1 ) · 𝐷 ) ) ) |
118 |
117
|
eqeq2d |
⊢ ( 𝑛 = ( 𝑚 + 1 ) → ( ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ↔ ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) = ( 𝐴 + ( ( 𝑚 + 1 ) · 𝐷 ) ) ) ) |
119 |
115 118
|
anbi12d |
⊢ ( 𝑛 = ( 𝑚 + 1 ) → ( ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ↔ ( ( 𝑚 + 1 ) ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) = ( 𝐴 + ( ( 𝑚 + 1 ) · 𝐷 ) ) ) ) ) |
120 |
114 119
|
spcev |
⊢ ( ( ( 𝑚 + 1 ) ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) = ( 𝐴 + ( ( 𝑚 + 1 ) · 𝐷 ) ) ) → ∃ 𝑛 ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) |
121 |
99 113 120
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ∃ 𝑛 ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) |
122 |
|
eqeq1 |
⊢ ( 𝑥 = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) → ( 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ↔ ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) |
123 |
122
|
anbi2d |
⊢ ( 𝑥 = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) → ( ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ↔ ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) ) |
124 |
123
|
exbidv |
⊢ ( 𝑥 = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) → ( ∃ 𝑛 ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ↔ ∃ 𝑛 ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) ) |
125 |
121 124
|
syl5ibrcom |
⊢ ( ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) ∧ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ) → ( 𝑥 = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) → ∃ 𝑛 ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) ) |
126 |
125
|
rexlimdva |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( ∃ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) 𝑥 = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) → ∃ 𝑛 ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) ) |
127 |
82 126
|
impbid |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( ∃ 𝑛 ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ↔ ∃ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) 𝑥 = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) ) ) |
128 |
|
nnaddcl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝐴 + 𝐷 ) ∈ ℕ ) |
129 |
128
|
3adant1 |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝐴 + 𝐷 ) ∈ ℕ ) |
130 |
|
vdwapval |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ ( 𝐴 + 𝐷 ) ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝑥 ∈ ( ( 𝐴 + 𝐷 ) ( AP ‘ 𝐾 ) 𝐷 ) ↔ ∃ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) 𝑥 = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) ) ) |
131 |
129 130
|
syld3an2 |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝑥 ∈ ( ( 𝐴 + 𝐷 ) ( AP ‘ 𝐾 ) 𝐷 ) ↔ ∃ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) 𝑥 = ( ( 𝐴 + 𝐷 ) + ( 𝑚 · 𝐷 ) ) ) ) |
132 |
127 131
|
bitr4d |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( ∃ 𝑛 ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ↔ 𝑥 ∈ ( ( 𝐴 + 𝐷 ) ( AP ‘ 𝐾 ) 𝐷 ) ) ) |
133 |
40 132
|
orbi12d |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( ( ∃ 𝑛 ( 𝑛 = 0 ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ∨ ∃ 𝑛 ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) ↔ ( 𝑥 ∈ { 𝐴 } ∨ 𝑥 ∈ ( ( 𝐴 + 𝐷 ) ( AP ‘ 𝐾 ) 𝐷 ) ) ) ) |
134 |
|
elun |
⊢ ( 𝑥 ∈ ( { 𝐴 } ∪ ( ( 𝐴 + 𝐷 ) ( AP ‘ 𝐾 ) 𝐷 ) ) ↔ ( 𝑥 ∈ { 𝐴 } ∨ 𝑥 ∈ ( ( 𝐴 + 𝐷 ) ( AP ‘ 𝐾 ) 𝐷 ) ) ) |
135 |
133 134
|
bitr4di |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( ( ∃ 𝑛 ( 𝑛 = 0 ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ∨ ∃ 𝑛 ( 𝑛 ∈ ( ( 0 + 1 ) ... 𝐾 ) ∧ 𝑥 = ( 𝐴 + ( 𝑛 · 𝐷 ) ) ) ) ↔ 𝑥 ∈ ( { 𝐴 } ∪ ( ( 𝐴 + 𝐷 ) ( AP ‘ 𝐾 ) 𝐷 ) ) ) ) |
136 |
24 135
|
bitrd |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝑥 ∈ ( 𝐴 ( AP ‘ ( 𝐾 + 1 ) ) 𝐷 ) ↔ 𝑥 ∈ ( { 𝐴 } ∪ ( ( 𝐴 + 𝐷 ) ( AP ‘ 𝐾 ) 𝐷 ) ) ) ) |
137 |
136
|
eqrdv |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝐴 ( AP ‘ ( 𝐾 + 1 ) ) 𝐷 ) = ( { 𝐴 } ∪ ( ( 𝐴 + 𝐷 ) ( AP ‘ 𝐾 ) 𝐷 ) ) ) |