Step |
Hyp |
Ref |
Expression |
1 |
|
seqhomo.1 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 ) |
2 |
|
seqhomo.2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 ) |
3 |
|
seqz.3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → ( 𝑍 + 𝑥 ) = 𝑍 ) |
4 |
|
seqz.4 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → ( 𝑥 + 𝑍 ) = 𝑍 ) |
5 |
|
seqz.5 |
⊢ ( 𝜑 → 𝐾 ∈ ( 𝑀 ... 𝑁 ) ) |
6 |
|
seqz.6 |
⊢ ( 𝜑 → 𝑁 ∈ 𝑉 ) |
7 |
|
seqz.7 |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐾 ) = 𝑍 ) |
8 |
|
elfzuz |
⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
9 |
5 8
|
syl |
⊢ ( 𝜑 → 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
10 |
5
|
elfzelzd |
⊢ ( 𝜑 → 𝐾 ∈ ℤ ) |
11 |
|
seq1 |
⊢ ( 𝐾 ∈ ℤ → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝐾 ) = ( 𝐹 ‘ 𝐾 ) ) |
12 |
10 11
|
syl |
⊢ ( 𝜑 → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝐾 ) = ( 𝐹 ‘ 𝐾 ) ) |
13 |
12 7
|
eqtrd |
⊢ ( 𝜑 → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝐾 ) = 𝑍 ) |
14 |
|
seqeq1 |
⊢ ( 𝐾 = 𝑀 → seq 𝐾 ( + , 𝐹 ) = seq 𝑀 ( + , 𝐹 ) ) |
15 |
14
|
fveq1d |
⊢ ( 𝐾 = 𝑀 → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝐾 ) = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝐾 ) ) |
16 |
15
|
eqeq1d |
⊢ ( 𝐾 = 𝑀 → ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝐾 ) = 𝑍 ↔ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝐾 ) = 𝑍 ) ) |
17 |
13 16
|
syl5ibcom |
⊢ ( 𝜑 → ( 𝐾 = 𝑀 → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝐾 ) = 𝑍 ) ) |
18 |
|
eluzel2 |
⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ ) |
19 |
9 18
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
20 |
|
seqm1 |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝐾 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝐾 − 1 ) ) + ( 𝐹 ‘ 𝐾 ) ) ) |
21 |
19 20
|
sylan |
⊢ ( ( 𝜑 ∧ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝐾 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝐾 − 1 ) ) + ( 𝐹 ‘ 𝐾 ) ) ) |
22 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( 𝐹 ‘ 𝐾 ) = 𝑍 ) |
23 |
22
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝐾 − 1 ) ) + ( 𝐹 ‘ 𝐾 ) ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝐾 − 1 ) ) + 𝑍 ) ) |
24 |
|
oveq1 |
⊢ ( 𝑥 = ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝐾 − 1 ) ) → ( 𝑥 + 𝑍 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝐾 − 1 ) ) + 𝑍 ) ) |
25 |
24
|
eqeq1d |
⊢ ( 𝑥 = ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝐾 − 1 ) ) → ( ( 𝑥 + 𝑍 ) = 𝑍 ↔ ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝐾 − 1 ) ) + 𝑍 ) = 𝑍 ) ) |
26 |
4
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ( 𝑥 + 𝑍 ) = 𝑍 ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ∀ 𝑥 ∈ 𝑆 ( 𝑥 + 𝑍 ) = 𝑍 ) |
28 |
|
eluzp1m1 |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( 𝐾 − 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) ) |
29 |
19 28
|
sylan |
⊢ ( ( 𝜑 ∧ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( 𝐾 − 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) ) |
30 |
|
fzssp1 |
⊢ ( 𝑀 ... ( 𝐾 − 1 ) ) ⊆ ( 𝑀 ... ( ( 𝐾 − 1 ) + 1 ) ) |
31 |
10
|
zcnd |
⊢ ( 𝜑 → 𝐾 ∈ ℂ ) |
32 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
33 |
|
npcan |
⊢ ( ( 𝐾 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝐾 − 1 ) + 1 ) = 𝐾 ) |
34 |
31 32 33
|
sylancl |
⊢ ( 𝜑 → ( ( 𝐾 − 1 ) + 1 ) = 𝐾 ) |
35 |
34
|
oveq2d |
⊢ ( 𝜑 → ( 𝑀 ... ( ( 𝐾 − 1 ) + 1 ) ) = ( 𝑀 ... 𝐾 ) ) |
36 |
30 35
|
sseqtrid |
⊢ ( 𝜑 → ( 𝑀 ... ( 𝐾 − 1 ) ) ⊆ ( 𝑀 ... 𝐾 ) ) |
37 |
|
elfzuz3 |
⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) |
38 |
5 37
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) |
39 |
|
fzss2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) → ( 𝑀 ... 𝐾 ) ⊆ ( 𝑀 ... 𝑁 ) ) |
40 |
38 39
|
syl |
⊢ ( 𝜑 → ( 𝑀 ... 𝐾 ) ⊆ ( 𝑀 ... 𝑁 ) ) |
41 |
36 40
|
sstrd |
⊢ ( 𝜑 → ( 𝑀 ... ( 𝐾 − 1 ) ) ⊆ ( 𝑀 ... 𝑁 ) ) |
42 |
41
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( 𝑀 ... ( 𝐾 − 1 ) ) ⊆ ( 𝑀 ... 𝑁 ) ) |
43 |
42
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ∧ 𝑥 ∈ ( 𝑀 ... ( 𝐾 − 1 ) ) ) → 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) |
44 |
2
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 ) |
45 |
43 44
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ∧ 𝑥 ∈ ( 𝑀 ... ( 𝐾 − 1 ) ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 ) |
46 |
1
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 ) |
47 |
29 45 46
|
seqcl |
⊢ ( ( 𝜑 ∧ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝐾 − 1 ) ) ∈ 𝑆 ) |
48 |
25 27 47
|
rspcdva |
⊢ ( ( 𝜑 ∧ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝐾 − 1 ) ) + 𝑍 ) = 𝑍 ) |
49 |
23 48
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝐾 − 1 ) ) + ( 𝐹 ‘ 𝐾 ) ) = 𝑍 ) |
50 |
21 49
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝐾 ) = 𝑍 ) |
51 |
50
|
ex |
⊢ ( 𝜑 → ( 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝐾 ) = 𝑍 ) ) |
52 |
|
uzp1 |
⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝐾 = 𝑀 ∨ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) |
53 |
9 52
|
syl |
⊢ ( 𝜑 → ( 𝐾 = 𝑀 ∨ 𝐾 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) |
54 |
17 51 53
|
mpjaod |
⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝐾 ) = 𝑍 ) |
55 |
54 7
|
eqtr4d |
⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝐾 ) = ( 𝐹 ‘ 𝐾 ) ) |
56 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
57 |
9 55 38 56
|
seqfveq2 |
⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) = ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑁 ) ) |
58 |
|
fvex |
⊢ ( 𝐹 ‘ 𝐾 ) ∈ V |
59 |
58
|
elsn |
⊢ ( ( 𝐹 ‘ 𝐾 ) ∈ { 𝑍 } ↔ ( 𝐹 ‘ 𝐾 ) = 𝑍 ) |
60 |
7 59
|
sylibr |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐾 ) ∈ { 𝑍 } ) |
61 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ { 𝑍 } ∧ 𝑦 ∈ 𝑆 ) ) → 𝑥 ∈ { 𝑍 } ) |
62 |
|
velsn |
⊢ ( 𝑥 ∈ { 𝑍 } ↔ 𝑥 = 𝑍 ) |
63 |
61 62
|
sylib |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ { 𝑍 } ∧ 𝑦 ∈ 𝑆 ) ) → 𝑥 = 𝑍 ) |
64 |
63
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ { 𝑍 } ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) = ( 𝑍 + 𝑦 ) ) |
65 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑍 + 𝑥 ) = ( 𝑍 + 𝑦 ) ) |
66 |
65
|
eqeq1d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑍 + 𝑥 ) = 𝑍 ↔ ( 𝑍 + 𝑦 ) = 𝑍 ) ) |
67 |
3
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ( 𝑍 + 𝑥 ) = 𝑍 ) |
68 |
67
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ { 𝑍 } ∧ 𝑦 ∈ 𝑆 ) ) → ∀ 𝑥 ∈ 𝑆 ( 𝑍 + 𝑥 ) = 𝑍 ) |
69 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ { 𝑍 } ∧ 𝑦 ∈ 𝑆 ) ) → 𝑦 ∈ 𝑆 ) |
70 |
66 68 69
|
rspcdva |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ { 𝑍 } ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑍 + 𝑦 ) = 𝑍 ) |
71 |
64 70
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ { 𝑍 } ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) = 𝑍 ) |
72 |
|
ovex |
⊢ ( 𝑥 + 𝑦 ) ∈ V |
73 |
72
|
elsn |
⊢ ( ( 𝑥 + 𝑦 ) ∈ { 𝑍 } ↔ ( 𝑥 + 𝑦 ) = 𝑍 ) |
74 |
71 73
|
sylibr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ { 𝑍 } ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ { 𝑍 } ) |
75 |
|
peano2uz |
⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝐾 + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) ) |
76 |
9 75
|
syl |
⊢ ( 𝜑 → ( 𝐾 + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) ) |
77 |
|
fzss1 |
⊢ ( ( 𝐾 + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝐾 + 1 ) ... 𝑁 ) ⊆ ( 𝑀 ... 𝑁 ) ) |
78 |
76 77
|
syl |
⊢ ( 𝜑 → ( ( 𝐾 + 1 ) ... 𝑁 ) ⊆ ( 𝑀 ... 𝑁 ) ) |
79 |
78
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) |
80 |
79 2
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 ) |
81 |
60 74 38 80
|
seqcl2 |
⊢ ( 𝜑 → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑁 ) ∈ { 𝑍 } ) |
82 |
|
elsni |
⊢ ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑁 ) ∈ { 𝑍 } → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑁 ) = 𝑍 ) |
83 |
81 82
|
syl |
⊢ ( 𝜑 → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑁 ) = 𝑍 ) |
84 |
57 83
|
eqtrd |
⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) = 𝑍 ) |