Step |
Hyp |
Ref |
Expression |
1 |
|
seqof.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
seqof.2 |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
3 |
|
seqof.3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑥 ) = ( 𝑧 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑥 ) ) ) |
4 |
|
fvex |
⊢ ( 𝐺 ‘ 𝑥 ) ∈ V |
5 |
4
|
rgenw |
⊢ ∀ 𝑧 ∈ 𝐴 ( 𝐺 ‘ 𝑥 ) ∈ V |
6 |
|
eqid |
⊢ ( 𝑧 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑥 ) ) = ( 𝑧 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑥 ) ) |
7 |
6
|
fnmpt |
⊢ ( ∀ 𝑧 ∈ 𝐴 ( 𝐺 ‘ 𝑥 ) ∈ V → ( 𝑧 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑥 ) ) Fn 𝐴 ) |
8 |
5 7
|
mp1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝑧 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑥 ) ) Fn 𝐴 ) |
9 |
3
|
fneq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( ( 𝐹 ‘ 𝑥 ) Fn 𝐴 ↔ ( 𝑧 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑥 ) ) Fn 𝐴 ) ) |
10 |
8 9
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑥 ) Fn 𝐴 ) |
11 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑥 ) ∈ V |
12 |
|
fneq1 |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑥 ) → ( 𝑧 Fn 𝐴 ↔ ( 𝐹 ‘ 𝑥 ) Fn 𝐴 ) ) |
13 |
11 12
|
elab |
⊢ ( ( 𝐹 ‘ 𝑥 ) ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ↔ ( 𝐹 ‘ 𝑥 ) Fn 𝐴 ) |
14 |
10 13
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ) |
15 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴 ) ) → 𝑥 Fn 𝐴 ) |
16 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴 ) ) → 𝑦 Fn 𝐴 ) |
17 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴 ) ) → 𝐴 ∈ 𝑉 ) |
18 |
|
inidm |
⊢ ( 𝐴 ∩ 𝐴 ) = 𝐴 |
19 |
15 16 17 17 18
|
offn |
⊢ ( ( 𝜑 ∧ ( 𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴 ) ) → ( 𝑥 ∘f + 𝑦 ) Fn 𝐴 ) |
20 |
19
|
ex |
⊢ ( 𝜑 → ( ( 𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴 ) → ( 𝑥 ∘f + 𝑦 ) Fn 𝐴 ) ) |
21 |
|
vex |
⊢ 𝑥 ∈ V |
22 |
|
fneq1 |
⊢ ( 𝑧 = 𝑥 → ( 𝑧 Fn 𝐴 ↔ 𝑥 Fn 𝐴 ) ) |
23 |
21 22
|
elab |
⊢ ( 𝑥 ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ↔ 𝑥 Fn 𝐴 ) |
24 |
|
vex |
⊢ 𝑦 ∈ V |
25 |
|
fneq1 |
⊢ ( 𝑧 = 𝑦 → ( 𝑧 Fn 𝐴 ↔ 𝑦 Fn 𝐴 ) ) |
26 |
24 25
|
elab |
⊢ ( 𝑦 ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ↔ 𝑦 Fn 𝐴 ) |
27 |
23 26
|
anbi12i |
⊢ ( ( 𝑥 ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ∧ 𝑦 ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ) ↔ ( 𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴 ) ) |
28 |
|
ovex |
⊢ ( 𝑥 ∘f + 𝑦 ) ∈ V |
29 |
|
fneq1 |
⊢ ( 𝑧 = ( 𝑥 ∘f + 𝑦 ) → ( 𝑧 Fn 𝐴 ↔ ( 𝑥 ∘f + 𝑦 ) Fn 𝐴 ) ) |
30 |
28 29
|
elab |
⊢ ( ( 𝑥 ∘f + 𝑦 ) ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ↔ ( 𝑥 ∘f + 𝑦 ) Fn 𝐴 ) |
31 |
20 27 30
|
3imtr4g |
⊢ ( 𝜑 → ( ( 𝑥 ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ∧ 𝑦 ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ) → ( 𝑥 ∘f + 𝑦 ) ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ) ) |
32 |
31
|
imp |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ∧ 𝑦 ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ) ) → ( 𝑥 ∘f + 𝑦 ) ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ) |
33 |
2 14 32
|
seqcl |
⊢ ( 𝜑 → ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ) |
34 |
|
fvex |
⊢ ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) ∈ V |
35 |
|
fneq1 |
⊢ ( 𝑧 = ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) → ( 𝑧 Fn 𝐴 ↔ ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) Fn 𝐴 ) ) |
36 |
34 35
|
elab |
⊢ ( ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ↔ ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) Fn 𝐴 ) |
37 |
33 36
|
sylib |
⊢ ( 𝜑 → ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) Fn 𝐴 ) |
38 |
|
dffn5 |
⊢ ( ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) Fn 𝐴 ↔ ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) = ( 𝑧 ∈ 𝐴 ↦ ( ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) ‘ 𝑧 ) ) ) |
39 |
37 38
|
sylib |
⊢ ( 𝜑 → ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) = ( 𝑧 ∈ 𝐴 ↦ ( ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) ‘ 𝑧 ) ) ) |
40 |
|
fveq1 |
⊢ ( 𝑤 = ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) → ( 𝑤 ‘ 𝑧 ) = ( ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) ‘ 𝑧 ) ) |
41 |
|
eqid |
⊢ ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) = ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) |
42 |
|
fvex |
⊢ ( ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) ‘ 𝑧 ) ∈ V |
43 |
40 41 42
|
fvmpt |
⊢ ( ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) ∈ V → ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) ) = ( ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) ‘ 𝑧 ) ) |
44 |
34 43
|
mp1i |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) ) = ( ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) ‘ 𝑧 ) ) |
45 |
32
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑥 ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ∧ 𝑦 ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ) ) → ( 𝑥 ∘f + 𝑦 ) ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ) |
46 |
14
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ) |
47 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
48 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴 ) ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑥 ‘ 𝑧 ) = ( 𝑥 ‘ 𝑧 ) ) |
49 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴 ) ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑦 ‘ 𝑧 ) = ( 𝑦 ‘ 𝑧 ) ) |
50 |
15 16 17 17 18 48 49
|
ofval |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴 ) ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑥 ∘f + 𝑦 ) ‘ 𝑧 ) = ( ( 𝑥 ‘ 𝑧 ) + ( 𝑦 ‘ 𝑧 ) ) ) |
51 |
50
|
an32s |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴 ) ) → ( ( 𝑥 ∘f + 𝑦 ) ‘ 𝑧 ) = ( ( 𝑥 ‘ 𝑧 ) + ( 𝑦 ‘ 𝑧 ) ) ) |
52 |
|
fveq1 |
⊢ ( 𝑤 = ( 𝑥 ∘f + 𝑦 ) → ( 𝑤 ‘ 𝑧 ) = ( ( 𝑥 ∘f + 𝑦 ) ‘ 𝑧 ) ) |
53 |
|
fvex |
⊢ ( ( 𝑥 ∘f + 𝑦 ) ‘ 𝑧 ) ∈ V |
54 |
52 41 53
|
fvmpt |
⊢ ( ( 𝑥 ∘f + 𝑦 ) ∈ V → ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ ( 𝑥 ∘f + 𝑦 ) ) = ( ( 𝑥 ∘f + 𝑦 ) ‘ 𝑧 ) ) |
55 |
28 54
|
ax-mp |
⊢ ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ ( 𝑥 ∘f + 𝑦 ) ) = ( ( 𝑥 ∘f + 𝑦 ) ‘ 𝑧 ) |
56 |
|
fveq1 |
⊢ ( 𝑤 = 𝑥 → ( 𝑤 ‘ 𝑧 ) = ( 𝑥 ‘ 𝑧 ) ) |
57 |
|
fvex |
⊢ ( 𝑥 ‘ 𝑧 ) ∈ V |
58 |
56 41 57
|
fvmpt |
⊢ ( 𝑥 ∈ V → ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ 𝑥 ) = ( 𝑥 ‘ 𝑧 ) ) |
59 |
58
|
elv |
⊢ ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ 𝑥 ) = ( 𝑥 ‘ 𝑧 ) |
60 |
|
fveq1 |
⊢ ( 𝑤 = 𝑦 → ( 𝑤 ‘ 𝑧 ) = ( 𝑦 ‘ 𝑧 ) ) |
61 |
|
fvex |
⊢ ( 𝑦 ‘ 𝑧 ) ∈ V |
62 |
60 41 61
|
fvmpt |
⊢ ( 𝑦 ∈ V → ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ 𝑦 ) = ( 𝑦 ‘ 𝑧 ) ) |
63 |
62
|
elv |
⊢ ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ 𝑦 ) = ( 𝑦 ‘ 𝑧 ) |
64 |
59 63
|
oveq12i |
⊢ ( ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ 𝑥 ) + ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ 𝑦 ) ) = ( ( 𝑥 ‘ 𝑧 ) + ( 𝑦 ‘ 𝑧 ) ) |
65 |
51 55 64
|
3eqtr4g |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴 ) ) → ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ ( 𝑥 ∘f + 𝑦 ) ) = ( ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ 𝑥 ) + ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ 𝑦 ) ) ) |
66 |
27 65
|
sylan2b |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑥 ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ∧ 𝑦 ∈ { 𝑧 ∣ 𝑧 Fn 𝐴 } ) ) → ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ ( 𝑥 ∘f + 𝑦 ) ) = ( ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ 𝑥 ) + ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ 𝑦 ) ) ) |
67 |
|
fveq1 |
⊢ ( 𝑤 = ( 𝐹 ‘ 𝑥 ) → ( 𝑤 ‘ 𝑧 ) = ( ( 𝐹 ‘ 𝑥 ) ‘ 𝑧 ) ) |
68 |
|
fvex |
⊢ ( ( 𝐹 ‘ 𝑥 ) ‘ 𝑧 ) ∈ V |
69 |
67 41 68
|
fvmpt |
⊢ ( ( 𝐹 ‘ 𝑥 ) ∈ V → ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ ( 𝐹 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) ‘ 𝑧 ) ) |
70 |
11 69
|
ax-mp |
⊢ ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ ( 𝐹 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) ‘ 𝑧 ) |
71 |
3
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑥 ) = ( 𝑧 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑥 ) ) ) |
72 |
71
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( ( 𝐹 ‘ 𝑥 ) ‘ 𝑧 ) = ( ( 𝑧 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑥 ) ) ‘ 𝑧 ) ) |
73 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → 𝑧 ∈ 𝐴 ) |
74 |
6
|
fvmpt2 |
⊢ ( ( 𝑧 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑥 ) ∈ V ) → ( ( 𝑧 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑥 ) ) ‘ 𝑧 ) = ( 𝐺 ‘ 𝑥 ) ) |
75 |
73 4 74
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( ( 𝑧 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑥 ) ) ‘ 𝑧 ) = ( 𝐺 ‘ 𝑥 ) ) |
76 |
72 75
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( ( 𝐹 ‘ 𝑥 ) ‘ 𝑧 ) = ( 𝐺 ‘ 𝑥 ) ) |
77 |
70 76
|
eqtrid |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐺 ‘ 𝑥 ) ) |
78 |
45 46 47 66 77
|
seqhomo |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑤 ∈ V ↦ ( 𝑤 ‘ 𝑧 ) ) ‘ ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) ) = ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) ) |
79 |
44 78
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) ‘ 𝑧 ) = ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) ) |
80 |
79
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑧 ∈ 𝐴 ↦ ( ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) ‘ 𝑧 ) ) = ( 𝑧 ∈ 𝐴 ↦ ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) ) ) |
81 |
39 80
|
eqtrd |
⊢ ( 𝜑 → ( seq 𝑀 ( ∘f + , 𝐹 ) ‘ 𝑁 ) = ( 𝑧 ∈ 𝐴 ↦ ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) ) ) |