Step | Hyp | Ref
| Expression |
1 | | seqexw.2 |
. . . 4
⊢ 𝑀 ∈ ℤ |
2 | | seqfn 13586 |
. . . 4
⊢ (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)) |
3 | 1, 2 | ax-mp 5 |
. . 3
⊢ seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀) |
4 | | fnfun 6479 |
. . 3
⊢ (seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀) → Fun seq𝑀( + , 𝐹)) |
5 | 3, 4 | ax-mp 5 |
. 2
⊢ Fun
seq𝑀( + , 𝐹) |
6 | 3 | fndmi 6482 |
. . 3
⊢ dom
seq𝑀( + , 𝐹) = (ℤ≥‘𝑀) |
7 | | fvex 6730 |
. . 3
⊢
(ℤ≥‘𝑀) ∈ V |
8 | 6, 7 | eqeltri 2834 |
. 2
⊢ dom
seq𝑀( + , 𝐹) ∈ V |
9 | | seqexw.1 |
. . . . 5
⊢ + ∈
V |
10 | 9 | rnex 7690 |
. . . 4
⊢ ran + ∈
V |
11 | | prex 5325 |
. . . 4
⊢ {∅,
(𝐹‘𝑀)} ∈ V |
12 | 10, 11 | unex 7531 |
. . 3
⊢ (ran
+ ∪
{∅, (𝐹‘𝑀)}) ∈ V |
13 | | fveq2 6717 |
. . . . . . 7
⊢ (𝑦 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑀)) |
14 | 13 | eleq1d 2822 |
. . . . . 6
⊢ (𝑦 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))) |
15 | | fveq2 6717 |
. . . . . . 7
⊢ (𝑦 = 𝑧 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑧)) |
16 | 15 | eleq1d 2822 |
. . . . . 6
⊢ (𝑦 = 𝑧 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))) |
17 | | fveq2 6717 |
. . . . . . 7
⊢ (𝑦 = (𝑧 + 1) → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘(𝑧 + 1))) |
18 | 17 | eleq1d 2822 |
. . . . . 6
⊢ (𝑦 = (𝑧 + 1) → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))) |
19 | | fveq2 6717 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑥)) |
20 | 19 | eleq1d 2822 |
. . . . . 6
⊢ (𝑦 = 𝑥 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))) |
21 | | seq1 13587 |
. . . . . . 7
⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
22 | | ssun2 4087 |
. . . . . . . 8
⊢ {∅,
(𝐹‘𝑀)} ⊆ (ran + ∪ {∅, (𝐹‘𝑀)}) |
23 | | fvex 6730 |
. . . . . . . . 9
⊢ (𝐹‘𝑀) ∈ V |
24 | 23 | prid2 4679 |
. . . . . . . 8
⊢ (𝐹‘𝑀) ∈ {∅, (𝐹‘𝑀)} |
25 | 22, 24 | sselii 3897 |
. . . . . . 7
⊢ (𝐹‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) |
26 | 21, 25 | eqeltrdi 2846 |
. . . . . 6
⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) |
27 | | seqp1 13589 |
. . . . . . . . 9
⊢ (𝑧 ∈
(ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1)))) |
28 | 27 | adantr 484 |
. . . . . . . 8
⊢ ((𝑧 ∈
(ℤ≥‘𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1)))) |
29 | | df-ov 7216 |
. . . . . . . . . 10
⊢
((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) = ( + ‘〈(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))〉) |
30 | | snsspr1 4727 |
. . . . . . . . . . . 12
⊢ {∅}
⊆ {∅, (𝐹‘𝑀)} |
31 | | unss2 4095 |
. . . . . . . . . . . 12
⊢
({∅} ⊆ {∅, (𝐹‘𝑀)} → (ran + ∪ {∅}) ⊆
(ran +
∪ {∅, (𝐹‘𝑀)})) |
32 | 30, 31 | ax-mp 5 |
. . . . . . . . . . 11
⊢ (ran
+ ∪
{∅}) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)}) |
33 | | fvrn0 6745 |
. . . . . . . . . . 11
⊢ ( +
‘〈(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))〉) ∈ (ran + ∪
{∅}) |
34 | 32, 33 | sselii 3897 |
. . . . . . . . . 10
⊢ ( +
‘〈(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))〉) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) |
35 | 29, 34 | eqeltri 2834 |
. . . . . . . . 9
⊢
((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) |
36 | 35 | a1i 11 |
. . . . . . . 8
⊢ ((𝑧 ∈
(ℤ≥‘𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) → ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) |
37 | 28, 36 | eqeltrd 2838 |
. . . . . . 7
⊢ ((𝑧 ∈
(ℤ≥‘𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) |
38 | 37 | ex 416 |
. . . . . 6
⊢ (𝑧 ∈
(ℤ≥‘𝑀) → ((seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))) |
39 | 14, 16, 18, 20, 26, 38 | uzind4 12502 |
. . . . 5
⊢ (𝑥 ∈
(ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) |
40 | 39 | rgen 3071 |
. . . 4
⊢
∀𝑥 ∈
(ℤ≥‘𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) |
41 | | fnfvrnss 6937 |
. . . 4
⊢
((seq𝑀( + , 𝐹) Fn
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (ℤ≥‘𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) → ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)})) |
42 | 3, 40, 41 | mp2an 692 |
. . 3
⊢ ran
seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)}) |
43 | 12, 42 | ssexi 5215 |
. 2
⊢ ran
seq𝑀( + , 𝐹) ∈ V |
44 | | funexw 7725 |
. 2
⊢ ((Fun
seq𝑀( + , 𝐹) ∧ dom seq𝑀( + , 𝐹) ∈ V ∧ ran seq𝑀( + , 𝐹) ∈ V) → seq𝑀( + , 𝐹) ∈ V) |
45 | 5, 8, 43, 44 | mp3an 1463 |
1
⊢ seq𝑀( + , 𝐹) ∈ V |