Step |
Hyp |
Ref |
Expression |
1 |
|
constr0.1 |
⊢ 𝐶 = rec ( ( 𝑠 ∈ V ↦ { 𝑥 ∈ ℂ ∣ ( ∃ 𝑎 ∈ 𝑠 ∃ 𝑏 ∈ 𝑠 ∃ 𝑐 ∈ 𝑠 ∃ 𝑑 ∈ 𝑠 ∃ 𝑡 ∈ ℝ ∃ 𝑟 ∈ ℝ ( 𝑥 = ( 𝑎 + ( 𝑡 · ( 𝑏 − 𝑎 ) ) ) ∧ 𝑥 = ( 𝑐 + ( 𝑟 · ( 𝑑 − 𝑐 ) ) ) ∧ ( ℑ ‘ ( ( ∗ ‘ ( 𝑏 − 𝑎 ) ) · ( 𝑑 − 𝑐 ) ) ) ≠ 0 ) ∨ ∃ 𝑎 ∈ 𝑠 ∃ 𝑏 ∈ 𝑠 ∃ 𝑐 ∈ 𝑠 ∃ 𝑒 ∈ 𝑠 ∃ 𝑓 ∈ 𝑠 ∃ 𝑡 ∈ ℝ ( 𝑥 = ( 𝑎 + ( 𝑡 · ( 𝑏 − 𝑎 ) ) ) ∧ ( abs ‘ ( 𝑥 − 𝑐 ) ) = ( abs ‘ ( 𝑒 − 𝑓 ) ) ) ∨ ∃ 𝑎 ∈ 𝑠 ∃ 𝑏 ∈ 𝑠 ∃ 𝑐 ∈ 𝑠 ∃ 𝑑 ∈ 𝑠 ∃ 𝑒 ∈ 𝑠 ∃ 𝑓 ∈ 𝑠 ( 𝑎 ≠ 𝑑 ∧ ( abs ‘ ( 𝑥 − 𝑎 ) ) = ( abs ‘ ( 𝑏 − 𝑐 ) ) ∧ ( abs ‘ ( 𝑥 − 𝑑 ) ) = ( abs ‘ ( 𝑒 − 𝑓 ) ) ) ) } ) , { 0 , 1 } ) |
2 |
|
constrsscn.1 |
⊢ ( 𝜑 → 𝑁 ∈ On ) |
3 |
|
fveq2 |
⊢ ( 𝑚 = ∅ → ( 𝐶 ‘ 𝑚 ) = ( 𝐶 ‘ ∅ ) ) |
4 |
3
|
sseq1d |
⊢ ( 𝑚 = ∅ → ( ( 𝐶 ‘ 𝑚 ) ⊆ ℂ ↔ ( 𝐶 ‘ ∅ ) ⊆ ℂ ) ) |
5 |
|
fveq2 |
⊢ ( 𝑚 = 𝑛 → ( 𝐶 ‘ 𝑚 ) = ( 𝐶 ‘ 𝑛 ) ) |
6 |
5
|
sseq1d |
⊢ ( 𝑚 = 𝑛 → ( ( 𝐶 ‘ 𝑚 ) ⊆ ℂ ↔ ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ) ) |
7 |
|
fveq2 |
⊢ ( 𝑚 = suc 𝑛 → ( 𝐶 ‘ 𝑚 ) = ( 𝐶 ‘ suc 𝑛 ) ) |
8 |
7
|
sseq1d |
⊢ ( 𝑚 = suc 𝑛 → ( ( 𝐶 ‘ 𝑚 ) ⊆ ℂ ↔ ( 𝐶 ‘ suc 𝑛 ) ⊆ ℂ ) ) |
9 |
|
fveq2 |
⊢ ( 𝑚 = 𝑁 → ( 𝐶 ‘ 𝑚 ) = ( 𝐶 ‘ 𝑁 ) ) |
10 |
9
|
sseq1d |
⊢ ( 𝑚 = 𝑁 → ( ( 𝐶 ‘ 𝑚 ) ⊆ ℂ ↔ ( 𝐶 ‘ 𝑁 ) ⊆ ℂ ) ) |
11 |
1
|
constr0 |
⊢ ( 𝐶 ‘ ∅ ) = { 0 , 1 } |
12 |
|
0cn |
⊢ 0 ∈ ℂ |
13 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
14 |
|
prssi |
⊢ ( ( 0 ∈ ℂ ∧ 1 ∈ ℂ ) → { 0 , 1 } ⊆ ℂ ) |
15 |
12 13 14
|
mp2an |
⊢ { 0 , 1 } ⊆ ℂ |
16 |
11 15
|
eqsstri |
⊢ ( 𝐶 ‘ ∅ ) ⊆ ℂ |
17 |
|
simpl |
⊢ ( ( 𝑛 ∈ On ∧ ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ) → 𝑛 ∈ On ) |
18 |
|
eqid |
⊢ ( 𝐶 ‘ 𝑛 ) = ( 𝐶 ‘ 𝑛 ) |
19 |
1 17 18
|
constrsuc |
⊢ ( ( 𝑛 ∈ On ∧ ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ) → ( 𝑥 ∈ ( 𝐶 ‘ suc 𝑛 ) ↔ ( 𝑥 ∈ ℂ ∧ ( ∃ 𝑎 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑡 ∈ ℝ ∃ 𝑟 ∈ ℝ ( 𝑥 = ( 𝑎 + ( 𝑡 · ( 𝑏 − 𝑎 ) ) ) ∧ 𝑥 = ( 𝑐 + ( 𝑟 · ( 𝑑 − 𝑐 ) ) ) ∧ ( ℑ ‘ ( ( ∗ ‘ ( 𝑏 − 𝑎 ) ) · ( 𝑑 − 𝑐 ) ) ) ≠ 0 ) ∨ ∃ 𝑎 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑒 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑓 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑡 ∈ ℝ ( 𝑥 = ( 𝑎 + ( 𝑡 · ( 𝑏 − 𝑎 ) ) ) ∧ ( abs ‘ ( 𝑥 − 𝑐 ) ) = ( abs ‘ ( 𝑒 − 𝑓 ) ) ) ∨ ∃ 𝑎 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑒 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑓 ∈ ( 𝐶 ‘ 𝑛 ) ( 𝑎 ≠ 𝑑 ∧ ( abs ‘ ( 𝑥 − 𝑎 ) ) = ( abs ‘ ( 𝑏 − 𝑐 ) ) ∧ ( abs ‘ ( 𝑥 − 𝑑 ) ) = ( abs ‘ ( 𝑒 − 𝑓 ) ) ) ) ) ) ) |
20 |
19
|
biimpa |
⊢ ( ( ( 𝑛 ∈ On ∧ ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ) ∧ 𝑥 ∈ ( 𝐶 ‘ suc 𝑛 ) ) → ( 𝑥 ∈ ℂ ∧ ( ∃ 𝑎 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑡 ∈ ℝ ∃ 𝑟 ∈ ℝ ( 𝑥 = ( 𝑎 + ( 𝑡 · ( 𝑏 − 𝑎 ) ) ) ∧ 𝑥 = ( 𝑐 + ( 𝑟 · ( 𝑑 − 𝑐 ) ) ) ∧ ( ℑ ‘ ( ( ∗ ‘ ( 𝑏 − 𝑎 ) ) · ( 𝑑 − 𝑐 ) ) ) ≠ 0 ) ∨ ∃ 𝑎 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑒 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑓 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑡 ∈ ℝ ( 𝑥 = ( 𝑎 + ( 𝑡 · ( 𝑏 − 𝑎 ) ) ) ∧ ( abs ‘ ( 𝑥 − 𝑐 ) ) = ( abs ‘ ( 𝑒 − 𝑓 ) ) ) ∨ ∃ 𝑎 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑒 ∈ ( 𝐶 ‘ 𝑛 ) ∃ 𝑓 ∈ ( 𝐶 ‘ 𝑛 ) ( 𝑎 ≠ 𝑑 ∧ ( abs ‘ ( 𝑥 − 𝑎 ) ) = ( abs ‘ ( 𝑏 − 𝑐 ) ) ∧ ( abs ‘ ( 𝑥 − 𝑑 ) ) = ( abs ‘ ( 𝑒 − 𝑓 ) ) ) ) ) ) |
21 |
20
|
simpld |
⊢ ( ( ( 𝑛 ∈ On ∧ ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ) ∧ 𝑥 ∈ ( 𝐶 ‘ suc 𝑛 ) ) → 𝑥 ∈ ℂ ) |
22 |
21
|
ex |
⊢ ( ( 𝑛 ∈ On ∧ ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ) → ( 𝑥 ∈ ( 𝐶 ‘ suc 𝑛 ) → 𝑥 ∈ ℂ ) ) |
23 |
22
|
ssrdv |
⊢ ( ( 𝑛 ∈ On ∧ ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ) → ( 𝐶 ‘ suc 𝑛 ) ⊆ ℂ ) |
24 |
23
|
ex |
⊢ ( 𝑛 ∈ On → ( ( 𝐶 ‘ 𝑛 ) ⊆ ℂ → ( 𝐶 ‘ suc 𝑛 ) ⊆ ℂ ) ) |
25 |
|
vex |
⊢ 𝑚 ∈ V |
26 |
25
|
a1i |
⊢ ( ( Lim 𝑚 ∧ ∀ 𝑛 ∈ 𝑚 ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ) → 𝑚 ∈ V ) |
27 |
|
simpl |
⊢ ( ( Lim 𝑚 ∧ ∀ 𝑛 ∈ 𝑚 ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ) → Lim 𝑚 ) |
28 |
1 26 27
|
constrlim |
⊢ ( ( Lim 𝑚 ∧ ∀ 𝑛 ∈ 𝑚 ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ) → ( 𝐶 ‘ 𝑚 ) = ∪ 𝑜 ∈ 𝑚 ( 𝐶 ‘ 𝑜 ) ) |
29 |
|
fveq2 |
⊢ ( 𝑛 = 𝑜 → ( 𝐶 ‘ 𝑛 ) = ( 𝐶 ‘ 𝑜 ) ) |
30 |
29
|
sseq1d |
⊢ ( 𝑛 = 𝑜 → ( ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ↔ ( 𝐶 ‘ 𝑜 ) ⊆ ℂ ) ) |
31 |
|
simplr |
⊢ ( ( ( Lim 𝑚 ∧ ∀ 𝑛 ∈ 𝑚 ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ) ∧ 𝑜 ∈ 𝑚 ) → ∀ 𝑛 ∈ 𝑚 ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ) |
32 |
|
simpr |
⊢ ( ( ( Lim 𝑚 ∧ ∀ 𝑛 ∈ 𝑚 ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ) ∧ 𝑜 ∈ 𝑚 ) → 𝑜 ∈ 𝑚 ) |
33 |
30 31 32
|
rspcdva |
⊢ ( ( ( Lim 𝑚 ∧ ∀ 𝑛 ∈ 𝑚 ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ) ∧ 𝑜 ∈ 𝑚 ) → ( 𝐶 ‘ 𝑜 ) ⊆ ℂ ) |
34 |
33
|
iunssd |
⊢ ( ( Lim 𝑚 ∧ ∀ 𝑛 ∈ 𝑚 ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ) → ∪ 𝑜 ∈ 𝑚 ( 𝐶 ‘ 𝑜 ) ⊆ ℂ ) |
35 |
28 34
|
eqsstrd |
⊢ ( ( Lim 𝑚 ∧ ∀ 𝑛 ∈ 𝑚 ( 𝐶 ‘ 𝑛 ) ⊆ ℂ ) → ( 𝐶 ‘ 𝑚 ) ⊆ ℂ ) |
36 |
35
|
ex |
⊢ ( Lim 𝑚 → ( ∀ 𝑛 ∈ 𝑚 ( 𝐶 ‘ 𝑛 ) ⊆ ℂ → ( 𝐶 ‘ 𝑚 ) ⊆ ℂ ) ) |
37 |
4 6 8 10 16 24 36
|
tfinds |
⊢ ( 𝑁 ∈ On → ( 𝐶 ‘ 𝑁 ) ⊆ ℂ ) |
38 |
2 37
|
syl |
⊢ ( 𝜑 → ( 𝐶 ‘ 𝑁 ) ⊆ ℂ ) |