Step | Hyp | Ref
| Expression |
1 | | caucvgprprlemlim.jk |
. . . . 5
⊢ (𝜑 → 𝐽 <N 𝐾) |
2 | | caucvgprprlemlim.jkq |
. . . . 5
⊢ (𝜑 → 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐽, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐽, 1o〉]
~Q ) <Q 𝑢}〉<P 𝑄) |
3 | 1, 2 | caucvgprprlemk 7603 |
. . . 4
⊢ (𝜑 → 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉<P 𝑄) |
4 | | ltrelpi 7244 |
. . . . . . . . . 10
⊢
<N ⊆ (N ×
N) |
5 | 4 | brel 4638 |
. . . . . . . . 9
⊢ (𝐽 <N
𝐾 → (𝐽 ∈ N ∧ 𝐾 ∈
N)) |
6 | 1, 5 | syl 14 |
. . . . . . . 8
⊢ (𝜑 → (𝐽 ∈ N ∧ 𝐾 ∈
N)) |
7 | 6 | simprd 113 |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈ N) |
8 | | nnnq 7342 |
. . . . . . . 8
⊢ (𝐾 ∈ N →
[〈𝐾,
1o〉] ~Q ∈
Q) |
9 | | recclnq 7312 |
. . . . . . . 8
⊢
([〈𝐾,
1o〉] ~Q ∈ Q →
(*Q‘[〈𝐾, 1o〉]
~Q ) ∈ Q) |
10 | 8, 9 | syl 14 |
. . . . . . 7
⊢ (𝐾 ∈ N →
(*Q‘[〈𝐾, 1o〉]
~Q ) ∈ Q) |
11 | 7, 10 | syl 14 |
. . . . . 6
⊢ (𝜑 →
(*Q‘[〈𝐾, 1o〉]
~Q ) ∈ Q) |
12 | | nqprlu 7467 |
. . . . . 6
⊢
((*Q‘[〈𝐾, 1o〉]
~Q ) ∈ Q → 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉 ∈
P) |
13 | 11, 12 | syl 14 |
. . . . 5
⊢ (𝜑 → 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉 ∈
P) |
14 | | caucvgprprlemlim.q |
. . . . 5
⊢ (𝜑 → 𝑄 ∈ P) |
15 | | caucvgprpr.f |
. . . . . 6
⊢ (𝜑 → 𝐹:N⟶P) |
16 | 15, 7 | ffvelrnd 5603 |
. . . . 5
⊢ (𝜑 → (𝐹‘𝐾) ∈ P) |
17 | | ltaprg 7539 |
. . . . 5
⊢
((〈{𝑙 ∣
𝑙
<Q (*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉 ∈ P ∧ 𝑄 ∈ P ∧
(𝐹‘𝐾) ∈ P) →
(〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉<P 𝑄 ↔ ((𝐹‘𝐾) +P 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)<P
((𝐹‘𝐾) +P 𝑄))) |
18 | 13, 14, 16, 17 | syl3anc 1220 |
. . . 4
⊢ (𝜑 → (〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉<P 𝑄 ↔ ((𝐹‘𝐾) +P 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)<P
((𝐹‘𝐾) +P 𝑄))) |
19 | 3, 18 | mpbid 146 |
. . 3
⊢ (𝜑 → ((𝐹‘𝐾) +P 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)<P
((𝐹‘𝐾) +P 𝑄)) |
20 | | addclpr 7457 |
. . . . 5
⊢ (((𝐹‘𝐾) ∈ P ∧ 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉 ∈ P) →
((𝐹‘𝐾) +P 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉) ∈
P) |
21 | 16, 13, 20 | syl2anc 409 |
. . . 4
⊢ (𝜑 → ((𝐹‘𝐾) +P 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉) ∈
P) |
22 | | addclpr 7457 |
. . . . 5
⊢ (((𝐹‘𝐾) ∈ P ∧ 𝑄 ∈ P) →
((𝐹‘𝐾) +P 𝑄) ∈
P) |
23 | 16, 14, 22 | syl2anc 409 |
. . . 4
⊢ (𝜑 → ((𝐹‘𝐾) +P 𝑄) ∈
P) |
24 | | ltdfpr 7426 |
. . . 4
⊢ ((((𝐹‘𝐾) +P 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉) ∈ P ∧ ((𝐹‘𝐾) +P 𝑄) ∈ P) →
(((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)<P
((𝐹‘𝐾) +P 𝑄) ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) |
25 | 21, 23, 24 | syl2anc 409 |
. . 3
⊢ (𝜑 → (((𝐹‘𝐾) +P 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)<P
((𝐹‘𝐾) +P 𝑄) ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) |
26 | 19, 25 | mpbid 146 |
. 2
⊢ (𝜑 → ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄)))) |
27 | | simprl 521 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → 𝑥 ∈ Q) |
28 | 7 | adantr 274 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → 𝐾 ∈ N) |
29 | | simprrl 529 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → 𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉))) |
30 | | breq1 3968 |
. . . . . . . . . . . 12
⊢ (𝑙 = 𝑝 → (𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q ) ↔ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q ))) |
31 | 30 | cbvabv 2282 |
. . . . . . . . . . 11
⊢ {𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )} = {𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )} |
32 | | breq2 3969 |
. . . . . . . . . . . 12
⊢ (𝑢 = 𝑞 →
((*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢 ↔
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞)) |
33 | 32 | cbvabv 2282 |
. . . . . . . . . . 11
⊢ {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢} = {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞} |
34 | 31, 33 | opeq12i 3746 |
. . . . . . . . . 10
⊢
〈{𝑙 ∣
𝑙
<Q (*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉 = 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉 |
35 | 34 | oveq2i 5835 |
. . . . . . . . 9
⊢ ((𝐹‘𝐾) +P 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉) = ((𝐹‘𝐾) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉) |
36 | 35 | fveq2i 5471 |
. . . . . . . 8
⊢
(2nd ‘((𝐹‘𝐾) +P 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) = (2nd ‘((𝐹‘𝐾) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉)) |
37 | 29, 36 | eleqtrdi 2250 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → 𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉))) |
38 | | nqprlu 7467 |
. . . . . . . . . . 11
⊢
((*Q‘[〈𝐾, 1o〉]
~Q ) ∈ Q → 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉 ∈
P) |
39 | 11, 38 | syl 14 |
. . . . . . . . . 10
⊢ (𝜑 → 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉 ∈
P) |
40 | | addclpr 7457 |
. . . . . . . . . 10
⊢ (((𝐹‘𝐾) ∈ P ∧ 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉 ∈ P) →
((𝐹‘𝐾) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉) ∈
P) |
41 | 16, 39, 40 | syl2anc 409 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐹‘𝐾) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉) ∈
P) |
42 | 41 | adantr 274 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → ((𝐹‘𝐾) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉) ∈
P) |
43 | | nqpru 7472 |
. . . . . . . 8
⊢ ((𝑥 ∈ Q ∧
((𝐹‘𝐾) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉) ∈ P) →
(𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉)) ↔ ((𝐹‘𝐾) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}〉)) |
44 | 27, 42, 43 | syl2anc 409 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → (𝑥 ∈ (2nd ‘((𝐹‘𝐾) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉)) ↔ ((𝐹‘𝐾) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}〉)) |
45 | 37, 44 | mpbid 146 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → ((𝐹‘𝐾) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}〉) |
46 | | fveq2 5468 |
. . . . . . . . 9
⊢ (𝑟 = 𝐾 → (𝐹‘𝑟) = (𝐹‘𝐾)) |
47 | | opeq1 3741 |
. . . . . . . . . . . . . 14
⊢ (𝑟 = 𝐾 → 〈𝑟, 1o〉 = 〈𝐾,
1o〉) |
48 | 47 | eceq1d 6516 |
. . . . . . . . . . . . 13
⊢ (𝑟 = 𝐾 → [〈𝑟, 1o〉]
~Q = [〈𝐾, 1o〉]
~Q ) |
49 | 48 | fveq2d 5472 |
. . . . . . . . . . . 12
⊢ (𝑟 = 𝐾 →
(*Q‘[〈𝑟, 1o〉]
~Q ) = (*Q‘[〈𝐾, 1o〉]
~Q )) |
50 | 49 | breq2d 3977 |
. . . . . . . . . . 11
⊢ (𝑟 = 𝐾 → (𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q ) ↔ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q ))) |
51 | 50 | abbidv 2275 |
. . . . . . . . . 10
⊢ (𝑟 = 𝐾 → {𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )} = {𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}) |
52 | 49 | breq1d 3975 |
. . . . . . . . . . 11
⊢ (𝑟 = 𝐾 →
((*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞 ↔
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞)) |
53 | 52 | abbidv 2275 |
. . . . . . . . . 10
⊢ (𝑟 = 𝐾 → {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞} = {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}) |
54 | 51, 53 | opeq12d 3749 |
. . . . . . . . 9
⊢ (𝑟 = 𝐾 → 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞}〉 = 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉) |
55 | 46, 54 | oveq12d 5842 |
. . . . . . . 8
⊢ (𝑟 = 𝐾 → ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞}〉) = ((𝐹‘𝐾) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉)) |
56 | 55 | breq1d 3975 |
. . . . . . 7
⊢ (𝑟 = 𝐾 → (((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}〉 ↔ ((𝐹‘𝐾) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}〉)) |
57 | 56 | rspcev 2816 |
. . . . . 6
⊢ ((𝐾 ∈ N ∧
((𝐹‘𝐾) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}〉) → ∃𝑟 ∈ N ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}〉) |
58 | 28, 45, 57 | syl2anc 409 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → ∃𝑟 ∈ N ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}〉) |
59 | | breq2 3969 |
. . . . . . . . . 10
⊢ (𝑢 = 𝑥 → (𝑝 <Q 𝑢 ↔ 𝑝 <Q 𝑥)) |
60 | 59 | abbidv 2275 |
. . . . . . . . 9
⊢ (𝑢 = 𝑥 → {𝑝 ∣ 𝑝 <Q 𝑢} = {𝑝 ∣ 𝑝 <Q 𝑥}) |
61 | | breq1 3968 |
. . . . . . . . . 10
⊢ (𝑢 = 𝑥 → (𝑢 <Q 𝑞 ↔ 𝑥 <Q 𝑞)) |
62 | 61 | abbidv 2275 |
. . . . . . . . 9
⊢ (𝑢 = 𝑥 → {𝑞 ∣ 𝑢 <Q 𝑞} = {𝑞 ∣ 𝑥 <Q 𝑞}) |
63 | 60, 62 | opeq12d 3749 |
. . . . . . . 8
⊢ (𝑢 = 𝑥 → 〈{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}〉 = 〈{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}〉) |
64 | 63 | breq2d 3977 |
. . . . . . 7
⊢ (𝑢 = 𝑥 → (((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}〉 ↔ ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}〉)) |
65 | 64 | rexbidv 2458 |
. . . . . 6
⊢ (𝑢 = 𝑥 → (∃𝑟 ∈ N ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}〉 ↔ ∃𝑟 ∈ N ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}〉)) |
66 | | caucvgprpr.lim |
. . . . . . . 8
⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑟 ∈ N
〈{𝑝 ∣ 𝑝 <Q
(𝑙
+Q (*Q‘[〈𝑟, 1o〉]
~Q ))}, {𝑞 ∣ (𝑙 +Q
(*Q‘[〈𝑟, 1o〉]
~Q )) <Q 𝑞}〉<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}〉}〉 |
67 | 66 | fveq2i 5471 |
. . . . . . 7
⊢
(2nd ‘𝐿) = (2nd ‘〈{𝑙 ∈ Q ∣
∃𝑟 ∈
N 〈{𝑝
∣ 𝑝
<Q (𝑙 +Q
(*Q‘[〈𝑟, 1o〉]
~Q ))}, {𝑞 ∣ (𝑙 +Q
(*Q‘[〈𝑟, 1o〉]
~Q )) <Q 𝑞}〉<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}〉}〉) |
68 | | nqex 7283 |
. . . . . . . . 9
⊢
Q ∈ V |
69 | 68 | rabex 4108 |
. . . . . . . 8
⊢ {𝑙 ∈ Q ∣
∃𝑟 ∈
N 〈{𝑝
∣ 𝑝
<Q (𝑙 +Q
(*Q‘[〈𝑟, 1o〉]
~Q ))}, {𝑞 ∣ (𝑙 +Q
(*Q‘[〈𝑟, 1o〉]
~Q )) <Q 𝑞}〉<P (𝐹‘𝑟)} ∈ V |
70 | 68 | rabex 4108 |
. . . . . . . 8
⊢ {𝑢 ∈ Q ∣
∃𝑟 ∈
N ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}〉} ∈
V |
71 | 69, 70 | op2nd 6095 |
. . . . . . 7
⊢
(2nd ‘〈{𝑙 ∈ Q ∣ ∃𝑟 ∈ N
〈{𝑝 ∣ 𝑝 <Q
(𝑙
+Q (*Q‘[〈𝑟, 1o〉]
~Q ))}, {𝑞 ∣ (𝑙 +Q
(*Q‘[〈𝑟, 1o〉]
~Q )) <Q 𝑞}〉<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}〉}〉) = {𝑢 ∈ Q ∣
∃𝑟 ∈
N ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}〉} |
72 | 67, 71 | eqtri 2178 |
. . . . . 6
⊢
(2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}〉} |
73 | 65, 72 | elrab2 2871 |
. . . . 5
⊢ (𝑥 ∈ (2nd
‘𝐿) ↔ (𝑥 ∈ Q ∧
∃𝑟 ∈
N ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q
(*Q‘[〈𝑟, 1o〉]
~Q )}, {𝑞 ∣
(*Q‘[〈𝑟, 1o〉]
~Q ) <Q 𝑞}〉)<P
〈{𝑝 ∣ 𝑝 <Q
𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}〉)) |
74 | 27, 58, 73 | sylanbrc 414 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → 𝑥 ∈ (2nd ‘𝐿)) |
75 | | simprrr 530 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))) |
76 | | rspe 2506 |
. . . 4
⊢ ((𝑥 ∈ Q ∧
(𝑥 ∈ (2nd
‘𝐿) ∧ 𝑥 ∈ (1st
‘((𝐹‘𝐾) +P
𝑄)))) → ∃𝑥 ∈ Q (𝑥 ∈ (2nd
‘𝐿) ∧ 𝑥 ∈ (1st
‘((𝐹‘𝐾) +P
𝑄)))) |
77 | 27, 74, 75, 76 | syl12anc 1218 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → ∃𝑥 ∈ Q (𝑥 ∈ (2nd
‘𝐿) ∧ 𝑥 ∈ (1st
‘((𝐹‘𝐾) +P
𝑄)))) |
78 | | caucvgprpr.cau |
. . . . . 6
⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N
𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝑛, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝑛, 1o〉]
~Q ) <Q 𝑢}〉) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P 〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝑛, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝑛, 1o〉]
~Q ) <Q 𝑢}〉)))) |
79 | | caucvgprpr.bnd |
. . . . . 6
⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚)) |
80 | 15, 78, 79, 66 | caucvgprprlemcl 7624 |
. . . . 5
⊢ (𝜑 → 𝐿 ∈ P) |
81 | 80 | adantr 274 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → 𝐿 ∈ P) |
82 | 23 | adantr 274 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → ((𝐹‘𝐾) +P 𝑄) ∈
P) |
83 | | ltdfpr 7426 |
. . . 4
⊢ ((𝐿 ∈ P ∧
((𝐹‘𝐾) +P 𝑄) ∈ P) →
(𝐿<P ((𝐹‘𝐾) +P 𝑄) ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd
‘𝐿) ∧ 𝑥 ∈ (1st
‘((𝐹‘𝐾) +P
𝑄))))) |
84 | 81, 82, 83 | syl2anc 409 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → (𝐿<P ((𝐹‘𝐾) +P 𝑄) ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd
‘𝐿) ∧ 𝑥 ∈ (1st
‘((𝐹‘𝐾) +P
𝑄))))) |
85 | 77, 84 | mpbird 166 |
. 2
⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd
‘((𝐹‘𝐾) +P
〈{𝑙 ∣ 𝑙 <Q
(*Q‘[〈𝐾, 1o〉]
~Q )}, {𝑢 ∣
(*Q‘[〈𝐾, 1o〉]
~Q ) <Q 𝑢}〉)) ∧ 𝑥 ∈ (1st ‘((𝐹‘𝐾) +P 𝑄))))) → 𝐿<P ((𝐹‘𝐾) +P 𝑄)) |
86 | 26, 85 | rexlimddv 2579 |
1
⊢ (𝜑 → 𝐿<P ((𝐹‘𝐾) +P 𝑄)) |