Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > caucvgsr | GIF version |
Description: A Cauchy sequence of
signed reals with a modulus of convergence
converges to a signed real. This is basically Corollary 11.2.13 of
[HoTT], p. (varies). The HoTT book
theorem has a modulus of
convergence (that is, a rate of convergence) specified by (11.2.9) in
HoTT whereas this theorem fixes the rate of convergence to say that
all terms after the nth term must be within 1 / 𝑛 of the nth term
(it should later be able to prove versions of this theorem with a
different fixed rate or a modulus of convergence supplied as a
hypothesis).
This is similar to caucvgprpr 7674 but is for signed reals rather than positive reals. Here is an outline of how we prove it: 1. Choose a lower bound for the sequence (see caucvgsrlembnd 7763). 2. Offset each element of the sequence so that each element of the resulting sequence is greater than one (greater than zero would not suffice, because the limit as well as the elements of the sequence need to be positive) (see caucvgsrlemofff 7759). 3. Since a signed real (element of R) which is greater than zero can be mapped to a positive real (element of P), perform that mapping on each element of the sequence and invoke caucvgprpr 7674 to get a limit (see caucvgsrlemgt1 7757). 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7757). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7762). (Contributed by Jim Kingdon, 20-Jun-2021.) |
Ref | Expression |
---|---|
caucvgsr.f | ⊢ (𝜑 → 𝐹:N⟶R) |
caucvgsr.cau | ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) |
Ref | Expression |
---|---|
caucvgsr | ⊢ (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0R <R 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹‘𝑘) +R 𝑥))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgsr.f | . 2 ⊢ (𝜑 → 𝐹:N⟶R) | |
2 | caucvgsr.cau | . 2 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) | |
3 | breq1 3992 | . . . . . . . . . . . . 13 ⊢ (𝑛 = 1o → (𝑛 <N 𝑘 ↔ 1o <N 𝑘)) | |
4 | fveq2 5496 | . . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1o → (𝐹‘𝑛) = (𝐹‘1o)) | |
5 | opeq1 3765 | . . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 1o → 〈𝑛, 1o〉 = 〈1o, 1o〉) | |
6 | 5 | eceq1d 6549 | . . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 1o → [〈𝑛, 1o〉] ~Q = [〈1o, 1o〉] ~Q ) |
7 | 6 | fveq2d 5500 | . . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 1o → (*Q‘[〈𝑛, 1o〉] ~Q ) = (*Q‘[〈1o, 1o〉] ~Q )) |
8 | 7 | breq2d 4001 | . . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1o → (𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q ) ↔ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q ))) |
9 | 8 | abbidv 2288 | . . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1o → {𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}) |
10 | 7 | breq1d 3999 | . . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1o → ((*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢 ↔ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢)) |
11 | 10 | abbidv 2288 | . . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1o → {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}) |
12 | 9, 11 | opeq12d 3773 | . . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1o → 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 = 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉) |
13 | 12 | oveq1d 5868 | . . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1o → (〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P) = (〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P)) |
14 | 13 | opeq1d 3771 | . . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1o → 〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉 = 〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉) |
15 | 14 | eceq1d 6549 | . . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1o → [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R = [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) |
16 | 15 | oveq2d 5869 | . . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1o → ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) = ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )) |
17 | 4, 16 | breq12d 4002 | . . . . . . . . . . . . . 14 ⊢ (𝑛 = 1o → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ↔ (𝐹‘1o) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ))) |
18 | 4, 15 | oveq12d 5871 | . . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1o → ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) = ((𝐹‘1o) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )) |
19 | 18 | breq2d 4001 | . . . . . . . . . . . . . 14 ⊢ (𝑛 = 1o → ((𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ↔ (𝐹‘𝑘) <R ((𝐹‘1o) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ))) |
20 | 17, 19 | anbi12d 470 | . . . . . . . . . . . . 13 ⊢ (𝑛 = 1o → (((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )) ↔ ((𝐹‘1o) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘1o) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) |
21 | 3, 20 | imbi12d 233 | . . . . . . . . . . . 12 ⊢ (𝑛 = 1o → ((𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ))) ↔ (1o <N 𝑘 → ((𝐹‘1o) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘1o) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ))))) |
22 | 21 | ralbidv 2470 | . . . . . . . . . . 11 ⊢ (𝑛 = 1o → (∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ))) ↔ ∀𝑘 ∈ N (1o <N 𝑘 → ((𝐹‘1o) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘1o) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ))))) |
23 | 1pi 7277 | . . . . . . . . . . . 12 ⊢ 1o ∈ N | |
24 | 23 | a1i 9 | . . . . . . . . . . 11 ⊢ (𝜑 → 1o ∈ N) |
25 | 22, 2, 24 | rspcdva 2839 | . . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ N (1o <N 𝑘 → ((𝐹‘1o) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘1o) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) |
26 | simpl 108 | . . . . . . . . . . . 12 ⊢ (((𝐹‘1o) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘1o) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )) → (𝐹‘1o) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )) | |
27 | 26 | imim2i 12 | . . . . . . . . . . 11 ⊢ ((1o <N 𝑘 → ((𝐹‘1o) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘1o) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ))) → (1o <N 𝑘 → (𝐹‘1o) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ))) |
28 | 27 | ralimi 2533 | . . . . . . . . . 10 ⊢ (∀𝑘 ∈ N (1o <N 𝑘 → ((𝐹‘1o) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘1o) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ))) → ∀𝑘 ∈ N (1o <N 𝑘 → (𝐹‘1o) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ))) |
29 | 25, 28 | syl 14 | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ N (1o <N 𝑘 → (𝐹‘1o) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ))) |
30 | breq2 3993 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (1o <N 𝑘 ↔ 1o <N 𝑚)) | |
31 | fveq2 5496 | . . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) | |
32 | 31 | oveq1d 5868 | . . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) = ((𝐹‘𝑚) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )) |
33 | 32 | breq2d 4001 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → ((𝐹‘1o) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ↔ (𝐹‘1o) <R ((𝐹‘𝑚) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ))) |
34 | 30, 33 | imbi12d 233 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ((1o <N 𝑘 → (𝐹‘1o) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )) ↔ (1o <N 𝑚 → (𝐹‘1o) <R ((𝐹‘𝑚) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) |
35 | 34 | rspcv 2830 | . . . . . . . . 9 ⊢ (𝑚 ∈ N → (∀𝑘 ∈ N (1o <N 𝑘 → (𝐹‘1o) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )) → (1o <N 𝑚 → (𝐹‘1o) <R ((𝐹‘𝑚) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) |
36 | 29, 35 | mpan9 279 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (1o <N 𝑚 → (𝐹‘1o) <R ((𝐹‘𝑚) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ))) |
37 | df-1nqqs 7313 | . . . . . . . . . . . . . . . . . . . 20 ⊢ 1Q = [〈1o, 1o〉] ~Q | |
38 | 37 | fveq2i 5499 | . . . . . . . . . . . . . . . . . . 19 ⊢ (*Q‘1Q) = (*Q‘[〈1o, 1o〉] ~Q ) |
39 | rec1nq 7357 | . . . . . . . . . . . . . . . . . . 19 ⊢ (*Q‘1Q) = 1Q | |
40 | 38, 39 | eqtr3i 2193 | . . . . . . . . . . . . . . . . . 18 ⊢ (*Q‘[〈1o, 1o〉] ~Q ) = 1Q |
41 | 40 | breq2i 3997 | . . . . . . . . . . . . . . . . 17 ⊢ (𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q ) ↔ 𝑙 <Q 1Q) |
42 | 41 | abbii 2286 | . . . . . . . . . . . . . . . 16 ⊢ {𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )} = {𝑙 ∣ 𝑙 <Q 1Q} |
43 | 40 | breq1i 3996 | . . . . . . . . . . . . . . . . 17 ⊢ ((*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢 ↔ 1Q <Q 𝑢) |
44 | 43 | abbii 2286 | . . . . . . . . . . . . . . . 16 ⊢ {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢} |
45 | 42, 44 | opeq12i 3770 | . . . . . . . . . . . . . . 15 ⊢ 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 = 〈{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}〉 |
46 | df-i1p 7429 | . . . . . . . . . . . . . . 15 ⊢ 1P = 〈{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}〉 | |
47 | 45, 46 | eqtr4i 2194 | . . . . . . . . . . . . . 14 ⊢ 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 = 1P |
48 | 47 | oveq1i 5863 | . . . . . . . . . . . . 13 ⊢ (〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P) = (1P +P 1P) |
49 | 48 | opeq1i 3768 | . . . . . . . . . . . 12 ⊢ 〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉 = 〈(1P +P 1P), 1P〉 |
50 | eceq1 6548 | . . . . . . . . . . . 12 ⊢ (〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉 = 〈(1P +P 1P), 1P〉 → [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R = [〈(1P +P 1P), 1P〉] ~R ) | |
51 | 49, 50 | ax-mp 5 | . . . . . . . . . . 11 ⊢ [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R = [〈(1P +P 1P), 1P〉] ~R |
52 | df-1r 7694 | . . . . . . . . . . 11 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
53 | 51, 52 | eqtr4i 2194 | . . . . . . . . . 10 ⊢ [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R = 1R |
54 | 53 | oveq2i 5864 | . . . . . . . . 9 ⊢ ((𝐹‘𝑚) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) = ((𝐹‘𝑚) +R 1R) |
55 | 54 | breq2i 3997 | . . . . . . . 8 ⊢ ((𝐹‘1o) <R ((𝐹‘𝑚) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ↔ (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R)) |
56 | 36, 55 | syl6ib 160 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (1o <N 𝑚 → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R))) |
57 | 56 | imp 123 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o <N 𝑚) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R)) |
58 | 1 | adantr 274 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 𝐹:N⟶R) |
59 | 23 | a1i 9 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 1o ∈ N) |
60 | 58, 59 | ffvelrnd 5632 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘1o) ∈ R) |
61 | ltadd1sr 7738 | . . . . . . . . 9 ⊢ ((𝐹‘1o) ∈ R → (𝐹‘1o) <R ((𝐹‘1o) +R 1R)) | |
62 | 60, 61 | syl 14 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘1o) <R ((𝐹‘1o) +R 1R)) |
63 | 62 | adantr 274 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o = 𝑚) → (𝐹‘1o) <R ((𝐹‘1o) +R 1R)) |
64 | fveq2 5496 | . . . . . . . . 9 ⊢ (1o = 𝑚 → (𝐹‘1o) = (𝐹‘𝑚)) | |
65 | 64 | oveq1d 5868 | . . . . . . . 8 ⊢ (1o = 𝑚 → ((𝐹‘1o) +R 1R) = ((𝐹‘𝑚) +R 1R)) |
66 | 65 | adantl 275 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o = 𝑚) → ((𝐹‘1o) +R 1R) = ((𝐹‘𝑚) +R 1R)) |
67 | 63, 66 | breqtrd 4015 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o = 𝑚) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R)) |
68 | nlt1pig 7303 | . . . . . . . . 9 ⊢ (𝑚 ∈ N → ¬ 𝑚 <N 1o) | |
69 | 68 | adantl 275 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ¬ 𝑚 <N 1o) |
70 | 69 | pm2.21d 614 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝑚 <N 1o → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R))) |
71 | 70 | imp 123 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 𝑚 <N 1o) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R)) |
72 | pitri3or 7284 | . . . . . . . 8 ⊢ ((1o ∈ N ∧ 𝑚 ∈ N) → (1o <N 𝑚 ∨ 1o = 𝑚 ∨ 𝑚 <N 1o)) | |
73 | 23, 72 | mpan 422 | . . . . . . 7 ⊢ (𝑚 ∈ N → (1o <N 𝑚 ∨ 1o = 𝑚 ∨ 𝑚 <N 1o)) |
74 | 73 | adantl 275 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (1o <N 𝑚 ∨ 1o = 𝑚 ∨ 𝑚 <N 1o)) |
75 | 57, 67, 71, 74 | mpjao3dan 1302 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R)) |
76 | ltasrg 7732 | . . . . . . 7 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → (𝑓 <R 𝑔 ↔ (ℎ +R 𝑓) <R (ℎ +R 𝑔))) | |
77 | 76 | adantl 275 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R)) → (𝑓 <R 𝑔 ↔ (ℎ +R 𝑓) <R (ℎ +R 𝑔))) |
78 | 1 | ffvelrnda 5631 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘𝑚) ∈ R) |
79 | 1sr 7713 | . . . . . . 7 ⊢ 1R ∈ R | |
80 | addclsr 7715 | . . . . . . 7 ⊢ (((𝐹‘𝑚) ∈ R ∧ 1R ∈ R) → ((𝐹‘𝑚) +R 1R) ∈ R) | |
81 | 78, 79, 80 | sylancl 411 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘𝑚) +R 1R) ∈ R) |
82 | m1r 7714 | . . . . . . 7 ⊢ -1R ∈ R | |
83 | 82 | a1i 9 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → -1R ∈ R) |
84 | addcomsrg 7717 | . . . . . . 7 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓)) | |
85 | 84 | adantl 275 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R)) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓)) |
86 | 77, 60, 81, 83, 85 | caovord2d 6022 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘1o) <R ((𝐹‘𝑚) +R 1R) ↔ ((𝐹‘1o) +R -1R) <R (((𝐹‘𝑚) +R 1R) +R -1R))) |
87 | 75, 86 | mpbid 146 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘1o) +R -1R) <R (((𝐹‘𝑚) +R 1R) +R -1R)) |
88 | 79 | a1i 9 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 1R ∈ R) |
89 | addasssrg 7718 | . . . . . 6 ⊢ (((𝐹‘𝑚) ∈ R ∧ 1R ∈ R ∧ -1R ∈ R) → (((𝐹‘𝑚) +R 1R) +R -1R) = ((𝐹‘𝑚) +R (1R +R -1R))) | |
90 | 78, 88, 83, 89 | syl3anc 1233 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (((𝐹‘𝑚) +R 1R) +R -1R) = ((𝐹‘𝑚) +R (1R +R -1R))) |
91 | addcomsrg 7717 | . . . . . . . . 9 ⊢ ((1R ∈ R ∧ -1R ∈ R) → (1R +R -1R) = (-1R +R 1R)) | |
92 | 79, 82, 91 | mp2an 424 | . . . . . . . 8 ⊢ (1R +R -1R) = (-1R +R 1R) |
93 | m1p1sr 7722 | . . . . . . . 8 ⊢ (-1R +R 1R) = 0R | |
94 | 92, 93 | eqtri 2191 | . . . . . . 7 ⊢ (1R +R -1R) = 0R |
95 | 94 | oveq2i 5864 | . . . . . 6 ⊢ ((𝐹‘𝑚) +R (1R +R -1R)) = ((𝐹‘𝑚) +R 0R) |
96 | 0idsr 7729 | . . . . . . 7 ⊢ ((𝐹‘𝑚) ∈ R → ((𝐹‘𝑚) +R 0R) = (𝐹‘𝑚)) | |
97 | 78, 96 | syl 14 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘𝑚) +R 0R) = (𝐹‘𝑚)) |
98 | 95, 97 | eqtrid 2215 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘𝑚) +R (1R +R -1R)) = (𝐹‘𝑚)) |
99 | 90, 98 | eqtrd 2203 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (((𝐹‘𝑚) +R 1R) +R -1R) = (𝐹‘𝑚)) |
100 | 87, 99 | breqtrd 4015 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘1o) +R -1R) <R (𝐹‘𝑚)) |
101 | 100 | ralrimiva 2543 | . 2 ⊢ (𝜑 → ∀𝑚 ∈ N ((𝐹‘1o) +R -1R) <R (𝐹‘𝑚)) |
102 | 1, 2, 101 | caucvgsrlembnd 7763 | 1 ⊢ (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0R <R 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹‘𝑘) +R 𝑥))))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ w3o 972 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 {cab 2156 ∀wral 2448 ∃wrex 2449 〈cop 3586 class class class wbr 3989 ⟶wf 5194 ‘cfv 5198 (class class class)co 5853 1oc1o 6388 [cec 6511 Ncnpi 7234 <N clti 7237 ~Q ceq 7241 1Qc1q 7243 *Qcrq 7246 <Q cltq 7247 1Pc1p 7254 +P cpp 7255 ~R cer 7258 Rcnr 7259 0Rc0r 7260 1Rc1r 7261 -1Rcm1r 7262 +R cplr 7263 <R cltr 7265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-eprel 4274 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-1o 6395 df-2o 6396 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-pli 7267 df-mi 7268 df-lti 7269 df-plpq 7306 df-mpq 7307 df-enq 7309 df-nqqs 7310 df-plqqs 7311 df-mqqs 7312 df-1nqqs 7313 df-rq 7314 df-ltnqqs 7315 df-enq0 7386 df-nq0 7387 df-0nq0 7388 df-plq0 7389 df-mq0 7390 df-inp 7428 df-i1p 7429 df-iplp 7430 df-imp 7431 df-iltp 7432 df-enr 7688 df-nr 7689 df-plr 7690 df-mr 7691 df-ltr 7692 df-0r 7693 df-1r 7694 df-m1r 7695 |
This theorem is referenced by: axcaucvglemres 7861 |
Copyright terms: Public domain | W3C validator |