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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 7544 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 7633). 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 7629). 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 7544 to get a limit (see caucvgsrlemgt1 7627). 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7627). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7632). (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 3940 | . . . . . . . . . . . . 13 ⊢ (𝑛 = 1o → (𝑛 <N 𝑘 ↔ 1o <N 𝑘)) | |
4 | fveq2 5429 | . . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1o → (𝐹‘𝑛) = (𝐹‘1o)) | |
5 | opeq1 3713 | . . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 1o → 〈𝑛, 1o〉 = 〈1o, 1o〉) | |
6 | 5 | eceq1d 6473 | . . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 1o → [〈𝑛, 1o〉] ~Q = [〈1o, 1o〉] ~Q ) |
7 | 6 | fveq2d 5433 | . . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 1o → (*Q‘[〈𝑛, 1o〉] ~Q ) = (*Q‘[〈1o, 1o〉] ~Q )) |
8 | 7 | breq2d 3949 | . . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1o → (𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q ) ↔ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q ))) |
9 | 8 | abbidv 2258 | . . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1o → {𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}) |
10 | 7 | breq1d 3947 | . . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1o → ((*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢 ↔ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢)) |
11 | 10 | abbidv 2258 | . . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1o → {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}) |
12 | 9, 11 | opeq12d 3721 | . . . . . . . . . . . . . . . . . . 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 5797 | . . . . . . . . . . . . . . . . . 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 3719 | . . . . . . . . . . . . . . . . 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 6473 | . . . . . . . . . . . . . . . 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 5798 | . . . . . . . . . . . . . . 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 3950 | . . . . . . . . . . . . . 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 5800 | . . . . . . . . . . . . . . 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 3949 | . . . . . . . . . . . . . 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 465 | . . . . . . . . . . . . 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 2438 | . . . . . . . . . . 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 7147 | . . . . . . . . . . . 12 ⊢ 1o ∈ N | |
24 | 23 | a1i 9 | . . . . . . . . . . 11 ⊢ (𝜑 → 1o ∈ N) |
25 | 22, 2, 24 | rspcdva 2798 | . . . . . . . . . 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 2498 | . . . . . . . . . 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 3941 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (1o <N 𝑘 ↔ 1o <N 𝑚)) | |
31 | fveq2 5429 | . . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) | |
32 | 31 | oveq1d 5797 | . . . . . . . . . . . 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 3949 | . . . . . . . . . . 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 2789 | . . . . . . . . 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 7183 | . . . . . . . . . . . . . . . . . . . 20 ⊢ 1Q = [〈1o, 1o〉] ~Q | |
38 | 37 | fveq2i 5432 | . . . . . . . . . . . . . . . . . . 19 ⊢ (*Q‘1Q) = (*Q‘[〈1o, 1o〉] ~Q ) |
39 | rec1nq 7227 | . . . . . . . . . . . . . . . . . . 19 ⊢ (*Q‘1Q) = 1Q | |
40 | 38, 39 | eqtr3i 2163 | . . . . . . . . . . . . . . . . . 18 ⊢ (*Q‘[〈1o, 1o〉] ~Q ) = 1Q |
41 | 40 | breq2i 3945 | . . . . . . . . . . . . . . . . 17 ⊢ (𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q ) ↔ 𝑙 <Q 1Q) |
42 | 41 | abbii 2256 | . . . . . . . . . . . . . . . 16 ⊢ {𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )} = {𝑙 ∣ 𝑙 <Q 1Q} |
43 | 40 | breq1i 3944 | . . . . . . . . . . . . . . . . 17 ⊢ ((*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢 ↔ 1Q <Q 𝑢) |
44 | 43 | abbii 2256 | . . . . . . . . . . . . . . . 16 ⊢ {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢} |
45 | 42, 44 | opeq12i 3718 | . . . . . . . . . . . . . . 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 7299 | . . . . . . . . . . . . . . 15 ⊢ 1P = 〈{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}〉 | |
47 | 45, 46 | eqtr4i 2164 | . . . . . . . . . . . . . 14 ⊢ 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 = 1P |
48 | 47 | oveq1i 5792 | . . . . . . . . . . . . 13 ⊢ (〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P) = (1P +P 1P) |
49 | 48 | opeq1i 3716 | . . . . . . . . . . . 12 ⊢ 〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉 = 〈(1P +P 1P), 1P〉 |
50 | eceq1 6472 | . . . . . . . . . . . 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 7564 | . . . . . . . . . . 11 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
53 | 51, 52 | eqtr4i 2164 | . . . . . . . . . 10 ⊢ [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R = 1R |
54 | 53 | oveq2i 5793 | . . . . . . . . 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 3945 | . . . . . . . 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 5564 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘1o) ∈ R) |
61 | ltadd1sr 7608 | . . . . . . . . 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 5429 | . . . . . . . . 9 ⊢ (1o = 𝑚 → (𝐹‘1o) = (𝐹‘𝑚)) | |
65 | 64 | oveq1d 5797 | . . . . . . . 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 3962 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o = 𝑚) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R)) |
68 | nlt1pig 7173 | . . . . . . . . 9 ⊢ (𝑚 ∈ N → ¬ 𝑚 <N 1o) | |
69 | 68 | adantl 275 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ¬ 𝑚 <N 1o) |
70 | 69 | pm2.21d 609 | . . . . . . 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 7154 | . . . . . . . 8 ⊢ ((1o ∈ N ∧ 𝑚 ∈ N) → (1o <N 𝑚 ∨ 1o = 𝑚 ∨ 𝑚 <N 1o)) | |
73 | 23, 72 | mpan 421 | . . . . . . 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 1286 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R)) |
76 | ltasrg 7602 | . . . . . . 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 5563 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘𝑚) ∈ R) |
79 | 1sr 7583 | . . . . . . 7 ⊢ 1R ∈ R | |
80 | addclsr 7585 | . . . . . . 7 ⊢ (((𝐹‘𝑚) ∈ R ∧ 1R ∈ R) → ((𝐹‘𝑚) +R 1R) ∈ R) | |
81 | 78, 79, 80 | sylancl 410 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘𝑚) +R 1R) ∈ R) |
82 | m1r 7584 | . . . . . . 7 ⊢ -1R ∈ R | |
83 | 82 | a1i 9 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → -1R ∈ R) |
84 | addcomsrg 7587 | . . . . . . 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 5948 | . . . . 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 7588 | . . . . . 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 1217 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (((𝐹‘𝑚) +R 1R) +R -1R) = ((𝐹‘𝑚) +R (1R +R -1R))) |
91 | addcomsrg 7587 | . . . . . . . . 9 ⊢ ((1R ∈ R ∧ -1R ∈ R) → (1R +R -1R) = (-1R +R 1R)) | |
92 | 79, 82, 91 | mp2an 423 | . . . . . . . 8 ⊢ (1R +R -1R) = (-1R +R 1R) |
93 | m1p1sr 7592 | . . . . . . . 8 ⊢ (-1R +R 1R) = 0R | |
94 | 92, 93 | eqtri 2161 | . . . . . . 7 ⊢ (1R +R -1R) = 0R |
95 | 94 | oveq2i 5793 | . . . . . 6 ⊢ ((𝐹‘𝑚) +R (1R +R -1R)) = ((𝐹‘𝑚) +R 0R) |
96 | 0idsr 7599 | . . . . . . 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 | syl5eq 2185 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘𝑚) +R (1R +R -1R)) = (𝐹‘𝑚)) |
99 | 90, 98 | eqtrd 2173 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (((𝐹‘𝑚) +R 1R) +R -1R) = (𝐹‘𝑚)) |
100 | 87, 99 | breqtrd 3962 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘1o) +R -1R) <R (𝐹‘𝑚)) |
101 | 100 | ralrimiva 2508 | . 2 ⊢ (𝜑 → ∀𝑚 ∈ N ((𝐹‘1o) +R -1R) <R (𝐹‘𝑚)) |
102 | 1, 2, 101 | caucvgsrlembnd 7633 | 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 962 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 {cab 2126 ∀wral 2417 ∃wrex 2418 〈cop 3535 class class class wbr 3937 ⟶wf 5127 ‘cfv 5131 (class class class)co 5782 1oc1o 6314 [cec 6435 Ncnpi 7104 <N clti 7107 ~Q ceq 7111 1Qc1q 7113 *Qcrq 7116 <Q cltq 7117 1Pc1p 7124 +P cpp 7125 ~R cer 7128 Rcnr 7129 0Rc0r 7130 1Rc1r 7131 -1Rcm1r 7132 +R cplr 7133 <R cltr 7135 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-2o 6322 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-enq0 7256 df-nq0 7257 df-0nq0 7258 df-plq0 7259 df-mq0 7260 df-inp 7298 df-i1p 7299 df-iplp 7300 df-imp 7301 df-iltp 7302 df-enr 7558 df-nr 7559 df-plr 7560 df-mr 7561 df-ltr 7562 df-0r 7563 df-1r 7564 df-m1r 7565 |
This theorem is referenced by: axcaucvglemres 7731 |
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