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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 7368 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 7443). 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 7439). 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 7368 to get a limit (see caucvgsrlemgt1 7437). 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7437). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7442). (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 3870 | . . . . . . . . . . . . 13 ⊢ (𝑛 = 1o → (𝑛 <N 𝑘 ↔ 1o <N 𝑘)) | |
4 | fveq2 5340 | . . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1o → (𝐹‘𝑛) = (𝐹‘1o)) | |
5 | opeq1 3644 | . . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 1o → 〈𝑛, 1o〉 = 〈1o, 1o〉) | |
6 | 5 | eceq1d 6368 | . . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 1o → [〈𝑛, 1o〉] ~Q = [〈1o, 1o〉] ~Q ) |
7 | 6 | fveq2d 5344 | . . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 1o → (*Q‘[〈𝑛, 1o〉] ~Q ) = (*Q‘[〈1o, 1o〉] ~Q )) |
8 | 7 | breq2d 3879 | . . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1o → (𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q ) ↔ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q ))) |
9 | 8 | abbidv 2212 | . . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1o → {𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}) |
10 | 7 | breq1d 3877 | . . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1o → ((*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢 ↔ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢)) |
11 | 10 | abbidv 2212 | . . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1o → {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}) |
12 | 9, 11 | opeq12d 3652 | . . . . . . . . . . . . . . . . . . 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 5705 | . . . . . . . . . . . . . . . . . 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 3650 | . . . . . . . . . . . . . . . . 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 6368 | . . . . . . . . . . . . . . . 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 5706 | . . . . . . . . . . . . . . 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 3880 | . . . . . . . . . . . . . 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 5708 | . . . . . . . . . . . . . . 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 3879 | . . . . . . . . . . . . . 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 458 | . . . . . . . . . . . . 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 2391 | . . . . . . . . . . 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 6971 | . . . . . . . . . . . 12 ⊢ 1o ∈ N | |
24 | 23 | a1i 9 | . . . . . . . . . . 11 ⊢ (𝜑 → 1o ∈ N) |
25 | 22, 2, 24 | rspcdva 2741 | . . . . . . . . . 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 2449 | . . . . . . . . . 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 3871 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (1o <N 𝑘 ↔ 1o <N 𝑚)) | |
31 | fveq2 5340 | . . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) | |
32 | 31 | oveq1d 5705 | . . . . . . . . . . . 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 3879 | . . . . . . . . . . 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 2732 | . . . . . . . . 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 276 | . . . . . . . 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 7007 | . . . . . . . . . . . . . . . . . . . 20 ⊢ 1Q = [〈1o, 1o〉] ~Q | |
38 | 37 | fveq2i 5343 | . . . . . . . . . . . . . . . . . . 19 ⊢ (*Q‘1Q) = (*Q‘[〈1o, 1o〉] ~Q ) |
39 | rec1nq 7051 | . . . . . . . . . . . . . . . . . . 19 ⊢ (*Q‘1Q) = 1Q | |
40 | 38, 39 | eqtr3i 2117 | . . . . . . . . . . . . . . . . . 18 ⊢ (*Q‘[〈1o, 1o〉] ~Q ) = 1Q |
41 | 40 | breq2i 3875 | . . . . . . . . . . . . . . . . 17 ⊢ (𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q ) ↔ 𝑙 <Q 1Q) |
42 | 41 | abbii 2210 | . . . . . . . . . . . . . . . 16 ⊢ {𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )} = {𝑙 ∣ 𝑙 <Q 1Q} |
43 | 40 | breq1i 3874 | . . . . . . . . . . . . . . . . 17 ⊢ ((*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢 ↔ 1Q <Q 𝑢) |
44 | 43 | abbii 2210 | . . . . . . . . . . . . . . . 16 ⊢ {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢} |
45 | 42, 44 | opeq12i 3649 | . . . . . . . . . . . . . . 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 7123 | . . . . . . . . . . . . . . 15 ⊢ 1P = 〈{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}〉 | |
47 | 45, 46 | eqtr4i 2118 | . . . . . . . . . . . . . 14 ⊢ 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 = 1P |
48 | 47 | oveq1i 5700 | . . . . . . . . . . . . 13 ⊢ (〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P) = (1P +P 1P) |
49 | 48 | opeq1i 3647 | . . . . . . . . . . . 12 ⊢ 〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉 = 〈(1P +P 1P), 1P〉 |
50 | eceq1 6367 | . . . . . . . . . . . 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 7 | . . . . . . . . . . 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 7375 | . . . . . . . . . . 11 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
53 | 51, 52 | eqtr4i 2118 | . . . . . . . . . 10 ⊢ [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈1o, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈1o, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R = 1R |
54 | 53 | oveq2i 5701 | . . . . . . . . 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 3875 | . . . . . . . 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 271 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 𝐹:N⟶R) |
59 | 23 | a1i 9 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 1o ∈ N) |
60 | 58, 59 | ffvelrnd 5474 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘1o) ∈ R) |
61 | ltadd1sr 7419 | . . . . . . . . 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 271 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o = 𝑚) → (𝐹‘1o) <R ((𝐹‘1o) +R 1R)) |
64 | fveq2 5340 | . . . . . . . . 9 ⊢ (1o = 𝑚 → (𝐹‘1o) = (𝐹‘𝑚)) | |
65 | 64 | oveq1d 5705 | . . . . . . . 8 ⊢ (1o = 𝑚 → ((𝐹‘1o) +R 1R) = ((𝐹‘𝑚) +R 1R)) |
66 | 65 | adantl 272 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o = 𝑚) → ((𝐹‘1o) +R 1R) = ((𝐹‘𝑚) +R 1R)) |
67 | 63, 66 | breqtrd 3891 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o = 𝑚) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R)) |
68 | nlt1pig 6997 | . . . . . . . . 9 ⊢ (𝑚 ∈ N → ¬ 𝑚 <N 1o) | |
69 | 68 | adantl 272 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ¬ 𝑚 <N 1o) |
70 | 69 | pm2.21d 587 | . . . . . . 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 6978 | . . . . . . . 8 ⊢ ((1o ∈ N ∧ 𝑚 ∈ N) → (1o <N 𝑚 ∨ 1o = 𝑚 ∨ 𝑚 <N 1o)) | |
73 | 23, 72 | mpan 416 | . . . . . . 7 ⊢ (𝑚 ∈ N → (1o <N 𝑚 ∨ 1o = 𝑚 ∨ 𝑚 <N 1o)) |
74 | 73 | adantl 272 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (1o <N 𝑚 ∨ 1o = 𝑚 ∨ 𝑚 <N 1o)) |
75 | 57, 67, 71, 74 | mpjao3dan 1250 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R)) |
76 | ltasrg 7413 | . . . . . . 7 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → (𝑓 <R 𝑔 ↔ (ℎ +R 𝑓) <R (ℎ +R 𝑔))) | |
77 | 76 | adantl 272 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R)) → (𝑓 <R 𝑔 ↔ (ℎ +R 𝑓) <R (ℎ +R 𝑔))) |
78 | 1 | ffvelrnda 5473 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘𝑚) ∈ R) |
79 | 1sr 7394 | . . . . . . 7 ⊢ 1R ∈ R | |
80 | addclsr 7396 | . . . . . . 7 ⊢ (((𝐹‘𝑚) ∈ R ∧ 1R ∈ R) → ((𝐹‘𝑚) +R 1R) ∈ R) | |
81 | 78, 79, 80 | sylancl 405 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘𝑚) +R 1R) ∈ R) |
82 | m1r 7395 | . . . . . . 7 ⊢ -1R ∈ R | |
83 | 82 | a1i 9 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → -1R ∈ R) |
84 | addcomsrg 7398 | . . . . . . 7 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓)) | |
85 | 84 | adantl 272 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R)) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓)) |
86 | 77, 60, 81, 83, 85 | caovord2d 5852 | . . . . 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 7399 | . . . . . 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 1181 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (((𝐹‘𝑚) +R 1R) +R -1R) = ((𝐹‘𝑚) +R (1R +R -1R))) |
91 | addcomsrg 7398 | . . . . . . . . 9 ⊢ ((1R ∈ R ∧ -1R ∈ R) → (1R +R -1R) = (-1R +R 1R)) | |
92 | 79, 82, 91 | mp2an 418 | . . . . . . . 8 ⊢ (1R +R -1R) = (-1R +R 1R) |
93 | m1p1sr 7403 | . . . . . . . 8 ⊢ (-1R +R 1R) = 0R | |
94 | 92, 93 | eqtri 2115 | . . . . . . 7 ⊢ (1R +R -1R) = 0R |
95 | 94 | oveq2i 5701 | . . . . . 6 ⊢ ((𝐹‘𝑚) +R (1R +R -1R)) = ((𝐹‘𝑚) +R 0R) |
96 | 0idsr 7410 | . . . . . . 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 2139 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘𝑚) +R (1R +R -1R)) = (𝐹‘𝑚)) |
99 | 90, 98 | eqtrd 2127 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (((𝐹‘𝑚) +R 1R) +R -1R) = (𝐹‘𝑚)) |
100 | 87, 99 | breqtrd 3891 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘1o) +R -1R) <R (𝐹‘𝑚)) |
101 | 100 | ralrimiva 2458 | . 2 ⊢ (𝜑 → ∀𝑚 ∈ N ((𝐹‘1o) +R -1R) <R (𝐹‘𝑚)) |
102 | 1, 2, 101 | caucvgsrlembnd 7443 | 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 926 ∧ w3a 927 = wceq 1296 ∈ wcel 1445 {cab 2081 ∀wral 2370 ∃wrex 2371 〈cop 3469 class class class wbr 3867 ⟶wf 5045 ‘cfv 5049 (class class class)co 5690 1oc1o 6212 [cec 6330 Ncnpi 6928 <N clti 6931 ~Q ceq 6935 1Qc1q 6937 *Qcrq 6940 <Q cltq 6941 1Pc1p 6948 +P cpp 6949 ~R cer 6952 Rcnr 6953 0Rc0r 6954 1Rc1r 6955 -1Rcm1r 6956 +R cplr 6957 <R cltr 6959 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-eprel 4140 df-id 4144 df-po 4147 df-iso 4148 df-iord 4217 df-on 4219 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-recs 6108 df-irdg 6173 df-1o 6219 df-2o 6220 df-oadd 6223 df-omul 6224 df-er 6332 df-ec 6334 df-qs 6338 df-ni 6960 df-pli 6961 df-mi 6962 df-lti 6963 df-plpq 7000 df-mpq 7001 df-enq 7003 df-nqqs 7004 df-plqqs 7005 df-mqqs 7006 df-1nqqs 7007 df-rq 7008 df-ltnqqs 7009 df-enq0 7080 df-nq0 7081 df-0nq0 7082 df-plq0 7083 df-mq0 7084 df-inp 7122 df-i1p 7123 df-iplp 7124 df-imp 7125 df-iltp 7126 df-enr 7369 df-nr 7370 df-plr 7371 df-mr 7372 df-ltr 7373 df-0r 7374 df-1r 7375 df-m1r 7376 |
This theorem is referenced by: axcaucvglemres 7531 |
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