Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > caucvgsrlemf | GIF version |
Description: Lemma for caucvgsr 7723. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-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 )))) |
caucvgsrlemgt1.gt1 | ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚)) |
caucvgsrlemf.xfr | ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [〈(𝑦 +P 1P), 1P〉] ~R )) |
Ref | Expression |
---|---|
caucvgsrlemf | ⊢ (𝜑 → 𝐺:N⟶P) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgsr.f | . . 3 ⊢ (𝜑 → 𝐹:N⟶R) | |
2 | caucvgsrlemgt1.gt1 | . . 3 ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚)) | |
3 | 1, 2 | caucvgsrlemcl 7710 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ N) → (℩𝑦 ∈ P (𝐹‘𝑥) = [〈(𝑦 +P 1P), 1P〉] ~R ) ∈ P) |
4 | caucvgsrlemf.xfr | . 2 ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [〈(𝑦 +P 1P), 1P〉] ~R )) | |
5 | 3, 4 | fmptd 5622 | 1 ⊢ (𝜑 → 𝐺:N⟶P) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1335 {cab 2143 ∀wral 2435 〈cop 3563 class class class wbr 3966 ↦ cmpt 4026 ⟶wf 5167 ‘cfv 5171 ℩crio 5780 (class class class)co 5825 1oc1o 6357 [cec 6479 Ncnpi 7193 <N clti 7196 ~Q ceq 7200 *Qcrq 7205 <Q cltq 7206 Pcnp 7212 1Pc1p 7213 +P cpp 7214 ~R cer 7217 Rcnr 7218 1Rc1r 7220 +R cplr 7222 <R cltr 7224 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-eprel 4250 df-id 4254 df-po 4257 df-iso 4258 df-iord 4327 df-on 4329 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-irdg 6318 df-1o 6364 df-2o 6365 df-oadd 6368 df-omul 6369 df-er 6481 df-ec 6483 df-qs 6487 df-ni 7225 df-pli 7226 df-mi 7227 df-lti 7228 df-plpq 7265 df-mpq 7266 df-enq 7268 df-nqqs 7269 df-plqqs 7270 df-mqqs 7271 df-1nqqs 7272 df-rq 7273 df-ltnqqs 7274 df-enq0 7345 df-nq0 7346 df-0nq0 7347 df-plq0 7348 df-mq0 7349 df-inp 7387 df-i1p 7388 df-iplp 7389 df-iltp 7391 df-enr 7647 df-nr 7648 df-ltr 7651 df-0r 7652 df-1r 7653 |
This theorem is referenced by: caucvgsrlemcau 7714 caucvgsrlembound 7715 caucvgsrlemgt1 7716 |
Copyright terms: Public domain | W3C validator |