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Mirrors > Home > ILE Home > Th. List > caucvgsrlemf | GIF version |
Description: Lemma for caucvgsr 7826. 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 7813 | . 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 5687 | 1 ⊢ (𝜑 → 𝐺:N⟶P) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 {cab 2175 ∀wral 2468 〈cop 3610 class class class wbr 4018 ↦ cmpt 4079 ⟶wf 5228 ‘cfv 5232 ℩crio 5847 (class class class)co 5892 1oc1o 6429 [cec 6552 Ncnpi 7296 <N clti 7299 ~Q ceq 7303 *Qcrq 7308 <Q cltq 7309 Pcnp 7315 1Pc1p 7316 +P cpp 7317 ~R cer 7320 Rcnr 7321 1Rc1r 7323 +R cplr 7325 <R cltr 7327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-eprel 4304 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-1st 6160 df-2nd 6161 df-recs 6325 df-irdg 6390 df-1o 6436 df-2o 6437 df-oadd 6440 df-omul 6441 df-er 6554 df-ec 6556 df-qs 6560 df-ni 7328 df-pli 7329 df-mi 7330 df-lti 7331 df-plpq 7368 df-mpq 7369 df-enq 7371 df-nqqs 7372 df-plqqs 7373 df-mqqs 7374 df-1nqqs 7375 df-rq 7376 df-ltnqqs 7377 df-enq0 7448 df-nq0 7449 df-0nq0 7450 df-plq0 7451 df-mq0 7452 df-inp 7490 df-i1p 7491 df-iplp 7492 df-iltp 7494 df-enr 7750 df-nr 7751 df-ltr 7754 df-0r 7755 df-1r 7756 |
This theorem is referenced by: caucvgsrlemcau 7817 caucvgsrlembound 7818 caucvgsrlemgt1 7819 |
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