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Mirrors > Home > ILE Home > Th. List > caucvgsrlemofff | GIF version |
Description: Lemma for caucvgsr 7862. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-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 )))) |
caucvgsrlembnd.bnd | ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚)) |
caucvgsrlembnd.offset | ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R))) |
Ref | Expression |
---|---|
caucvgsrlemofff | ⊢ (𝜑 → 𝐺:N⟶R) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgsr.f | . . . . 5 ⊢ (𝜑 → 𝐹:N⟶R) | |
2 | 1 | ffvelcdmda 5693 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ N) → (𝐹‘𝑎) ∈ R) |
3 | 1sr 7811 | . . . 4 ⊢ 1R ∈ R | |
4 | addclsr 7813 | . . . 4 ⊢ (((𝐹‘𝑎) ∈ R ∧ 1R ∈ R) → ((𝐹‘𝑎) +R 1R) ∈ R) | |
5 | 2, 3, 4 | sylancl 413 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ N) → ((𝐹‘𝑎) +R 1R) ∈ R) |
6 | caucvgsrlembnd.bnd | . . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚)) | |
7 | 6 | caucvgsrlemasr 7850 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ R) |
8 | 7 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ N) → 𝐴 ∈ R) |
9 | m1r 7812 | . . . 4 ⊢ -1R ∈ R | |
10 | mulclsr 7814 | . . . 4 ⊢ ((𝐴 ∈ R ∧ -1R ∈ R) → (𝐴 ·R -1R) ∈ R) | |
11 | 8, 9, 10 | sylancl 413 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ N) → (𝐴 ·R -1R) ∈ R) |
12 | addclsr 7813 | . . 3 ⊢ ((((𝐹‘𝑎) +R 1R) ∈ R ∧ (𝐴 ·R -1R) ∈ R) → (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)) ∈ R) | |
13 | 5, 11, 12 | syl2anc 411 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ N) → (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)) ∈ R) |
14 | caucvgsrlembnd.offset | . 2 ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R))) | |
15 | 13, 14 | fmptd 5712 | 1 ⊢ (𝜑 → 𝐺:N⟶R) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 {cab 2179 ∀wral 2472 〈cop 3621 class class class wbr 4029 ↦ cmpt 4090 ⟶wf 5250 ‘cfv 5254 (class class class)co 5918 1oc1o 6462 [cec 6585 Ncnpi 7332 <N clti 7335 ~Q ceq 7339 *Qcrq 7344 <Q cltq 7345 1Pc1p 7352 +P cpp 7353 ~R cer 7356 Rcnr 7357 1Rc1r 7359 -1Rcm1r 7360 +R cplr 7361 ·R cmr 7362 <R cltr 7363 |
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 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 |
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 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-eprel 4320 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-1o 6469 df-2o 6470 df-oadd 6473 df-omul 6474 df-er 6587 df-ec 6589 df-qs 6593 df-ni 7364 df-pli 7365 df-mi 7366 df-lti 7367 df-plpq 7404 df-mpq 7405 df-enq 7407 df-nqqs 7408 df-plqqs 7409 df-mqqs 7410 df-1nqqs 7411 df-rq 7412 df-ltnqqs 7413 df-enq0 7484 df-nq0 7485 df-0nq0 7486 df-plq0 7487 df-mq0 7488 df-inp 7526 df-i1p 7527 df-iplp 7528 df-imp 7529 df-enr 7786 df-nr 7787 df-plr 7788 df-mr 7789 df-ltr 7790 df-1r 7792 df-m1r 7793 |
This theorem is referenced by: caucvgsrlemoffcau 7858 caucvgsrlemoffgt1 7859 caucvgsrlemoffres 7860 |
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