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Mirrors > Home > ILE Home > Th. List > caucvgsrlemofff | GIF version |
Description: Lemma for caucvgsr 7734. 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 | ffvelrnda 5614 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ N) → (𝐹‘𝑎) ∈ R) |
3 | 1sr 7683 | . . . 4 ⊢ 1R ∈ R | |
4 | addclsr 7685 | . . . 4 ⊢ (((𝐹‘𝑎) ∈ R ∧ 1R ∈ R) → ((𝐹‘𝑎) +R 1R) ∈ R) | |
5 | 2, 3, 4 | sylancl 410 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ N) → ((𝐹‘𝑎) +R 1R) ∈ R) |
6 | caucvgsrlembnd.bnd | . . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚)) | |
7 | 6 | caucvgsrlemasr 7722 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ R) |
8 | 7 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ N) → 𝐴 ∈ R) |
9 | m1r 7684 | . . . 4 ⊢ -1R ∈ R | |
10 | mulclsr 7686 | . . . 4 ⊢ ((𝐴 ∈ R ∧ -1R ∈ R) → (𝐴 ·R -1R) ∈ R) | |
11 | 8, 9, 10 | sylancl 410 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ N) → (𝐴 ·R -1R) ∈ R) |
12 | addclsr 7685 | . . 3 ⊢ ((((𝐹‘𝑎) +R 1R) ∈ R ∧ (𝐴 ·R -1R) ∈ R) → (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)) ∈ R) | |
13 | 5, 11, 12 | syl2anc 409 | . 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 5633 | 1 ⊢ (𝜑 → 𝐺:N⟶R) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 {cab 2150 ∀wral 2442 〈cop 3573 class class class wbr 3976 ↦ cmpt 4037 ⟶wf 5178 ‘cfv 5182 (class class class)co 5836 1oc1o 6368 [cec 6490 Ncnpi 7204 <N clti 7207 ~Q ceq 7211 *Qcrq 7216 <Q cltq 7217 1Pc1p 7224 +P cpp 7225 ~R cer 7228 Rcnr 7229 1Rc1r 7231 -1Rcm1r 7232 +R cplr 7233 ·R cmr 7234 <R cltr 7235 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-2o 6376 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 df-enq0 7356 df-nq0 7357 df-0nq0 7358 df-plq0 7359 df-mq0 7360 df-inp 7398 df-i1p 7399 df-iplp 7400 df-imp 7401 df-enr 7658 df-nr 7659 df-plr 7660 df-mr 7661 df-ltr 7662 df-1r 7664 df-m1r 7665 |
This theorem is referenced by: caucvgsrlemoffcau 7730 caucvgsrlemoffgt1 7731 caucvgsrlemoffres 7732 |
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