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Mirrors > Home > ILE Home > Th. List > caucvgsrlemofff | GIF version |
Description: Lemma for caucvgsr 7603. 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 5548 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ N) → (𝐹‘𝑎) ∈ R) |
3 | 1sr 7552 | . . . 4 ⊢ 1R ∈ R | |
4 | addclsr 7554 | . . . 4 ⊢ (((𝐹‘𝑎) ∈ R ∧ 1R ∈ R) → ((𝐹‘𝑎) +R 1R) ∈ R) | |
5 | 2, 3, 4 | sylancl 409 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ N) → ((𝐹‘𝑎) +R 1R) ∈ R) |
6 | caucvgsrlembnd.bnd | . . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚)) | |
7 | 6 | caucvgsrlemasr 7591 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ R) |
8 | 7 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ N) → 𝐴 ∈ R) |
9 | m1r 7553 | . . . 4 ⊢ -1R ∈ R | |
10 | mulclsr 7555 | . . . 4 ⊢ ((𝐴 ∈ R ∧ -1R ∈ R) → (𝐴 ·R -1R) ∈ R) | |
11 | 8, 9, 10 | sylancl 409 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ N) → (𝐴 ·R -1R) ∈ R) |
12 | addclsr 7554 | . . 3 ⊢ ((((𝐹‘𝑎) +R 1R) ∈ R ∧ (𝐴 ·R -1R) ∈ R) → (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)) ∈ R) | |
13 | 5, 11, 12 | syl2anc 408 | . 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 5567 | 1 ⊢ (𝜑 → 𝐺:N⟶R) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 {cab 2123 ∀wral 2414 〈cop 3525 class class class wbr 3924 ↦ cmpt 3984 ⟶wf 5114 ‘cfv 5118 (class class class)co 5767 1oc1o 6299 [cec 6420 Ncnpi 7073 <N clti 7076 ~Q ceq 7080 *Qcrq 7085 <Q cltq 7086 1Pc1p 7093 +P cpp 7094 ~R cer 7097 Rcnr 7098 1Rc1r 7100 -1Rcm1r 7101 +R cplr 7102 ·R cmr 7103 <R cltr 7104 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-2o 6307 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-pli 7106 df-mi 7107 df-lti 7108 df-plpq 7145 df-mpq 7146 df-enq 7148 df-nqqs 7149 df-plqqs 7150 df-mqqs 7151 df-1nqqs 7152 df-rq 7153 df-ltnqqs 7154 df-enq0 7225 df-nq0 7226 df-0nq0 7227 df-plq0 7228 df-mq0 7229 df-inp 7267 df-i1p 7268 df-iplp 7269 df-imp 7270 df-enr 7527 df-nr 7528 df-plr 7529 df-mr 7530 df-ltr 7531 df-1r 7533 df-m1r 7534 |
This theorem is referenced by: caucvgsrlemoffcau 7599 caucvgsrlemoffgt1 7600 caucvgsrlemoffres 7601 |
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