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Mirrors > Home > ILE Home > Th. List > caucvgsrlemoffval | GIF version |
Description: Lemma for caucvgsr 7578. 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 |
---|---|
caucvgsrlemoffval | ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ((𝐺‘𝐽) +R 𝐴) = ((𝐹‘𝐽) +R 1R)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgsrlembnd.offset | . . . . 5 ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R))) | |
2 | 1 | a1i 9 | . . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)))) |
3 | fveq2 5389 | . . . . . . 7 ⊢ (𝑎 = 𝐽 → (𝐹‘𝑎) = (𝐹‘𝐽)) | |
4 | 3 | oveq1d 5757 | . . . . . 6 ⊢ (𝑎 = 𝐽 → ((𝐹‘𝑎) +R 1R) = ((𝐹‘𝐽) +R 1R)) |
5 | 4 | oveq1d 5757 | . . . . 5 ⊢ (𝑎 = 𝐽 → (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)) = (((𝐹‘𝐽) +R 1R) +R (𝐴 ·R -1R))) |
6 | 5 | adantl 275 | . . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ N) ∧ 𝑎 = 𝐽) → (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)) = (((𝐹‘𝐽) +R 1R) +R (𝐴 ·R -1R))) |
7 | simpr 109 | . . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → 𝐽 ∈ N) | |
8 | caucvgsr.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:N⟶R) | |
9 | 8 | ffvelrnda 5523 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (𝐹‘𝐽) ∈ R) |
10 | 1sr 7527 | . . . . . 6 ⊢ 1R ∈ R | |
11 | addclsr 7529 | . . . . . 6 ⊢ (((𝐹‘𝐽) ∈ R ∧ 1R ∈ R) → ((𝐹‘𝐽) +R 1R) ∈ R) | |
12 | 9, 10, 11 | sylancl 409 | . . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ((𝐹‘𝐽) +R 1R) ∈ R) |
13 | caucvgsrlembnd.bnd | . . . . . . . 8 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚)) | |
14 | 13 | caucvgsrlemasr 7566 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ R) |
15 | 14 | adantr 274 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → 𝐴 ∈ R) |
16 | m1r 7528 | . . . . . 6 ⊢ -1R ∈ R | |
17 | mulclsr 7530 | . . . . . 6 ⊢ ((𝐴 ∈ R ∧ -1R ∈ R) → (𝐴 ·R -1R) ∈ R) | |
18 | 15, 16, 17 | sylancl 409 | . . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (𝐴 ·R -1R) ∈ R) |
19 | addclsr 7529 | . . . . 5 ⊢ ((((𝐹‘𝐽) +R 1R) ∈ R ∧ (𝐴 ·R -1R) ∈ R) → (((𝐹‘𝐽) +R 1R) +R (𝐴 ·R -1R)) ∈ R) | |
20 | 12, 18, 19 | syl2anc 408 | . . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (((𝐹‘𝐽) +R 1R) +R (𝐴 ·R -1R)) ∈ R) |
21 | 2, 6, 7, 20 | fvmptd 5470 | . . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (𝐺‘𝐽) = (((𝐹‘𝐽) +R 1R) +R (𝐴 ·R -1R))) |
22 | 21 | oveq1d 5757 | . 2 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ((𝐺‘𝐽) +R 𝐴) = ((((𝐹‘𝐽) +R 1R) +R (𝐴 ·R -1R)) +R 𝐴)) |
23 | addasssrg 7532 | . . 3 ⊢ ((((𝐹‘𝐽) +R 1R) ∈ R ∧ (𝐴 ·R -1R) ∈ R ∧ 𝐴 ∈ R) → ((((𝐹‘𝐽) +R 1R) +R (𝐴 ·R -1R)) +R 𝐴) = (((𝐹‘𝐽) +R 1R) +R ((𝐴 ·R -1R) +R 𝐴))) | |
24 | 12, 18, 15, 23 | syl3anc 1201 | . 2 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ((((𝐹‘𝐽) +R 1R) +R (𝐴 ·R -1R)) +R 𝐴) = (((𝐹‘𝐽) +R 1R) +R ((𝐴 ·R -1R) +R 𝐴))) |
25 | addcomsrg 7531 | . . . . . 6 ⊢ (((𝐴 ·R -1R) ∈ R ∧ 𝐴 ∈ R) → ((𝐴 ·R -1R) +R 𝐴) = (𝐴 +R (𝐴 ·R -1R))) | |
26 | 18, 15, 25 | syl2anc 408 | . . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ((𝐴 ·R -1R) +R 𝐴) = (𝐴 +R (𝐴 ·R -1R))) |
27 | pn0sr 7547 | . . . . . 6 ⊢ (𝐴 ∈ R → (𝐴 +R (𝐴 ·R -1R)) = 0R) | |
28 | 15, 27 | syl 14 | . . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (𝐴 +R (𝐴 ·R -1R)) = 0R) |
29 | 26, 28 | eqtrd 2150 | . . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ((𝐴 ·R -1R) +R 𝐴) = 0R) |
30 | 29 | oveq2d 5758 | . . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (((𝐹‘𝐽) +R 1R) +R ((𝐴 ·R -1R) +R 𝐴)) = (((𝐹‘𝐽) +R 1R) +R 0R)) |
31 | 0idsr 7543 | . . . 4 ⊢ (((𝐹‘𝐽) +R 1R) ∈ R → (((𝐹‘𝐽) +R 1R) +R 0R) = ((𝐹‘𝐽) +R 1R)) | |
32 | 12, 31 | syl 14 | . . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (((𝐹‘𝐽) +R 1R) +R 0R) = ((𝐹‘𝐽) +R 1R)) |
33 | 30, 32 | eqtrd 2150 | . 2 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (((𝐹‘𝐽) +R 1R) +R ((𝐴 ·R -1R) +R 𝐴)) = ((𝐹‘𝐽) +R 1R)) |
34 | 22, 24, 33 | 3eqtrd 2154 | 1 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ((𝐺‘𝐽) +R 𝐴) = ((𝐹‘𝐽) +R 1R)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 {cab 2103 ∀wral 2393 〈cop 3500 class class class wbr 3899 ↦ cmpt 3959 ⟶wf 5089 ‘cfv 5093 (class class class)co 5742 1oc1o 6274 [cec 6395 Ncnpi 7048 <N clti 7051 ~Q ceq 7055 *Qcrq 7060 <Q cltq 7061 1Pc1p 7068 +P cpp 7069 ~R cer 7072 Rcnr 7073 0Rc0r 7074 1Rc1r 7075 -1Rcm1r 7076 +R cplr 7077 ·R cmr 7078 <R cltr 7079 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-eprel 4181 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-2o 6282 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-pli 7081 df-mi 7082 df-lti 7083 df-plpq 7120 df-mpq 7121 df-enq 7123 df-nqqs 7124 df-plqqs 7125 df-mqqs 7126 df-1nqqs 7127 df-rq 7128 df-ltnqqs 7129 df-enq0 7200 df-nq0 7201 df-0nq0 7202 df-plq0 7203 df-mq0 7204 df-inp 7242 df-i1p 7243 df-iplp 7244 df-imp 7245 df-enr 7502 df-nr 7503 df-plr 7504 df-mr 7505 df-ltr 7506 df-0r 7507 df-1r 7508 df-m1r 7509 |
This theorem is referenced by: caucvgsrlemoffcau 7574 caucvgsrlemoffgt1 7575 caucvgsrlemoffres 7576 |
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