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Mirrors > Home > ILE Home > Th. List > caucvgsrlemoffgt1 | GIF version |
Description: Lemma for caucvgsr 7789. 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 |
---|---|
caucvgsrlemoffgt1 | ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐺‘𝑚)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgsrlembnd.bnd | . . . . . . 7 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚)) | |
2 | 1 | r19.21bi 2565 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 𝐴 <R (𝐹‘𝑚)) |
3 | ltasrg 7757 | . . . . . . . 8 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → (𝑓 <R 𝑔 ↔ (ℎ +R 𝑓) <R (ℎ +R 𝑔))) | |
4 | 3 | adantl 277 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R)) → (𝑓 <R 𝑔 ↔ (ℎ +R 𝑓) <R (ℎ +R 𝑔))) |
5 | 1 | caucvgsrlemasr 7777 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ R) |
6 | 5 | adantr 276 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 𝐴 ∈ R) |
7 | caucvgsr.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹:N⟶R) | |
8 | 7 | ffvelcdmda 5647 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘𝑚) ∈ R) |
9 | 1sr 7738 | . . . . . . . 8 ⊢ 1R ∈ R | |
10 | 9 | a1i 9 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 1R ∈ R) |
11 | addcomsrg 7742 | . . . . . . . 8 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓)) | |
12 | 11 | adantl 277 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R)) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓)) |
13 | 4, 6, 8, 10, 12 | caovord2d 6038 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐴 <R (𝐹‘𝑚) ↔ (𝐴 +R 1R) <R ((𝐹‘𝑚) +R 1R))) |
14 | 2, 13 | mpbid 147 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐴 +R 1R) <R ((𝐹‘𝑚) +R 1R)) |
15 | caucvgsr.cau | . . . . . 6 ⊢ (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 )))) | |
16 | caucvgsrlembnd.offset | . . . . . 6 ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R))) | |
17 | 7, 15, 1, 16 | caucvgsrlemoffval 7783 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐺‘𝑚) +R 𝐴) = ((𝐹‘𝑚) +R 1R)) |
18 | 14, 17 | breqtrrd 4028 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐴 +R 1R) <R ((𝐺‘𝑚) +R 𝐴)) |
19 | 7, 15, 1, 16 | caucvgsrlemofff 7784 | . . . . . 6 ⊢ (𝜑 → 𝐺:N⟶R) |
20 | 19 | ffvelcdmda 5647 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐺‘𝑚) ∈ R) |
21 | addcomsrg 7742 | . . . . 5 ⊢ (((𝐺‘𝑚) ∈ R ∧ 𝐴 ∈ R) → ((𝐺‘𝑚) +R 𝐴) = (𝐴 +R (𝐺‘𝑚))) | |
22 | 20, 6, 21 | syl2anc 411 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐺‘𝑚) +R 𝐴) = (𝐴 +R (𝐺‘𝑚))) |
23 | 18, 22 | breqtrd 4026 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐴 +R 1R) <R (𝐴 +R (𝐺‘𝑚))) |
24 | ltasrg 7757 | . . . 4 ⊢ ((1R ∈ R ∧ (𝐺‘𝑚) ∈ R ∧ 𝐴 ∈ R) → (1R <R (𝐺‘𝑚) ↔ (𝐴 +R 1R) <R (𝐴 +R (𝐺‘𝑚)))) | |
25 | 10, 20, 6, 24 | syl3anc 1238 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (1R <R (𝐺‘𝑚) ↔ (𝐴 +R 1R) <R (𝐴 +R (𝐺‘𝑚)))) |
26 | 23, 25 | mpbird 167 | . 2 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 1R <R (𝐺‘𝑚)) |
27 | 26 | ralrimiva 2550 | 1 ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐺‘𝑚)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 {cab 2163 ∀wral 2455 〈cop 3594 class class class wbr 4000 ↦ cmpt 4061 ⟶wf 5208 ‘cfv 5212 (class class class)co 5869 1oc1o 6404 [cec 6527 Ncnpi 7259 <N clti 7262 ~Q ceq 7266 *Qcrq 7271 <Q cltq 7272 1Pc1p 7279 +P cpp 7280 ~R cer 7283 Rcnr 7284 1Rc1r 7286 -1Rcm1r 7287 +R cplr 7288 ·R cmr 7289 <R cltr 7290 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-eprel 4286 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-1o 6411 df-2o 6412 df-oadd 6415 df-omul 6416 df-er 6529 df-ec 6531 df-qs 6535 df-ni 7291 df-pli 7292 df-mi 7293 df-lti 7294 df-plpq 7331 df-mpq 7332 df-enq 7334 df-nqqs 7335 df-plqqs 7336 df-mqqs 7337 df-1nqqs 7338 df-rq 7339 df-ltnqqs 7340 df-enq0 7411 df-nq0 7412 df-0nq0 7413 df-plq0 7414 df-mq0 7415 df-inp 7453 df-i1p 7454 df-iplp 7455 df-imp 7456 df-iltp 7457 df-enr 7713 df-nr 7714 df-plr 7715 df-mr 7716 df-ltr 7717 df-0r 7718 df-1r 7719 df-m1r 7720 |
This theorem is referenced by: caucvgsrlemoffres 7787 |
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