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Mirrors > Home > ILE Home > Th. List > caucvgsrlemoffgt1 | GIF version |
Description: Lemma for caucvgsr 7040. 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 2450 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 𝐴 <R (𝐹‘𝑚)) |
3 | ltasrg 7009 | . . . . . . . 8 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → (𝑓 <R 𝑔 ↔ (ℎ +R 𝑓) <R (ℎ +R 𝑔))) | |
4 | 3 | adantl 271 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R)) → (𝑓 <R 𝑔 ↔ (ℎ +R 𝑓) <R (ℎ +R 𝑔))) |
5 | 1 | caucvgsrlemasr 7028 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ R) |
6 | 5 | adantr 270 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 𝐴 ∈ R) |
7 | caucvgsr.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹:N⟶R) | |
8 | 7 | ffvelrnda 5334 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘𝑚) ∈ R) |
9 | 1sr 6990 | . . . . . . . 8 ⊢ 1R ∈ R | |
10 | 9 | a1i 9 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 1R ∈ R) |
11 | addcomsrg 6994 | . . . . . . . 8 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓)) | |
12 | 11 | adantl 271 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R)) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓)) |
13 | 4, 6, 8, 10, 12 | caovord2d 5701 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐴 <R (𝐹‘𝑚) ↔ (𝐴 +R 1R) <R ((𝐹‘𝑚) +R 1R))) |
14 | 2, 13 | mpbid 145 | . . . . 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 7034 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐺‘𝑚) +R 𝐴) = ((𝐹‘𝑚) +R 1R)) |
18 | 14, 17 | breqtrrd 3819 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐴 +R 1R) <R ((𝐺‘𝑚) +R 𝐴)) |
19 | 7, 15, 1, 16 | caucvgsrlemofff 7035 | . . . . . 6 ⊢ (𝜑 → 𝐺:N⟶R) |
20 | 19 | ffvelrnda 5334 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐺‘𝑚) ∈ R) |
21 | addcomsrg 6994 | . . . . 5 ⊢ (((𝐺‘𝑚) ∈ R ∧ 𝐴 ∈ R) → ((𝐺‘𝑚) +R 𝐴) = (𝐴 +R (𝐺‘𝑚))) | |
22 | 20, 6, 21 | syl2anc 403 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐺‘𝑚) +R 𝐴) = (𝐴 +R (𝐺‘𝑚))) |
23 | 18, 22 | breqtrd 3817 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐴 +R 1R) <R (𝐴 +R (𝐺‘𝑚))) |
24 | ltasrg 7009 | . . . 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 1170 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (1R <R (𝐺‘𝑚) ↔ (𝐴 +R 1R) <R (𝐴 +R (𝐺‘𝑚)))) |
26 | 23, 25 | mpbird 165 | . 2 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 1R <R (𝐺‘𝑚)) |
27 | 26 | ralrimiva 2435 | 1 ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐺‘𝑚)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 {cab 2068 ∀wral 2349 〈cop 3409 class class class wbr 3793 ↦ cmpt 3847 ⟶wf 4928 ‘cfv 4932 (class class class)co 5543 1𝑜c1o 6058 [cec 6170 Ncnpi 6524 <N clti 6527 ~Q ceq 6531 *Qcrq 6536 <Q cltq 6537 1Pc1p 6544 +P cpp 6545 ~R cer 6548 Rcnr 6549 1Rc1r 6551 -1Rcm1r 6552 +R cplr 6553 ·R cmr 6554 <R cltr 6555 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-coll 3901 ax-sep 3904 ax-nul 3912 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-iinf 4337 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-csb 2910 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-nul 3259 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-int 3645 df-iun 3688 df-br 3794 df-opab 3848 df-mpt 3849 df-tr 3884 df-eprel 4052 df-id 4056 df-po 4059 df-iso 4060 df-iord 4129 df-on 4131 df-suc 4134 df-iom 4340 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-rn 4382 df-res 4383 df-ima 4384 df-iota 4897 df-fun 4934 df-fn 4935 df-f 4936 df-f1 4937 df-fo 4938 df-f1o 4939 df-fv 4940 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-1st 5798 df-2nd 5799 df-recs 5954 df-irdg 6019 df-1o 6065 df-2o 6066 df-oadd 6069 df-omul 6070 df-er 6172 df-ec 6174 df-qs 6178 df-ni 6556 df-pli 6557 df-mi 6558 df-lti 6559 df-plpq 6596 df-mpq 6597 df-enq 6599 df-nqqs 6600 df-plqqs 6601 df-mqqs 6602 df-1nqqs 6603 df-rq 6604 df-ltnqqs 6605 df-enq0 6676 df-nq0 6677 df-0nq0 6678 df-plq0 6679 df-mq0 6680 df-inp 6718 df-i1p 6719 df-iplp 6720 df-imp 6721 df-iltp 6722 df-enr 6965 df-nr 6966 df-plr 6967 df-mr 6968 df-ltr 6969 df-0r 6970 df-1r 6971 df-m1r 6972 |
This theorem is referenced by: caucvgsrlemoffres 7038 |
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