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Mirrors > Home > ILE Home > Th. List > caucvgprprlemrnd | GIF version |
Description: Lemma for caucvgprpr 6810. The putative limit is rounded. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Ref | Expression |
---|---|
caucvgprpr.f | ⊢ (𝜑 → 𝐹:N⟶P) |
caucvgprpr.cau | ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1𝑜〉] ~Q ) <Q 𝑢}〉) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1𝑜〉] ~Q ) <Q 𝑢}〉)))) |
caucvgprpr.bnd | ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚)) |
caucvgprpr.lim | ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑟 ∈ N 〈{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[〈𝑟, 1𝑜〉] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[〈𝑟, 1𝑜〉] ~Q )) <Q 𝑞}〉<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑟, 1𝑜〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑟, 1𝑜〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}〉}〉 |
Ref | Expression |
---|---|
caucvgprprlemrnd | ⊢ (𝜑 → (∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿))) ∧ ∀𝑡 ∈ Q (𝑡 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgprpr.f | . . . . . 6 ⊢ (𝜑 → 𝐹:N⟶P) | |
2 | caucvgprpr.cau | . . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1𝑜〉] ~Q ) <Q 𝑢}〉) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1𝑜〉] ~Q ) <Q 𝑢}〉)))) | |
3 | caucvgprpr.bnd | . . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚)) | |
4 | caucvgprpr.lim | . . . . . 6 ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑟 ∈ N 〈{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[〈𝑟, 1𝑜〉] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[〈𝑟, 1𝑜〉] ~Q )) <Q 𝑞}〉<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑟, 1𝑜〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑟, 1𝑜〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}〉}〉 | |
5 | 1, 2, 3, 4 | caucvgprprlemopl 6795 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿))) |
6 | 5 | ex 108 | . . . 4 ⊢ (𝜑 → (𝑠 ∈ (1st ‘𝐿) → ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)))) |
7 | 1, 2, 3, 4 | caucvgprprlemlol 6796 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿)) |
8 | 7 | 3expib 1107 | . . . . 5 ⊢ (𝜑 → ((𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿))) |
9 | 8 | rexlimdvw 2436 | . . . 4 ⊢ (𝜑 → (∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿))) |
10 | 6, 9 | impbid 120 | . . 3 ⊢ (𝜑 → (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)))) |
11 | 10 | ralrimivw 2393 | . 2 ⊢ (𝜑 → ∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)))) |
12 | 1, 2, 3, 4 | caucvgprprlemopu 6797 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿))) |
13 | 12 | ex 108 | . . . 4 ⊢ (𝜑 → (𝑡 ∈ (2nd ‘𝐿) → ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)))) |
14 | 1, 2, 3, 4 | caucvgprprlemupu 6798 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑡 ∈ (2nd ‘𝐿)) |
15 | 14 | 3expib 1107 | . . . . 5 ⊢ (𝜑 → ((𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑡 ∈ (2nd ‘𝐿))) |
16 | 15 | rexlimdvw 2436 | . . . 4 ⊢ (𝜑 → (∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑡 ∈ (2nd ‘𝐿))) |
17 | 13, 16 | impbid 120 | . . 3 ⊢ (𝜑 → (𝑡 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)))) |
18 | 17 | ralrimivw 2393 | . 2 ⊢ (𝜑 → ∀𝑡 ∈ Q (𝑡 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)))) |
19 | 11, 18 | jca 290 | 1 ⊢ (𝜑 → (∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿))) ∧ ∀𝑡 ∈ Q (𝑡 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿))))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 {cab 2026 ∀wral 2306 ∃wrex 2307 {crab 2310 〈cop 3378 class class class wbr 3764 ⟶wf 4898 ‘cfv 4902 (class class class)co 5512 1st c1st 5765 2nd c2nd 5766 1𝑜c1o 5994 [cec 6104 Ncnpi 6370 <N clti 6373 ~Q ceq 6377 Qcnq 6378 +Q cplq 6380 *Qcrq 6382 <Q cltq 6383 Pcnp 6389 +P cpp 6391 <P cltp 6393 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-iplp 6566 df-iltp 6568 |
This theorem is referenced by: caucvgprprlemcl 6802 |
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