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Mirrors > Home > ILE Home > Th. List > caucvgprprlemupu | GIF version |
Description: Lemma for caucvgprpr 6810. The upper cut of the putative limit is upper. (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 |
---|---|
caucvgprprlemupu | ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑡 ∈ (2nd ‘𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelnq 6463 | . . . . 5 ⊢ <Q ⊆ (Q × Q) | |
2 | 1 | brel 4392 | . . . 4 ⊢ (𝑠 <Q 𝑡 → (𝑠 ∈ Q ∧ 𝑡 ∈ Q)) |
3 | 2 | simprd 107 | . . 3 ⊢ (𝑠 <Q 𝑡 → 𝑡 ∈ Q) |
4 | 3 | 3ad2ant2 926 | . 2 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑡 ∈ Q) |
5 | 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}〉}〉 | |
6 | 5 | caucvgprprlemelu 6784 | . . . . 5 ⊢ (𝑠 ∈ (2nd ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑏 ∈ N ((𝐹‘𝑏) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑏, 1𝑜〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑏, 1𝑜〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}〉)) |
7 | 6 | simprbi 260 | . . . 4 ⊢ (𝑠 ∈ (2nd ‘𝐿) → ∃𝑏 ∈ N ((𝐹‘𝑏) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑏, 1𝑜〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑏, 1𝑜〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}〉) |
8 | 7 | 3ad2ant3 927 | . . 3 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)) → ∃𝑏 ∈ N ((𝐹‘𝑏) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑏, 1𝑜〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑏, 1𝑜〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}〉) |
9 | ltnqpri 6692 | . . . . . 6 ⊢ (𝑠 <Q 𝑡 → 〈{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}〉<P 〈{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}〉) | |
10 | 9 | 3ad2ant2 926 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 〈{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}〉<P 〈{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}〉) |
11 | ltsopr 6694 | . . . . . . 7 ⊢ <P Or P | |
12 | ltrelpr 6603 | . . . . . . 7 ⊢ <P ⊆ (P × P) | |
13 | 11, 12 | sotri 4720 | . . . . . 6 ⊢ ((((𝐹‘𝑏) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑏, 1𝑜〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑏, 1𝑜〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}〉 ∧ 〈{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}〉<P 〈{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}〉) → ((𝐹‘𝑏) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑏, 1𝑜〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑏, 1𝑜〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}〉) |
14 | 13 | expcom 109 | . . . . 5 ⊢ (〈{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}〉<P 〈{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}〉 → (((𝐹‘𝑏) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑏, 1𝑜〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑏, 1𝑜〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}〉 → ((𝐹‘𝑏) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑏, 1𝑜〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑏, 1𝑜〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}〉)) |
15 | 10, 14 | syl 14 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)) → (((𝐹‘𝑏) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑏, 1𝑜〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑏, 1𝑜〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}〉 → ((𝐹‘𝑏) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑏, 1𝑜〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑏, 1𝑜〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}〉)) |
16 | 15 | reximdv 2420 | . . 3 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)) → (∃𝑏 ∈ N ((𝐹‘𝑏) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑏, 1𝑜〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑏, 1𝑜〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}〉 → ∃𝑏 ∈ N ((𝐹‘𝑏) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑏, 1𝑜〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑏, 1𝑜〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}〉)) |
17 | 8, 16 | mpd 13 | . 2 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)) → ∃𝑏 ∈ N ((𝐹‘𝑏) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑏, 1𝑜〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑏, 1𝑜〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}〉) |
18 | 5 | caucvgprprlemelu 6784 | . 2 ⊢ (𝑡 ∈ (2nd ‘𝐿) ↔ (𝑡 ∈ Q ∧ ∃𝑏 ∈ N ((𝐹‘𝑏) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑏, 1𝑜〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑏, 1𝑜〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}〉)) |
19 | 4, 17, 18 | sylanbrc 394 | 1 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑡 ∈ (2nd ‘𝐿)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∧ w3a 885 = 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 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-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-inp 6564 df-iltp 6568 |
This theorem is referenced by: caucvgprprlemrnd 6799 |
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