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Mirrors > Home > ILE Home > Th. List > caucvgprprlemupu | GIF version |
Description: Lemma for caucvgprpr 7644. 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 7297 | . . . . 5 ⊢ <Q ⊆ (Q × Q) | |
2 | 1 | brel 4650 | . . . 4 ⊢ (𝑠 <Q 𝑡 → (𝑠 ∈ Q ∧ 𝑡 ∈ Q)) |
3 | 2 | simprd 113 | . . 3 ⊢ (𝑠 <Q 𝑡 → 𝑡 ∈ Q) |
4 | 3 | 3ad2ant2 1008 | . 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 7618 | . . . . 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 273 | . . . 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 1009 | . . 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 7526 | . . . . . 6 ⊢ (𝑠 <Q 𝑡 → 〈{𝑝 ∣ 𝑝 <Q 𝑠}, {𝑞 ∣ 𝑠 <Q 𝑞}〉<P 〈{𝑝 ∣ 𝑝 <Q 𝑡}, {𝑞 ∣ 𝑡 <Q 𝑞}〉) | |
10 | 9 | 3ad2ant2 1008 | . . . . 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 7528 | . . . . . . 7 ⊢ <P Or P | |
12 | ltrelpr 7437 | . . . . . . 7 ⊢ <P ⊆ (P × P) | |
13 | 11, 12 | sotri 4993 | . . . . . 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 115 | . . . . 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 2565 | . . 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 7618 | . 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 414 | 1 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑡 ∈ (2nd ‘𝐿)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 967 = wceq 1342 ∈ wcel 2135 {cab 2150 ∀wral 2442 ∃wrex 2443 {crab 2446 〈cop 3573 class class class wbr 3976 ⟶wf 5178 ‘cfv 5182 (class class class)co 5836 2nd c2nd 6099 1oc1o 6368 [cec 6490 Ncnpi 7204 <N clti 7207 ~Q ceq 7211 Qcnq 7212 +Q cplq 7214 *Qcrq 7216 <Q cltq 7217 Pcnp 7223 +P cpp 7225 <P cltp 7227 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 df-inp 7398 df-iltp 7402 |
This theorem is referenced by: caucvgprprlemrnd 7633 |
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