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Mirrors > Home > ILE Home > Th. List > caucvgprprlemrnd | GIF version |
Description: Lemma for caucvgprpr 7544. 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 7529 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿))) |
6 | 5 | ex 114 | . . . 4 ⊢ (𝜑 → (𝑠 ∈ (1st ‘𝐿) → ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)))) |
7 | 1, 2, 3, 4 | caucvgprprlemlol 7530 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿)) |
8 | 7 | 3expib 1185 | . . . . 5 ⊢ (𝜑 → ((𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿))) |
9 | 8 | rexlimdvw 2556 | . . . 4 ⊢ (𝜑 → (∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿))) |
10 | 6, 9 | impbid 128 | . . 3 ⊢ (𝜑 → (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)))) |
11 | 10 | ralrimivw 2509 | . 2 ⊢ (𝜑 → ∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)))) |
12 | 1, 2, 3, 4 | caucvgprprlemopu 7531 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿))) |
13 | 12 | ex 114 | . . . 4 ⊢ (𝜑 → (𝑡 ∈ (2nd ‘𝐿) → ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)))) |
14 | 1, 2, 3, 4 | caucvgprprlemupu 7532 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑡 ∈ (2nd ‘𝐿)) |
15 | 14 | 3expib 1185 | . . . . 5 ⊢ (𝜑 → ((𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑡 ∈ (2nd ‘𝐿))) |
16 | 15 | rexlimdvw 2556 | . . . 4 ⊢ (𝜑 → (∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑡 ∈ (2nd ‘𝐿))) |
17 | 13, 16 | impbid 128 | . . 3 ⊢ (𝜑 → (𝑡 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)))) |
18 | 17 | ralrimivw 2509 | . 2 ⊢ (𝜑 → ∀𝑡 ∈ Q (𝑡 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)))) |
19 | 11, 18 | jca 304 | 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 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 {cab 2126 ∀wral 2417 ∃wrex 2418 {crab 2421 〈cop 3535 class class class wbr 3937 ⟶wf 5127 ‘cfv 5131 (class class class)co 5782 1st c1st 6044 2nd c2nd 6045 1oc1o 6314 [cec 6435 Ncnpi 7104 <N clti 7107 ~Q ceq 7111 Qcnq 7112 +Q cplq 7114 *Qcrq 7116 <Q cltq 7117 Pcnp 7123 +P cpp 7125 <P cltp 7127 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-2o 6322 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-enq0 7256 df-nq0 7257 df-0nq0 7258 df-plq0 7259 df-mq0 7260 df-inp 7298 df-iplp 7300 df-iltp 7302 |
This theorem is referenced by: caucvgprprlemcl 7536 |
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