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Mirrors > Home > ILE Home > Th. List > caucvgprprlemm | GIF version |
Description: Lemma for caucvgprpr 6902. The putative limit is inhabited. (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 |
---|---|
caucvgprprlemm | ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑡 ∈ Q 𝑡 ∈ (2nd ‘𝐿))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgprpr.f | . . 3 ⊢ (𝜑 → 𝐹:N⟶P) | |
2 | caucvgprpr.cau | . . 3 ⊢ (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 | . . 3 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚)) | |
4 | caucvgprpr.lim | . . 3 ⊢ 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 | caucvgprprlemml 6884 | . 2 ⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿)) |
6 | 1, 2, 3, 4 | caucvgprprlemmu 6885 | . 2 ⊢ (𝜑 → ∃𝑡 ∈ Q 𝑡 ∈ (2nd ‘𝐿)) |
7 | 5, 6 | jca 300 | 1 ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑡 ∈ Q 𝑡 ∈ (2nd ‘𝐿))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 {cab 2067 ∀wral 2348 ∃wrex 2349 {crab 2352 〈cop 3401 class class class wbr 3785 ⟶wf 4918 ‘cfv 4922 (class class class)co 5532 1st c1st 5785 2nd c2nd 5786 1𝑜c1o 6017 [cec 6127 Ncnpi 6462 <N clti 6465 ~Q ceq 6469 Qcnq 6470 +Q cplq 6472 *Qcrq 6474 <Q cltq 6475 Pcnp 6481 +P cpp 6483 <P cltp 6485 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-iplp 6658 df-iltp 6660 |
This theorem is referenced by: caucvgprprlemcl 6894 |
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