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Mirrors > Home > ILE Home > Th. List > cauappcvgprlemupu | GIF version |
Description: Lemma for cauappcvgpr 7470. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 4-Aug-2020.) |
Ref | Expression |
---|---|
cauappcvgpr.f | ⊢ (𝜑 → 𝐹:Q⟶Q) |
cauappcvgpr.app | ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) |
cauappcvgpr.bnd | ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) |
cauappcvgpr.lim | ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 |
Ref | Expression |
---|---|
cauappcvgprlemupu | ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑟 ∈ (2nd ‘𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelnq 7173 | . . . . 5 ⊢ <Q ⊆ (Q × Q) | |
2 | 1 | brel 4591 | . . . 4 ⊢ (𝑠 <Q 𝑟 → (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) |
3 | 2 | simprd 113 | . . 3 ⊢ (𝑠 <Q 𝑟 → 𝑟 ∈ Q) |
4 | 3 | 3ad2ant2 1003 | . 2 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑟 ∈ Q) |
5 | breq2 3933 | . . . . . . 7 ⊢ (𝑢 = 𝑠 → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠)) | |
6 | 5 | rexbidv 2438 | . . . . . 6 ⊢ (𝑢 = 𝑠 → (∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠)) |
7 | cauappcvgpr.lim | . . . . . . . 8 ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 | |
8 | 7 | fveq2i 5424 | . . . . . . 7 ⊢ (2nd ‘𝐿) = (2nd ‘〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉) |
9 | nqex 7171 | . . . . . . . . 9 ⊢ Q ∈ V | |
10 | 9 | rabex 4072 | . . . . . . . 8 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V |
11 | 9 | rabex 4072 | . . . . . . . 8 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V |
12 | 10, 11 | op2nd 6045 | . . . . . . 7 ⊢ (2nd ‘〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} |
13 | 8, 12 | eqtri 2160 | . . . . . 6 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} |
14 | 6, 13 | elrab2 2843 | . . . . 5 ⊢ (𝑠 ∈ (2nd ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠)) |
15 | 14 | simprbi 273 | . . . 4 ⊢ (𝑠 ∈ (2nd ‘𝐿) → ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) |
16 | 15 | 3ad2ant3 1004 | . . 3 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)) → ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) |
17 | ltsonq 7206 | . . . . . . 7 ⊢ <Q Or Q | |
18 | 17, 1 | sotri 4934 | . . . . . 6 ⊢ ((((𝐹‘𝑞) +Q 𝑞) <Q 𝑠 ∧ 𝑠 <Q 𝑟) → ((𝐹‘𝑞) +Q 𝑞) <Q 𝑟) |
19 | 18 | expcom 115 | . . . . 5 ⊢ (𝑠 <Q 𝑟 → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑠 → ((𝐹‘𝑞) +Q 𝑞) <Q 𝑟)) |
20 | 19 | 3ad2ant2 1003 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)) → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑠 → ((𝐹‘𝑞) +Q 𝑞) <Q 𝑟)) |
21 | 20 | reximdv 2533 | . . 3 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)) → (∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠 → ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑟)) |
22 | 16, 21 | mpd 13 | . 2 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)) → ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑟) |
23 | breq2 3933 | . . . 4 ⊢ (𝑢 = 𝑟 → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑟)) | |
24 | 23 | rexbidv 2438 | . . 3 ⊢ (𝑢 = 𝑟 → (∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑟)) |
25 | 24, 13 | elrab2 2843 | . 2 ⊢ (𝑟 ∈ (2nd ‘𝐿) ↔ (𝑟 ∈ Q ∧ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑟)) |
26 | 4, 22, 25 | sylanbrc 413 | 1 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑟 ∈ (2nd ‘𝐿)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 {crab 2420 〈cop 3530 class class class wbr 3929 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 2nd c2nd 6037 Qcnq 7088 +Q cplq 7090 <Q cltq 7093 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-mi 7114 df-lti 7115 df-enq 7155 df-nqqs 7156 df-ltnqqs 7161 |
This theorem is referenced by: cauappcvgprlemrnd 7458 |
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