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Mirrors > Home > ILE Home > Th. List > cauappcvgprlemupu | GIF version |
Description: Lemma for cauappcvgpr 7594. 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 7297 | . . . . 5 ⊢ <Q ⊆ (Q × Q) | |
2 | 1 | brel 4650 | . . . 4 ⊢ (𝑠 <Q 𝑟 → (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) |
3 | 2 | simprd 113 | . . 3 ⊢ (𝑠 <Q 𝑟 → 𝑟 ∈ Q) |
4 | 3 | 3ad2ant2 1008 | . 2 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑟 ∈ Q) |
5 | breq2 3980 | . . . . . . 7 ⊢ (𝑢 = 𝑠 → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠)) | |
6 | 5 | rexbidv 2465 | . . . . . 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 5483 | . . . . . . 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 7295 | . . . . . . . . 9 ⊢ Q ∈ V | |
10 | 9 | rabex 4120 | . . . . . . . 8 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V |
11 | 9 | rabex 4120 | . . . . . . . 8 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V |
12 | 10, 11 | op2nd 6107 | . . . . . . 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 2185 | . . . . . 6 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} |
14 | 6, 13 | elrab2 2880 | . . . . 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 1009 | . . 3 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)) → ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) |
17 | ltsonq 7330 | . . . . . . 7 ⊢ <Q Or Q | |
18 | 17, 1 | sotri 4993 | . . . . . 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 1008 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)) → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑠 → ((𝐹‘𝑞) +Q 𝑞) <Q 𝑟)) |
21 | 20 | reximdv 2565 | . . 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 3980 | . . . 4 ⊢ (𝑢 = 𝑟 → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑟)) | |
24 | 23 | rexbidv 2465 | . . 3 ⊢ (𝑢 = 𝑟 → (∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑟)) |
25 | 24, 13 | elrab2 2880 | . 2 ⊢ (𝑟 ∈ (2nd ‘𝐿) ↔ (𝑟 ∈ Q ∧ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑟)) |
26 | 4, 22, 25 | sylanbrc 414 | 1 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑟 ∈ (2nd ‘𝐿)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 967 = wceq 1342 ∈ wcel 2135 ∀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 Qcnq 7212 +Q cplq 7214 <Q cltq 7217 |
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-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-mi 7238 df-lti 7239 df-enq 7279 df-nqqs 7280 df-ltnqqs 7285 |
This theorem is referenced by: cauappcvgprlemrnd 7582 |
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