![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > cauappcvgprlemcl | GIF version |
Description: Lemma for cauappcvgpr 6991. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-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 |
---|---|
cauappcvgprlemcl | ⊢ (𝜑 → 𝐿 ∈ P) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cauappcvgpr.f | . . . 4 ⊢ (𝜑 → 𝐹:Q⟶Q) | |
2 | cauappcvgpr.app | . . . 4 ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) | |
3 | cauappcvgpr.bnd | . . . 4 ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) | |
4 | cauappcvgpr.lim | . . . 4 ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 | |
5 | 1, 2, 3, 4 | cauappcvgprlemm 6974 | . . 3 ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))) |
6 | ssrab2 3089 | . . . . . 6 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ⊆ Q | |
7 | nqex 6692 | . . . . . . 7 ⊢ Q ∈ V | |
8 | 7 | elpw2 3953 | . . . . . 6 ⊢ ({𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ 𝒫 Q ↔ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ⊆ Q) |
9 | 6, 8 | mpbir 144 | . . . . 5 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ 𝒫 Q |
10 | ssrab2 3089 | . . . . . 6 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ⊆ Q | |
11 | 7 | elpw2 3953 | . . . . . 6 ⊢ ({𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ 𝒫 Q ↔ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ⊆ Q) |
12 | 10, 11 | mpbir 144 | . . . . 5 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ 𝒫 Q |
13 | opelxpi 4423 | . . . . 5 ⊢ (({𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ 𝒫 Q ∧ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ 𝒫 Q) → 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 ∈ (𝒫 Q × 𝒫 Q)) | |
14 | 9, 12, 13 | mp2an 417 | . . . 4 ⊢ 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 ∈ (𝒫 Q × 𝒫 Q) |
15 | 4, 14 | eqeltri 2155 | . . 3 ⊢ 𝐿 ∈ (𝒫 Q × 𝒫 Q) |
16 | 5, 15 | jctil 305 | . 2 ⊢ (𝜑 → (𝐿 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))) |
17 | 1, 2, 3, 4 | cauappcvgprlemrnd 6979 | . . 3 ⊢ (𝜑 → (∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿))))) |
18 | 1, 2, 3, 4 | cauappcvgprlemdisj 6980 | . . 3 ⊢ (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿))) |
19 | 1, 2, 3, 4 | cauappcvgprlemloc 6981 | . . 3 ⊢ (𝜑 → ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿)))) |
20 | 17, 18, 19 | 3jca 1119 | . 2 ⊢ (𝜑 → ((∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))) ∧ ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)) ∧ ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))) |
21 | elnp1st2nd 6805 | . 2 ⊢ (𝐿 ∈ P ↔ ((𝐿 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))) ∧ ((∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))) ∧ ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)) ∧ ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿)))))) | |
22 | 16, 20, 21 | sylanbrc 408 | 1 ⊢ (𝜑 → 𝐿 ∈ P) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 662 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 ∀wral 2353 ∃wrex 2354 {crab 2357 ⊆ wss 2983 𝒫 cpw 3401 〈cop 3420 class class class wbr 3806 × cxp 4390 ⟶wf 4949 ‘cfv 4953 (class class class)co 5565 1st c1st 5818 2nd c2nd 5819 Qcnq 6609 +Q cplq 6611 <Q cltq 6614 Pcnp 6620 |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3914 ax-sep 3917 ax-nul 3925 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-setind 4309 ax-iinf 4358 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2613 df-sbc 2826 df-csb 2919 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-nul 3269 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-int 3658 df-iun 3701 df-br 3807 df-opab 3861 df-mpt 3862 df-tr 3897 df-eprel 4073 df-id 4077 df-po 4080 df-iso 4081 df-iord 4150 df-on 4152 df-suc 4155 df-iom 4361 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-rn 4403 df-res 4404 df-ima 4405 df-iota 4918 df-fun 4955 df-fn 4956 df-f 4957 df-f1 4958 df-fo 4959 df-f1o 4960 df-fv 4961 df-ov 5568 df-oprab 5569 df-mpt2 5570 df-1st 5820 df-2nd 5821 df-recs 5976 df-irdg 6041 df-1o 6087 df-oadd 6091 df-omul 6092 df-er 6195 df-ec 6197 df-qs 6201 df-ni 6633 df-pli 6634 df-mi 6635 df-lti 6636 df-plpq 6673 df-mpq 6674 df-enq 6676 df-nqqs 6677 df-plqqs 6678 df-mqqs 6679 df-1nqqs 6680 df-rq 6681 df-ltnqqs 6682 df-inp 6795 |
This theorem is referenced by: cauappcvgprlemladdfu 6983 cauappcvgprlemladdfl 6984 cauappcvgprlemladdru 6985 cauappcvgprlemladdrl 6986 cauappcvgprlemladd 6987 cauappcvgprlem1 6988 cauappcvgprlem2 6989 cauappcvgpr 6991 |
Copyright terms: Public domain | W3C validator |