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Mirrors > Home > ILE Home > Th. List > caucvgsrlembound | GIF version |
Description: Lemma for caucvgsr 7610. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Ref | Expression |
---|---|
caucvgsr.f | ⊢ (𝜑 → 𝐹:N⟶R) |
caucvgsr.cau | ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) |
caucvgsrlemgt1.gt1 | ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚)) |
caucvgsrlemf.xfr | ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [〈(𝑦 +P 1P), 1P〉] ~R )) |
Ref | Expression |
---|---|
caucvgsrlembound | ⊢ (𝜑 → ∀𝑚 ∈ N 1P<P (𝐺‘𝑚)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgsrlemgt1.gt1 | . . . . . . 7 ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚)) | |
2 | fveq2 5421 | . . . . . . . . 9 ⊢ (𝑚 = 𝑤 → (𝐹‘𝑚) = (𝐹‘𝑤)) | |
3 | 2 | breq2d 3941 | . . . . . . . 8 ⊢ (𝑚 = 𝑤 → (1R <R (𝐹‘𝑚) ↔ 1R <R (𝐹‘𝑤))) |
4 | 3 | cbvralv 2654 | . . . . . . 7 ⊢ (∀𝑚 ∈ N 1R <R (𝐹‘𝑚) ↔ ∀𝑤 ∈ N 1R <R (𝐹‘𝑤)) |
5 | 1, 4 | sylib 121 | . . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ N 1R <R (𝐹‘𝑤)) |
6 | 5 | r19.21bi 2520 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → 1R <R (𝐹‘𝑤)) |
7 | df-1r 7540 | . . . . . . 7 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
8 | 7 | eqcomi 2143 | . . . . . 6 ⊢ [〈(1P +P 1P), 1P〉] ~R = 1R |
9 | 8 | a1i 9 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → [〈(1P +P 1P), 1P〉] ~R = 1R) |
10 | caucvgsr.f | . . . . . 6 ⊢ (𝜑 → 𝐹:N⟶R) | |
11 | caucvgsr.cau | . . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) | |
12 | caucvgsrlemf.xfr | . . . . . 6 ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [〈(𝑦 +P 1P), 1P〉] ~R )) | |
13 | 10, 11, 1, 12 | caucvgsrlemfv 7599 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → [〈((𝐺‘𝑤) +P 1P), 1P〉] ~R = (𝐹‘𝑤)) |
14 | 6, 9, 13 | 3brtr4d 3960 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → [〈(1P +P 1P), 1P〉] ~R <R [〈((𝐺‘𝑤) +P 1P), 1P〉] ~R ) |
15 | 1pr 7362 | . . . . 5 ⊢ 1P ∈ P | |
16 | 10, 11, 1, 12 | caucvgsrlemf 7600 | . . . . . 6 ⊢ (𝜑 → 𝐺:N⟶P) |
17 | 16 | ffvelrnda 5555 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → (𝐺‘𝑤) ∈ P) |
18 | prsrlt 7595 | . . . . 5 ⊢ ((1P ∈ P ∧ (𝐺‘𝑤) ∈ P) → (1P<P (𝐺‘𝑤) ↔ [〈(1P +P 1P), 1P〉] ~R <R [〈((𝐺‘𝑤) +P 1P), 1P〉] ~R )) | |
19 | 15, 17, 18 | sylancr 410 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → (1P<P (𝐺‘𝑤) ↔ [〈(1P +P 1P), 1P〉] ~R <R [〈((𝐺‘𝑤) +P 1P), 1P〉] ~R )) |
20 | 14, 19 | mpbird 166 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → 1P<P (𝐺‘𝑤)) |
21 | 20 | ralrimiva 2505 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ N 1P<P (𝐺‘𝑤)) |
22 | fveq2 5421 | . . . 4 ⊢ (𝑤 = 𝑚 → (𝐺‘𝑤) = (𝐺‘𝑚)) | |
23 | 22 | breq2d 3941 | . . 3 ⊢ (𝑤 = 𝑚 → (1P<P (𝐺‘𝑤) ↔ 1P<P (𝐺‘𝑚))) |
24 | 23 | cbvralv 2654 | . 2 ⊢ (∀𝑤 ∈ N 1P<P (𝐺‘𝑤) ↔ ∀𝑚 ∈ N 1P<P (𝐺‘𝑚)) |
25 | 21, 24 | sylib 121 | 1 ⊢ (𝜑 → ∀𝑚 ∈ N 1P<P (𝐺‘𝑚)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 {cab 2125 ∀wral 2416 〈cop 3530 class class class wbr 3929 ↦ cmpt 3989 ⟶wf 5119 ‘cfv 5123 ℩crio 5729 (class class class)co 5774 1oc1o 6306 [cec 6427 Ncnpi 7080 <N clti 7083 ~Q ceq 7087 *Qcrq 7092 <Q cltq 7093 Pcnp 7099 1Pc1p 7100 +P cpp 7101 <P cltp 7103 ~R cer 7104 Rcnr 7105 1Rc1r 7107 +R cplr 7109 <R cltr 7111 |
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-rmo 2424 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-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-2o 6314 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-enq0 7232 df-nq0 7233 df-0nq0 7234 df-plq0 7235 df-mq0 7236 df-inp 7274 df-i1p 7275 df-iplp 7276 df-iltp 7278 df-enr 7534 df-nr 7535 df-ltr 7538 df-0r 7539 df-1r 7540 |
This theorem is referenced by: caucvgsrlemgt1 7603 |
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