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Mirrors > Home > ILE Home > Th. List > caucvgsrlembound | GIF version |
Description: Lemma for caucvgsr 7408. 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 5318 | . . . . . . . . 9 ⊢ (𝑚 = 𝑤 → (𝐹‘𝑚) = (𝐹‘𝑤)) | |
3 | 2 | breq2d 3863 | . . . . . . . 8 ⊢ (𝑚 = 𝑤 → (1R <R (𝐹‘𝑚) ↔ 1R <R (𝐹‘𝑤))) |
4 | 3 | cbvralv 2591 | . . . . . . 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 2462 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → 1R <R (𝐹‘𝑤)) |
7 | df-1r 7339 | . . . . . . 7 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
8 | 7 | eqcomi 2093 | . . . . . 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 7397 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → [〈((𝐺‘𝑤) +P 1P), 1P〉] ~R = (𝐹‘𝑤)) |
14 | 6, 9, 13 | 3brtr4d 3881 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → [〈(1P +P 1P), 1P〉] ~R <R [〈((𝐺‘𝑤) +P 1P), 1P〉] ~R ) |
15 | 1pr 7174 | . . . . 5 ⊢ 1P ∈ P | |
16 | 10, 11, 1, 12 | caucvgsrlemf 7398 | . . . . . 6 ⊢ (𝜑 → 𝐺:N⟶P) |
17 | 16 | ffvelrnda 5448 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → (𝐺‘𝑤) ∈ P) |
18 | prsrlt 7393 | . . . . 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 406 | . . . 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 2447 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ N 1P<P (𝐺‘𝑤)) |
22 | fveq2 5318 | . . . 4 ⊢ (𝑤 = 𝑚 → (𝐺‘𝑤) = (𝐺‘𝑚)) | |
23 | 22 | breq2d 3863 | . . 3 ⊢ (𝑤 = 𝑚 → (1P<P (𝐺‘𝑤) ↔ 1P<P (𝐺‘𝑚))) |
24 | 23 | cbvralv 2591 | . 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 1290 ∈ wcel 1439 {cab 2075 ∀wral 2360 〈cop 3453 class class class wbr 3851 ↦ cmpt 3905 ⟶wf 5024 ‘cfv 5028 ℩crio 5621 (class class class)co 5666 1oc1o 6188 [cec 6304 Ncnpi 6892 <N clti 6895 ~Q ceq 6899 *Qcrq 6904 <Q cltq 6905 Pcnp 6911 1Pc1p 6912 +P cpp 6913 <P cltp 6915 ~R cer 6916 Rcnr 6917 1Rc1r 6919 +R cplr 6921 <R cltr 6923 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-eprel 4125 df-id 4129 df-po 4132 df-iso 4133 df-iord 4202 df-on 4204 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-1o 6195 df-2o 6196 df-oadd 6199 df-omul 6200 df-er 6306 df-ec 6308 df-qs 6312 df-ni 6924 df-pli 6925 df-mi 6926 df-lti 6927 df-plpq 6964 df-mpq 6965 df-enq 6967 df-nqqs 6968 df-plqqs 6969 df-mqqs 6970 df-1nqqs 6971 df-rq 6972 df-ltnqqs 6973 df-enq0 7044 df-nq0 7045 df-0nq0 7046 df-plq0 7047 df-mq0 7048 df-inp 7086 df-i1p 7087 df-iplp 7088 df-iltp 7090 df-enr 7333 df-nr 7334 df-ltr 7337 df-0r 7338 df-1r 7339 |
This theorem is referenced by: caucvgsrlemgt1 7401 |
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