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Mirrors > Home > ILE Home > Th. List > caucvgsrlembound | GIF version |
Description: Lemma for caucvgsr 7814. 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 5527 | . . . . . . . . 9 ⊢ (𝑚 = 𝑤 → (𝐹‘𝑚) = (𝐹‘𝑤)) | |
3 | 2 | breq2d 4027 | . . . . . . . 8 ⊢ (𝑚 = 𝑤 → (1R <R (𝐹‘𝑚) ↔ 1R <R (𝐹‘𝑤))) |
4 | 3 | cbvralv 2715 | . . . . . . 7 ⊢ (∀𝑚 ∈ N 1R <R (𝐹‘𝑚) ↔ ∀𝑤 ∈ N 1R <R (𝐹‘𝑤)) |
5 | 1, 4 | sylib 122 | . . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ N 1R <R (𝐹‘𝑤)) |
6 | 5 | r19.21bi 2575 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → 1R <R (𝐹‘𝑤)) |
7 | df-1r 7744 | . . . . . . 7 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R | |
8 | 7 | eqcomi 2191 | . . . . . 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 7803 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → [⟨((𝐺‘𝑤) +P 1P), 1P⟩] ~R = (𝐹‘𝑤)) |
14 | 6, 9, 13 | 3brtr4d 4047 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → [⟨(1P +P 1P), 1P⟩] ~R <R [⟨((𝐺‘𝑤) +P 1P), 1P⟩] ~R ) |
15 | 1pr 7566 | . . . . 5 ⊢ 1P ∈ P | |
16 | 10, 11, 1, 12 | caucvgsrlemf 7804 | . . . . . 6 ⊢ (𝜑 → 𝐺:N⟶P) |
17 | 16 | ffvelcdmda 5664 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → (𝐺‘𝑤) ∈ P) |
18 | prsrlt 7799 | . . . . 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 414 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → (1P<P (𝐺‘𝑤) ↔ [⟨(1P +P 1P), 1P⟩] ~R <R [⟨((𝐺‘𝑤) +P 1P), 1P⟩] ~R )) |
20 | 14, 19 | mpbird 167 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ N) → 1P<P (𝐺‘𝑤)) |
21 | 20 | ralrimiva 2560 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ N 1P<P (𝐺‘𝑤)) |
22 | fveq2 5527 | . . . 4 ⊢ (𝑤 = 𝑚 → (𝐺‘𝑤) = (𝐺‘𝑚)) | |
23 | 22 | breq2d 4027 | . . 3 ⊢ (𝑤 = 𝑚 → (1P<P (𝐺‘𝑤) ↔ 1P<P (𝐺‘𝑚))) |
24 | 23 | cbvralv 2715 | . 2 ⊢ (∀𝑤 ∈ N 1P<P (𝐺‘𝑤) ↔ ∀𝑚 ∈ N 1P<P (𝐺‘𝑚)) |
25 | 21, 24 | sylib 122 | 1 ⊢ (𝜑 → ∀𝑚 ∈ N 1P<P (𝐺‘𝑚)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1363 ∈ wcel 2158 {cab 2173 ∀wral 2465 ⟨cop 3607 class class class wbr 4015 ↦ cmpt 4076 ⟶wf 5224 ‘cfv 5228 ℩crio 5843 (class class class)co 5888 1oc1o 6423 [cec 6546 Ncnpi 7284 <N clti 7287 ~Q ceq 7291 *Qcrq 7296 <Q cltq 7297 Pcnp 7303 1Pc1p 7304 +P cpp 7305 <P cltp 7307 ~R cer 7308 Rcnr 7309 1Rc1r 7311 +R cplr 7313 <R cltr 7315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-eprel 4301 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-irdg 6384 df-1o 6430 df-2o 6431 df-oadd 6434 df-omul 6435 df-er 6548 df-ec 6550 df-qs 6554 df-ni 7316 df-pli 7317 df-mi 7318 df-lti 7319 df-plpq 7356 df-mpq 7357 df-enq 7359 df-nqqs 7360 df-plqqs 7361 df-mqqs 7362 df-1nqqs 7363 df-rq 7364 df-ltnqqs 7365 df-enq0 7436 df-nq0 7437 df-0nq0 7438 df-plq0 7439 df-mq0 7440 df-inp 7478 df-i1p 7479 df-iplp 7480 df-iltp 7482 df-enr 7738 df-nr 7739 df-ltr 7742 df-0r 7743 df-1r 7744 |
This theorem is referenced by: caucvgsrlemgt1 7807 |
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