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Mirrors > Home > ILE Home > Th. List > caucvgsrlemcl | GIF version |
Description: Lemma for caucvgsr 6886. Terms of the sequence from caucvgsrlemgt1 6879 can be mapped to positive reals. (Contributed by Jim Kingdon, 2-Jul-2021.) |
Ref | Expression |
---|---|
caucvgsrlemcl.f | ⊢ (𝜑 → 𝐹:N⟶R) |
caucvgsrlemcl.gt1 | ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚)) |
Ref | Expression |
---|---|
caucvgsrlemcl | ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (℩𝑦 ∈ P (𝐹‘𝐴) = [〈(𝑦 +P 1P), 1P〉] ~R ) ∈ P) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgsrlemcl.f | . . . . 5 ⊢ (𝜑 → 𝐹:N⟶R) | |
2 | 1 | ffvelrnda 5302 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (𝐹‘𝐴) ∈ R) |
3 | 0lt1sr 6850 | . . . . 5 ⊢ 0R <R 1R | |
4 | caucvgsrlemcl.gt1 | . . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚)) | |
5 | fveq2 5178 | . . . . . . . 8 ⊢ (𝑚 = 𝐴 → (𝐹‘𝑚) = (𝐹‘𝐴)) | |
6 | 5 | breq2d 3776 | . . . . . . 7 ⊢ (𝑚 = 𝐴 → (1R <R (𝐹‘𝑚) ↔ 1R <R (𝐹‘𝐴))) |
7 | 6 | rspcv 2652 | . . . . . 6 ⊢ (𝐴 ∈ N → (∀𝑚 ∈ N 1R <R (𝐹‘𝑚) → 1R <R (𝐹‘𝐴))) |
8 | 4, 7 | mpan9 265 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → 1R <R (𝐹‘𝐴)) |
9 | ltsosr 6849 | . . . . . 6 ⊢ <R Or R | |
10 | ltrelsr 6823 | . . . . . 6 ⊢ <R ⊆ (R × R) | |
11 | 9, 10 | sotri 4720 | . . . . 5 ⊢ ((0R <R 1R ∧ 1R <R (𝐹‘𝐴)) → 0R <R (𝐹‘𝐴)) |
12 | 3, 8, 11 | sylancr 393 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → 0R <R (𝐹‘𝐴)) |
13 | srpospr 6867 | . . . 4 ⊢ (((𝐹‘𝐴) ∈ R ∧ 0R <R (𝐹‘𝐴)) → ∃!𝑦 ∈ P [〈(𝑦 +P 1P), 1P〉] ~R = (𝐹‘𝐴)) | |
14 | 2, 12, 13 | syl2anc 391 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → ∃!𝑦 ∈ P [〈(𝑦 +P 1P), 1P〉] ~R = (𝐹‘𝐴)) |
15 | eqcom 2042 | . . . 4 ⊢ ([〈(𝑦 +P 1P), 1P〉] ~R = (𝐹‘𝐴) ↔ (𝐹‘𝐴) = [〈(𝑦 +P 1P), 1P〉] ~R ) | |
16 | 15 | reubii 2495 | . . 3 ⊢ (∃!𝑦 ∈ P [〈(𝑦 +P 1P), 1P〉] ~R = (𝐹‘𝐴) ↔ ∃!𝑦 ∈ P (𝐹‘𝐴) = [〈(𝑦 +P 1P), 1P〉] ~R ) |
17 | 14, 16 | sylib 127 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → ∃!𝑦 ∈ P (𝐹‘𝐴) = [〈(𝑦 +P 1P), 1P〉] ~R ) |
18 | riotacl 5482 | . 2 ⊢ (∃!𝑦 ∈ P (𝐹‘𝐴) = [〈(𝑦 +P 1P), 1P〉] ~R → (℩𝑦 ∈ P (𝐹‘𝐴) = [〈(𝑦 +P 1P), 1P〉] ~R ) ∈ P) | |
19 | 17, 18 | syl 14 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (℩𝑦 ∈ P (𝐹‘𝐴) = [〈(𝑦 +P 1P), 1P〉] ~R ) ∈ P) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ∀wral 2306 ∃!wreu 2308 〈cop 3378 class class class wbr 3764 ⟶wf 4898 ‘cfv 4902 ℩crio 5467 (class class class)co 5512 [cec 6104 Ncnpi 6370 Pcnp 6389 1Pc1p 6390 +P cpp 6391 ~R cer 6394 Rcnr 6395 0Rc0r 6396 1Rc1r 6397 <R cltr 6401 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 |
This theorem is referenced by: caucvgsrlemfv 6875 caucvgsrlemf 6876 |
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