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Mirrors > Home > ILE Home > Th. List > caucvgsrlemcl | GIF version |
Description: Lemma for caucvgsr 7610. Terms of the sequence from caucvgsrlemgt1 7603 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 5555 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (𝐹‘𝐴) ∈ R) |
3 | 0lt1sr 7573 | . . . . 5 ⊢ 0R <R 1R | |
4 | caucvgsrlemcl.gt1 | . . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚)) | |
5 | fveq2 5421 | . . . . . . . 8 ⊢ (𝑚 = 𝐴 → (𝐹‘𝑚) = (𝐹‘𝐴)) | |
6 | 5 | breq2d 3941 | . . . . . . 7 ⊢ (𝑚 = 𝐴 → (1R <R (𝐹‘𝑚) ↔ 1R <R (𝐹‘𝐴))) |
7 | 6 | rspcv 2785 | . . . . . 6 ⊢ (𝐴 ∈ N → (∀𝑚 ∈ N 1R <R (𝐹‘𝑚) → 1R <R (𝐹‘𝐴))) |
8 | 4, 7 | mpan9 279 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → 1R <R (𝐹‘𝐴)) |
9 | ltsosr 7572 | . . . . . 6 ⊢ <R Or R | |
10 | ltrelsr 7546 | . . . . . 6 ⊢ <R ⊆ (R × R) | |
11 | 9, 10 | sotri 4934 | . . . . 5 ⊢ ((0R <R 1R ∧ 1R <R (𝐹‘𝐴)) → 0R <R (𝐹‘𝐴)) |
12 | 3, 8, 11 | sylancr 410 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → 0R <R (𝐹‘𝐴)) |
13 | srpospr 7591 | . . . 4 ⊢ (((𝐹‘𝐴) ∈ R ∧ 0R <R (𝐹‘𝐴)) → ∃!𝑦 ∈ P [〈(𝑦 +P 1P), 1P〉] ~R = (𝐹‘𝐴)) | |
14 | 2, 12, 13 | syl2anc 408 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → ∃!𝑦 ∈ P [〈(𝑦 +P 1P), 1P〉] ~R = (𝐹‘𝐴)) |
15 | eqcom 2141 | . . . 4 ⊢ ([〈(𝑦 +P 1P), 1P〉] ~R = (𝐹‘𝐴) ↔ (𝐹‘𝐴) = [〈(𝑦 +P 1P), 1P〉] ~R ) | |
16 | 15 | reubii 2616 | . . 3 ⊢ (∃!𝑦 ∈ P [〈(𝑦 +P 1P), 1P〉] ~R = (𝐹‘𝐴) ↔ ∃!𝑦 ∈ P (𝐹‘𝐴) = [〈(𝑦 +P 1P), 1P〉] ~R ) |
17 | 14, 16 | sylib 121 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → ∃!𝑦 ∈ P (𝐹‘𝐴) = [〈(𝑦 +P 1P), 1P〉] ~R ) |
18 | riotacl 5744 | . 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 103 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ∃!wreu 2418 〈cop 3530 class class class wbr 3929 ⟶wf 5119 ‘cfv 5123 ℩crio 5729 (class class class)co 5774 [cec 6427 Ncnpi 7080 Pcnp 7099 1Pc1p 7100 +P cpp 7101 ~R cer 7104 Rcnr 7105 0Rc0r 7106 1Rc1r 7107 <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: caucvgsrlemfv 7599 caucvgsrlemf 7600 |
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