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Mirrors > Home > ILE Home > Th. List > caucvgsrlemcl | GIF version |
Description: Lemma for caucvgsr 7634. Terms of the sequence from caucvgsrlemgt1 7627 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 5563 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (𝐹‘𝐴) ∈ R) |
3 | 0lt1sr 7597 | . . . . 5 ⊢ 0R <R 1R | |
4 | caucvgsrlemcl.gt1 | . . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚)) | |
5 | fveq2 5429 | . . . . . . . 8 ⊢ (𝑚 = 𝐴 → (𝐹‘𝑚) = (𝐹‘𝐴)) | |
6 | 5 | breq2d 3949 | . . . . . . 7 ⊢ (𝑚 = 𝐴 → (1R <R (𝐹‘𝑚) ↔ 1R <R (𝐹‘𝐴))) |
7 | 6 | rspcv 2789 | . . . . . 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 7596 | . . . . . 6 ⊢ <R Or R | |
10 | ltrelsr 7570 | . . . . . 6 ⊢ <R ⊆ (R × R) | |
11 | 9, 10 | sotri 4942 | . . . . 5 ⊢ ((0R <R 1R ∧ 1R <R (𝐹‘𝐴)) → 0R <R (𝐹‘𝐴)) |
12 | 3, 8, 11 | sylancr 411 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → 0R <R (𝐹‘𝐴)) |
13 | srpospr 7615 | . . . 4 ⊢ (((𝐹‘𝐴) ∈ R ∧ 0R <R (𝐹‘𝐴)) → ∃!𝑦 ∈ P [〈(𝑦 +P 1P), 1P〉] ~R = (𝐹‘𝐴)) | |
14 | 2, 12, 13 | syl2anc 409 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → ∃!𝑦 ∈ P [〈(𝑦 +P 1P), 1P〉] ~R = (𝐹‘𝐴)) |
15 | eqcom 2142 | . . . 4 ⊢ ([〈(𝑦 +P 1P), 1P〉] ~R = (𝐹‘𝐴) ↔ (𝐹‘𝐴) = [〈(𝑦 +P 1P), 1P〉] ~R ) | |
16 | 15 | reubii 2619 | . . 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 5752 | . 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 1332 ∈ wcel 1481 ∀wral 2417 ∃!wreu 2419 〈cop 3535 class class class wbr 3937 ⟶wf 5127 ‘cfv 5131 ℩crio 5737 (class class class)co 5782 [cec 6435 Ncnpi 7104 Pcnp 7123 1Pc1p 7124 +P cpp 7125 ~R cer 7128 Rcnr 7129 0Rc0r 7130 1Rc1r 7131 <R cltr 7135 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-2o 6322 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-enq0 7256 df-nq0 7257 df-0nq0 7258 df-plq0 7259 df-mq0 7260 df-inp 7298 df-i1p 7299 df-iplp 7300 df-iltp 7302 df-enr 7558 df-nr 7559 df-ltr 7562 df-0r 7563 df-1r 7564 |
This theorem is referenced by: caucvgsrlemfv 7623 caucvgsrlemf 7624 |
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