Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > caucvgsrlemcl | GIF version |
Description: Lemma for caucvgsr 7722. Terms of the sequence from caucvgsrlemgt1 7715 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 5602 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (𝐹‘𝐴) ∈ R) |
3 | 0lt1sr 7685 | . . . . 5 ⊢ 0R <R 1R | |
4 | caucvgsrlemcl.gt1 | . . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚)) | |
5 | fveq2 5468 | . . . . . . . 8 ⊢ (𝑚 = 𝐴 → (𝐹‘𝑚) = (𝐹‘𝐴)) | |
6 | 5 | breq2d 3977 | . . . . . . 7 ⊢ (𝑚 = 𝐴 → (1R <R (𝐹‘𝑚) ↔ 1R <R (𝐹‘𝐴))) |
7 | 6 | rspcv 2812 | . . . . . 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 7684 | . . . . . 6 ⊢ <R Or R | |
10 | ltrelsr 7658 | . . . . . 6 ⊢ <R ⊆ (R × R) | |
11 | 9, 10 | sotri 4981 | . . . . 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 7703 | . . . 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 2159 | . . . 4 ⊢ ([〈(𝑦 +P 1P), 1P〉] ~R = (𝐹‘𝐴) ↔ (𝐹‘𝐴) = [〈(𝑦 +P 1P), 1P〉] ~R ) | |
16 | 15 | reubii 2642 | . . 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 5794 | . 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 1335 ∈ wcel 2128 ∀wral 2435 ∃!wreu 2437 〈cop 3563 class class class wbr 3965 ⟶wf 5166 ‘cfv 5170 ℩crio 5779 (class class class)co 5824 [cec 6478 Ncnpi 7192 Pcnp 7211 1Pc1p 7212 +P cpp 7213 ~R cer 7216 Rcnr 7217 0Rc0r 7218 1Rc1r 7219 <R cltr 7223 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-eprel 4249 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-1o 6363 df-2o 6364 df-oadd 6367 df-omul 6368 df-er 6480 df-ec 6482 df-qs 6486 df-ni 7224 df-pli 7225 df-mi 7226 df-lti 7227 df-plpq 7264 df-mpq 7265 df-enq 7267 df-nqqs 7268 df-plqqs 7269 df-mqqs 7270 df-1nqqs 7271 df-rq 7272 df-ltnqqs 7273 df-enq0 7344 df-nq0 7345 df-0nq0 7346 df-plq0 7347 df-mq0 7348 df-inp 7386 df-i1p 7387 df-iplp 7388 df-iltp 7390 df-enr 7646 df-nr 7647 df-ltr 7650 df-0r 7651 df-1r 7652 |
This theorem is referenced by: caucvgsrlemfv 7711 caucvgsrlemf 7712 |
Copyright terms: Public domain | W3C validator |