Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > caucvgsrlemcl | GIF version |
Description: Lemma for caucvgsr 7764. Terms of the sequence from caucvgsrlemgt1 7757 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 5631 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (𝐹‘𝐴) ∈ R) |
3 | 0lt1sr 7727 | . . . . 5 ⊢ 0R <R 1R | |
4 | caucvgsrlemcl.gt1 | . . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚)) | |
5 | fveq2 5496 | . . . . . . . 8 ⊢ (𝑚 = 𝐴 → (𝐹‘𝑚) = (𝐹‘𝐴)) | |
6 | 5 | breq2d 4001 | . . . . . . 7 ⊢ (𝑚 = 𝐴 → (1R <R (𝐹‘𝑚) ↔ 1R <R (𝐹‘𝐴))) |
7 | 6 | rspcv 2830 | . . . . . 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 7726 | . . . . . 6 ⊢ <R Or R | |
10 | ltrelsr 7700 | . . . . . 6 ⊢ <R ⊆ (R × R) | |
11 | 9, 10 | sotri 5006 | . . . . 5 ⊢ ((0R <R 1R ∧ 1R <R (𝐹‘𝐴)) → 0R <R (𝐹‘𝐴)) |
12 | 3, 8, 11 | sylancr 412 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → 0R <R (𝐹‘𝐴)) |
13 | srpospr 7745 | . . . 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 2172 | . . . 4 ⊢ ([〈(𝑦 +P 1P), 1P〉] ~R = (𝐹‘𝐴) ↔ (𝐹‘𝐴) = [〈(𝑦 +P 1P), 1P〉] ~R ) | |
16 | 15 | reubii 2655 | . . 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 5823 | . 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 1348 ∈ wcel 2141 ∀wral 2448 ∃!wreu 2450 〈cop 3586 class class class wbr 3989 ⟶wf 5194 ‘cfv 5198 ℩crio 5808 (class class class)co 5853 [cec 6511 Ncnpi 7234 Pcnp 7253 1Pc1p 7254 +P cpp 7255 ~R cer 7258 Rcnr 7259 0Rc0r 7260 1Rc1r 7261 <R cltr 7265 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-eprel 4274 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-1o 6395 df-2o 6396 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-pli 7267 df-mi 7268 df-lti 7269 df-plpq 7306 df-mpq 7307 df-enq 7309 df-nqqs 7310 df-plqqs 7311 df-mqqs 7312 df-1nqqs 7313 df-rq 7314 df-ltnqqs 7315 df-enq0 7386 df-nq0 7387 df-0nq0 7388 df-plq0 7389 df-mq0 7390 df-inp 7428 df-i1p 7429 df-iplp 7430 df-iltp 7432 df-enr 7688 df-nr 7689 df-ltr 7692 df-0r 7693 df-1r 7694 |
This theorem is referenced by: caucvgsrlemfv 7753 caucvgsrlemf 7754 |
Copyright terms: Public domain | W3C validator |