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Mirrors > Home > ILE Home > Th. List > seqex | GIF version |
Description: Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
Ref | Expression |
---|---|
seqex | ⊢ seq𝑀( + , 𝐹) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-seqfrec 10464 | . 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) | |
2 | frecex 6413 | . . 3 ⊢ frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) ∈ V | |
3 | 2 | rnex 4909 | . 2 ⊢ ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) ∈ V |
4 | 1, 3 | eqeltri 2262 | 1 ⊢ seq𝑀( + , 𝐹) ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 Vcvv 2752 〈cop 3610 ran crn 4642 ‘cfv 5231 (class class class)co 5891 ∈ cmpo 5893 freccfrec 6409 1c1 7830 + caddc 7832 ℤ≥cuz 9546 seqcseq 10463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-iord 4381 df-on 4383 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-recs 6324 df-frec 6410 df-seqfrec 10464 |
This theorem is referenced by: seq3shft 10865 clim2ser 11363 clim2ser2 11364 isermulc2 11366 iser3shft 11372 fsum3cvg 11404 sumrbdc 11405 isumclim3 11449 sumnul 11450 isumadd 11457 trireciplem 11526 geolim 11537 geolim2 11538 geo2lim 11542 geoisum1c 11546 mertensabs 11563 clim2prod 11565 clim2divap 11566 ntrivcvgap 11574 fproddccvg 11598 prodrbdclem2 11599 fprodntrivap 11610 efcj 11699 eftlub 11716 eflegeo 11727 nninfdc 12472 mulgfvalg 13029 trilpolemisumle 15184 trilpolemeq1 15186 |
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