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Mirrors > Home > MPE Home > Th. List > seqex | Structured version Visualization version 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-seq 13916 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) | |
2 | rdgfun 8366 | . . 3 ⊢ Fun rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) | |
3 | omex 9587 | . . 3 ⊢ ω ∈ V | |
4 | funimaexg 6591 | . . 3 ⊢ ((Fun rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ∧ ω ∈ V) → (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) ∈ V) | |
5 | 2, 3, 4 | mp2an 691 | . 2 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) ∈ V |
6 | 1, 5 | eqeltri 2830 | 1 ⊢ seq𝑀( + , 𝐹) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3447 ⟨cop 4596 “ cima 5640 Fun wfun 6494 ‘cfv 6500 (class class class)co 7361 ∈ cmpo 7363 ωcom 7806 reccrdg 8359 1c1 11060 + caddc 11062 seqcseq 13915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 ax-inf2 9585 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-seq 13916 |
This theorem is referenced by: seqshft 14979 clim2ser 15548 clim2ser2 15549 isermulc2 15551 isershft 15557 isercoll 15561 isercoll2 15562 iseralt 15578 fsumcvg 15605 sumrb 15606 isumclim3 15652 isumadd 15660 cvgcmp 15709 cvgcmpce 15711 trireciplem 15755 geolim 15763 geolim2 15764 geo2lim 15768 geomulcvg 15769 geoisum1c 15773 cvgrat 15776 mertens 15779 clim2prod 15781 clim2div 15782 ntrivcvg 15790 ntrivcvgfvn0 15792 ntrivcvgmullem 15794 fprodcvg 15821 prodrblem2 15822 fprodntriv 15833 iprodclim3 15891 iprodmul 15894 efcj 15982 eftlub 15999 eflegeo 16011 rpnnen2lem5 16108 mulgfvalALT 18883 ovoliunnul 24894 ioombl1lem4 24948 vitalilem5 24999 dvnfval 25309 aaliou3lem3 25727 dvradcnv 25803 pserulm 25804 abelthlem6 25818 abelthlem7 25820 abelthlem9 25822 logtayllem 26037 logtayl 26038 atantayl 26310 leibpilem2 26314 leibpi 26315 log2tlbnd 26318 zetacvg 26387 lgamgulm2 26408 lgamcvglem 26412 lgamcvg2 26427 dchrisumlem3 26862 dchrisum0re 26884 esumcvgsum 32751 sseqval 33052 iprodgam 34378 faclim 34382 knoppcnlem6 35014 knoppcnlem9 35017 knoppndvlem4 35031 knoppndvlem6 35033 knoppf 35051 geomcau 36268 dvradcnv2 42719 binomcxplemnotnn0 42728 sumnnodd 43961 stirlinglem5 44409 stirlinglem7 44411 fourierdlem112 44549 sge0isum 44758 itcoval 46837 |
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