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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 14043 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | |
| 2 | rdgfun 8456 | . . 3 ⊢ Fun rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) | |
| 3 | omex 9683 | . . 3 ⊢ ω ∈ V | |
| 4 | funimaexg 6653 | . . 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 692 | . 2 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) ∈ V |
| 6 | 1, 5 | eqeltri 2837 | 1 ⊢ seq𝑀( + , 𝐹) ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 〈cop 4632 “ cima 5688 Fun wfun 6555 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ωcom 7887 reccrdg 8449 1c1 11156 + caddc 11158 seqcseq 14042 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-seq 14043 |
| This theorem is referenced by: seqshft 15124 clim2ser 15691 clim2ser2 15692 isermulc2 15694 isershft 15700 isercoll 15704 isercoll2 15705 iseralt 15721 fsumcvg 15748 sumrb 15749 isumclim3 15795 isumadd 15803 cvgcmp 15852 cvgcmpce 15854 trireciplem 15898 geolim 15906 geolim2 15907 geo2lim 15911 geomulcvg 15912 geoisum1c 15916 cvgrat 15919 mertens 15922 clim2prod 15924 clim2div 15925 ntrivcvg 15933 ntrivcvgfvn0 15935 ntrivcvgmullem 15937 fprodcvg 15966 prodrblem2 15967 fprodntriv 15978 iprodclim3 16036 iprodmul 16039 efcj 16128 eftlub 16145 eflegeo 16157 rpnnen2lem5 16254 mulgfvalALT 19088 ovoliunnul 25542 ioombl1lem4 25596 vitalilem5 25647 dvnfval 25958 aaliou3lem3 26386 dvradcnv 26464 pserulm 26465 abelthlem6 26480 abelthlem7 26482 abelthlem9 26484 logtayllem 26701 logtayl 26702 atantayl 26980 leibpilem2 26984 leibpi 26985 log2tlbnd 26988 zetacvg 27058 lgamgulm2 27079 lgamcvglem 27083 lgamcvg2 27098 dchrisumlem3 27535 dchrisum0re 27557 esumcvgsum 34089 sseqval 34390 iprodgam 35742 faclim 35746 knoppcnlem6 36499 knoppcnlem9 36502 knoppndvlem4 36516 knoppndvlem6 36518 knoppf 36536 geomcau 37766 dvradcnv2 44366 binomcxplemnotnn0 44375 sumnnodd 45645 stirlinglem5 46093 stirlinglem7 46095 fourierdlem112 46233 sge0isum 46442 itcoval 48582 |
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