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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 13009 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | |
2 | rdgfun 7665 | . . 3 ⊢ Fun rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) | |
3 | omex 8704 | . . 3 ⊢ ω ∈ V | |
4 | funimaexg 6115 | . . 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 664 | . 2 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) ∈ V |
6 | 1, 5 | eqeltri 2846 | 1 ⊢ seq𝑀( + , 𝐹) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2145 Vcvv 3351 〈cop 4322 “ cima 5252 Fun wfun 6025 ‘cfv 6031 (class class class)co 6793 ↦ cmpt2 6795 ωcom 7212 reccrdg 7658 1c1 10139 + caddc 10141 seqcseq 13008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-om 7213 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-seq 13009 |
This theorem is referenced by: seqshft 14033 clim2ser 14593 clim2ser2 14594 isermulc2 14596 isershft 14602 isercoll 14606 isercoll2 14607 iseralt 14623 fsumcvg 14651 sumrb 14652 isumclim3 14698 isumadd 14706 cvgcmp 14755 cvgcmpce 14757 trireciplem 14801 geolim 14808 geolim2 14809 geo2lim 14813 geomulcvg 14814 geoisum1c 14818 cvgrat 14822 mertens 14825 clim2prod 14827 clim2div 14828 ntrivcvg 14836 ntrivcvgfvn0 14838 ntrivcvgmullem 14840 fprodcvg 14867 prodrblem2 14868 fprodntriv 14879 iprodclim3 14937 iprodmul 14940 efcj 15028 eftlub 15045 eflegeo 15057 rpnnen2lem5 15153 mulgfval 17750 ovoliunnul 23495 ioombl1lem4 23549 vitalilem5 23600 dvnfval 23905 aaliou3lem3 24319 dvradcnv 24395 pserulm 24396 abelthlem6 24410 abelthlem7 24412 abelthlem9 24414 logtayllem 24626 logtayl 24627 atantayl 24885 leibpilem2 24889 leibpi 24890 log2tlbnd 24893 zetacvg 24962 lgamgulm2 24983 lgamcvglem 24987 lgamcvg2 25002 dchrisumlem3 25401 dchrisum0re 25423 esumcvgsum 30490 sseqval 30790 iprodgam 31966 faclim 31970 knoppcnlem6 32825 knoppcnlem9 32828 knoppndvlem4 32843 knoppndvlem6 32845 knoppf 32863 geomcau 33887 dvradcnv2 39072 binomcxplemnotnn0 39081 sumnnodd 40380 stirlinglem5 40812 stirlinglem7 40814 fourierdlem112 40952 sge0isum 41161 |
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