![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > seqfn | Structured version Visualization version GIF version |
Description: The sequence builder function is a function. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
seqfn | ⊢ (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqeq1 13968 | . . 3 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → seq𝑀( + , 𝐹) = seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)) | |
2 | fveq2 6891 | . . 3 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → (ℤ≥‘𝑀) = (ℤ≥‘if(𝑀 ∈ ℤ, 𝑀, 0))) | |
3 | 1, 2 | fneq12d 6644 | . 2 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → (seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀) ↔ seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹) Fn (ℤ≥‘if(𝑀 ∈ ℤ, 𝑀, 0)))) |
4 | 0z 12568 | . . . 4 ⊢ 0 ∈ ℤ | |
5 | 4 | elimel 4597 | . . 3 ⊢ if(𝑀 ∈ ℤ, 𝑀, 0) ∈ ℤ |
6 | eqid 2732 | . . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), if(𝑀 ∈ ℤ, 𝑀, 0)) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), if(𝑀 ∈ ℤ, 𝑀, 0)) ↾ ω) | |
7 | fvex 6904 | . . 3 ⊢ (𝐹‘if(𝑀 ∈ ℤ, 𝑀, 0)) ∈ V | |
8 | eqid 2732 | . . 3 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨if(𝑀 ∈ ℤ, 𝑀, 0), (𝐹‘if(𝑀 ∈ ℤ, 𝑀, 0))⟩) ↾ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨if(𝑀 ∈ ℤ, 𝑀, 0), (𝐹‘if(𝑀 ∈ ℤ, 𝑀, 0))⟩) ↾ ω) | |
9 | 8 | seqval 13976 | . . 3 ⊢ seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹) = ran (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨if(𝑀 ∈ ℤ, 𝑀, 0), (𝐹‘if(𝑀 ∈ ℤ, 𝑀, 0))⟩) ↾ ω) |
10 | 5, 6, 7, 8, 9 | uzrdgfni 13922 | . 2 ⊢ seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹) Fn (ℤ≥‘if(𝑀 ∈ ℤ, 𝑀, 0)) |
11 | 3, 10 | dedth 4586 | 1 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ifcif 4528 ⟨cop 4634 ↦ cmpt 5231 ↾ cres 5678 Fn wfn 6538 ‘cfv 6543 (class class class)co 7408 ∈ cmpo 7410 ωcom 7854 reccrdg 8408 0cc0 11109 1c1 11110 + caddc 11112 ℤcz 12557 ℤ≥cuz 12821 seqcseq 13965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-seq 13966 |
This theorem is referenced by: seqexw 13981 seqf2 13986 seqfeq2 13990 seqfeq 13992 seqfeq3 14017 ser0f 14020 facnn 14234 fac0 14235 seqshft 15031 prodf1f 15837 efcvgfsum 16028 seq1st 16507 prmrec 16854 gsumpropd2lem 18597 mulgfval 18951 ovolunlem1 25013 ovoliunlem1 25018 volsup 25072 mtest 25915 mtestbdd 25916 pserulm 25933 pserdvlem2 25939 emcllem5 26501 lgamgulm2 26537 lgamcvglem 26541 gamcvg2lem 26560 esumfsup 33063 esumpcvgval 33071 esumcvg 33079 esumcvgsum 33081 esumsup 33082 sseqfv1 33383 sseqfn 33384 sseqfv2 33388 faclimlem1 34708 knoppcnlem8 35371 knoppcnlem11 35374 mblfinlem2 36521 ovoliunnfl 36525 voliunnfl 36527 |
Copyright terms: Public domain | W3C validator |