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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 13995 | . . 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 6643 | . 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 12593 | . . . 4 ⊢ 0 ∈ ℤ | |
5 | 4 | elimel 4593 | . . 3 ⊢ if(𝑀 ∈ ℤ, 𝑀, 0) ∈ ℤ |
6 | eqid 2728 | . . 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 2728 | . . 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 14003 | . . 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 13949 | . 2 ⊢ seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹) Fn (ℤ≥‘if(𝑀 ∈ ℤ, 𝑀, 0)) |
11 | 3, 10 | dedth 4582 | 1 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ifcif 4524 ⟨cop 4630 ↦ cmpt 5225 ↾ cres 5674 Fn wfn 6537 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 ωcom 7864 reccrdg 8423 0cc0 11132 1c1 11133 + caddc 11135 ℤcz 12582 ℤ≥cuz 12846 seqcseq 13992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-n0 12497 df-z 12583 df-uz 12847 df-seq 13993 |
This theorem is referenced by: seqexw 14008 seqf2 14012 seqfeq2 14016 seqfeq 14018 seqfeq3 14043 ser0f 14046 facnn 14260 fac0 14261 seqshft 15058 prodf1f 15864 efcvgfsum 16056 seq1st 16535 prmrec 16884 gsumpropd2lem 18632 mulgfval 19018 ovolunlem1 25419 ovoliunlem1 25424 volsup 25478 mtest 26333 mtestbdd 26334 pserulm 26351 pserdvlem2 26358 emcllem5 26925 lgamgulm2 26961 lgamcvglem 26965 gamcvg2lem 26984 esumfsup 33683 esumpcvgval 33691 esumcvg 33699 esumcvgsum 33701 esumsup 33702 sseqfv1 34003 sseqfn 34004 sseqfv2 34008 faclimlem1 35331 knoppcnlem8 35969 knoppcnlem11 35972 mblfinlem2 37125 ovoliunnfl 37129 voliunnfl 37131 |
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