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Mirrors > Home > MPE Home > Th. List > seqcl | Structured version Visualization version GIF version |
Description: Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.) |
Ref | Expression |
---|---|
seqcl.1 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
seqcl.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) |
seqcl.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
Ref | Expression |
---|---|
seqcl | ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqcl.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | eluzfz1 12512 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
4 | seqcl.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) | |
5 | 4 | ralrimiva 3092 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) ∈ 𝑆) |
6 | fveq2 6340 | . . . . 5 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀)) | |
7 | 6 | eleq1d 2812 | . . . 4 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆)) |
8 | 7 | rspcv 3433 | . . 3 ⊢ (𝑀 ∈ (𝑀...𝑁) → (∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) ∈ 𝑆 → (𝐹‘𝑀) ∈ 𝑆)) |
9 | 3, 5, 8 | sylc 65 | . 2 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑆) |
10 | seqcl.3 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
11 | eluzel2 11855 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
12 | 1, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
13 | fzp1ss 12556 | . . . . 5 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
15 | 14 | sselda 3732 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → 𝑥 ∈ (𝑀...𝑁)) |
16 | 15, 4 | syldan 488 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) |
17 | 9, 10, 1, 16 | seqcl2 12984 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1620 ∈ wcel 2127 ∀wral 3038 ⊆ wss 3703 ‘cfv 6037 (class class class)co 6801 1c1 10100 + caddc 10102 ℤcz 11540 ℤ≥cuz 11850 ...cfz 12490 seqcseq 12966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-iun 4662 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-om 7219 df-1st 7321 df-2nd 7322 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-er 7899 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-nn 11184 df-n0 11456 df-z 11541 df-uz 11851 df-fz 12491 df-seq 12967 |
This theorem is referenced by: sermono 12998 seqsplit 12999 seqcaopr2 13002 seqf1olem2a 13004 seqf1olem2 13006 seqid3 13010 seqhomo 13013 seqz 13014 seqdistr 13017 serge0 13020 serle 13021 seqof 13023 seqcoll 13411 seqcoll2 13412 fsumcl2lem 14632 prodfn0 14796 prodfrec 14797 prodfdiv 14798 fprodcl2lem 14850 eulerthlem2 15660 gsumwsubmcl 17547 mulgnnsubcl 17725 gsumzcl2 18482 gsumzaddlem 18492 gsummptfzcl 18539 lgscllem 25199 lgsval4a 25214 lgsneg 25216 lgsdir 25227 lgsdilem2 25228 lgsdi 25229 lgsne0 25230 gsumncl 30894 faclim 31910 knoppcnlem8 32767 mblfinlem2 33729 fmul01 40284 fmulcl 40285 fmuldfeq 40287 fmul01lt1lem1 40288 fmul01lt1lem2 40289 stoweidlem3 40692 stoweidlem42 40731 stoweidlem48 40737 wallispilem4 40757 wallispi 40759 wallispi2lem1 40760 wallispi2 40762 stirlinglem5 40767 stirlinglem7 40769 stirlinglem10 40772 sge0isum 41116 |
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