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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 | fveq2 6846 | . . . 4 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀)) | |
2 | 1 | eleq1d 2819 | . . 3 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆)) |
3 | seqcl.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) | |
4 | 3 | ralrimiva 3140 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) ∈ 𝑆) |
5 | seqcl.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
6 | eluzfz1 13457 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
8 | 2, 4, 7 | rspcdva 3584 | . 2 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑆) |
9 | seqcl.3 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
10 | eluzel2 12776 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
11 | 5, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
12 | fzp1ss 13501 | . . . . 5 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
14 | 13 | sselda 3948 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → 𝑥 ∈ (𝑀...𝑁)) |
15 | 14, 3 | syldan 592 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) |
16 | 8, 9, 5, 15 | seqcl2 13935 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3914 ‘cfv 6500 (class class class)co 7361 1c1 11060 + caddc 11062 ℤcz 12507 ℤ≥cuz 12771 ...cfz 13433 seqcseq 13915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-seq 13916 |
This theorem is referenced by: sermono 13949 seqsplit 13950 seqcaopr2 13953 seqf1olem2a 13955 seqf1olem2 13957 seqid3 13961 seqhomo 13964 seqz 13965 seqdistr 13968 serge0 13971 serle 13972 seqof 13974 seqcoll 14372 seqcoll2 14373 fsumcl2lem 15624 prodfn0 15787 prodfrec 15788 prodfdiv 15789 fprodcl2lem 15841 eulerthlem2 16662 gsumwsubmcl 18655 mulgnnsubcl 18896 gsumzcl2 19695 gsumzaddlem 19706 gsummptfzcl 19754 lgscllem 26675 lgsval4a 26690 lgsneg 26692 lgsdir 26703 lgsdilem2 26704 lgsdi 26705 lgsne0 26706 gsumncl 33216 faclim 34382 knoppcnlem8 35016 mblfinlem2 36166 fmul01 43911 fmulcl 43912 fmuldfeq 43914 fmul01lt1lem1 43915 fmul01lt1lem2 43916 stoweidlem3 44334 stoweidlem42 44373 stoweidlem48 44379 wallispilem4 44399 wallispi 44401 wallispi2lem1 44402 wallispi2 44404 stirlinglem5 44409 stirlinglem7 44411 stirlinglem10 44414 sge0isum 44758 |
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