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| 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 6905 | . . . 4 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀)) | |
| 2 | 1 | eleq1d 2825 | . . 3 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆)) | 
| 3 | seqcl.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) | |
| 4 | 3 | ralrimiva 3145 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) ∈ 𝑆) | 
| 5 | seqcl.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
| 6 | eluzfz1 13572 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
| 7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) | 
| 8 | 2, 4, 7 | rspcdva 3622 | . 2 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑆) | 
| 9 | seqcl.3 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
| 10 | eluzel2 12884 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
| 11 | 5, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | 
| 12 | fzp1ss 13616 | . . . . 5 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) | |
| 13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) | 
| 14 | 13 | sselda 3982 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → 𝑥 ∈ (𝑀...𝑁)) | 
| 15 | 14, 3 | syldan 591 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) | 
| 16 | 8, 9, 5, 15 | seqcl2 14062 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ‘cfv 6560 (class class class)co 7432 1c1 11157 + caddc 11159 ℤcz 12615 ℤ≥cuz 12879 ...cfz 13548 seqcseq 14043 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-seq 14044 | 
| This theorem is referenced by: sermono 14076 seqsplit 14077 seqcaopr2 14080 seqf1olem2a 14082 seqf1olem2 14084 seqid3 14088 seqhomo 14091 seqz 14092 seqdistr 14095 serge0 14098 serle 14099 seqof 14101 seqcoll 14504 seqcoll2 14505 fsumcl2lem 15768 prodfn0 15931 prodfrec 15932 prodfdiv 15933 fprodcl2lem 15987 eulerthlem2 16820 gsumwsubmcl 18851 mulgnnsubcl 19105 gsumzcl2 19929 gsumzaddlem 19940 gsummptfzcl 19988 lgscllem 27349 lgsval4a 27364 lgsneg 27366 lgsdir 27377 lgsdilem2 27378 lgsdi 27379 lgsne0 27380 gsumncl 34556 faclim 35747 knoppcnlem8 36502 mblfinlem2 37666 fmul01 45600 fmulcl 45601 fmuldfeq 45603 fmul01lt1lem1 45604 fmul01lt1lem2 45605 stoweidlem3 46023 stoweidlem42 46062 stoweidlem48 46068 wallispilem4 46088 wallispi 46090 wallispi2lem1 46091 wallispi2 46093 stirlinglem5 46098 stirlinglem7 46100 stirlinglem10 46103 sge0isum 46447 | 
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