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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 6897 | . . . 4 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀)) | |
2 | 1 | eleq1d 2814 | . . 3 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆)) |
3 | seqcl.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) | |
4 | 3 | ralrimiva 3143 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) ∈ 𝑆) |
5 | seqcl.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
6 | eluzfz1 13540 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
8 | 2, 4, 7 | rspcdva 3610 | . 2 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑆) |
9 | seqcl.3 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
10 | eluzel2 12857 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
11 | 5, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
12 | fzp1ss 13584 | . . . . 5 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
14 | 13 | sselda 3980 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → 𝑥 ∈ (𝑀...𝑁)) |
15 | 14, 3 | syldan 590 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) |
16 | 8, 9, 5, 15 | seqcl2 14017 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ‘cfv 6548 (class class class)co 7420 1c1 11139 + caddc 11141 ℤcz 12588 ℤ≥cuz 12852 ...cfz 13516 seqcseq 13998 |
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 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-seq 13999 |
This theorem is referenced by: sermono 14031 seqsplit 14032 seqcaopr2 14035 seqf1olem2a 14037 seqf1olem2 14039 seqid3 14043 seqhomo 14046 seqz 14047 seqdistr 14050 serge0 14053 serle 14054 seqof 14056 seqcoll 14457 seqcoll2 14458 fsumcl2lem 15709 prodfn0 15872 prodfrec 15873 prodfdiv 15874 fprodcl2lem 15926 eulerthlem2 16750 gsumwsubmcl 18788 mulgnnsubcl 19040 gsumzcl2 19864 gsumzaddlem 19875 gsummptfzcl 19923 lgscllem 27236 lgsval4a 27251 lgsneg 27253 lgsdir 27264 lgsdilem2 27265 lgsdi 27266 lgsne0 27267 gsumncl 34172 faclim 35340 knoppcnlem8 35975 mblfinlem2 37131 fmul01 44968 fmulcl 44969 fmuldfeq 44971 fmul01lt1lem1 44972 fmul01lt1lem2 44973 stoweidlem3 45391 stoweidlem42 45430 stoweidlem48 45436 wallispilem4 45456 wallispi 45458 wallispi2lem1 45459 wallispi2 45461 stirlinglem5 45466 stirlinglem7 45468 stirlinglem10 45471 sge0isum 45815 |
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