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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 6717 | . . . 4 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀)) | |
2 | 1 | eleq1d 2822 | . . 3 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆)) |
3 | seqcl.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) | |
4 | 3 | ralrimiva 3105 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) ∈ 𝑆) |
5 | seqcl.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
6 | eluzfz1 13119 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
8 | 2, 4, 7 | rspcdva 3539 | . 2 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑆) |
9 | seqcl.3 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
10 | eluzel2 12443 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
11 | 5, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
12 | fzp1ss 13163 | . . . . 5 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
14 | 13 | sselda 3901 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → 𝑥 ∈ (𝑀...𝑁)) |
15 | 14, 3 | syldan 594 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) |
16 | 8, 9, 5, 15 | seqcl2 13594 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ⊆ wss 3866 ‘cfv 6380 (class class class)co 7213 1c1 10730 + caddc 10732 ℤcz 12176 ℤ≥cuz 12438 ...cfz 13095 seqcseq 13574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-seq 13575 |
This theorem is referenced by: sermono 13608 seqsplit 13609 seqcaopr2 13612 seqf1olem2a 13614 seqf1olem2 13616 seqid3 13620 seqhomo 13623 seqz 13624 seqdistr 13627 serge0 13630 serle 13631 seqof 13633 seqcoll 14030 seqcoll2 14031 fsumcl2lem 15295 prodfn0 15458 prodfrec 15459 prodfdiv 15460 fprodcl2lem 15512 eulerthlem2 16335 gsumwsubmcl 18263 mulgnnsubcl 18504 gsumzcl2 19295 gsumzaddlem 19306 gsummptfzcl 19354 lgscllem 26185 lgsval4a 26200 lgsneg 26202 lgsdir 26213 lgsdilem2 26214 lgsdi 26215 lgsne0 26216 gsumncl 32231 faclim 33430 knoppcnlem8 34417 mblfinlem2 35552 fmul01 42796 fmulcl 42797 fmuldfeq 42799 fmul01lt1lem1 42800 fmul01lt1lem2 42801 stoweidlem3 43219 stoweidlem42 43258 stoweidlem48 43264 wallispilem4 43284 wallispi 43286 wallispi2lem1 43287 wallispi2 43289 stirlinglem5 43294 stirlinglem7 43296 stirlinglem10 43299 sge0isum 43640 |
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