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Mirrors > Home > MPE Home > Th. List > seqf | Structured version Visualization version GIF version |
Description: Range of the recursive sequence builder (special case of seqf2 13881). (Contributed by Mario Carneiro, 24-Jun-2013.) |
Ref | Expression |
---|---|
seqf.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
seqf.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
seqf.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → (𝐹‘𝑥) ∈ 𝑆) |
seqf.4 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
Ref | Expression |
---|---|
seqf | ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6839 | . . . 4 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀)) | |
2 | 1 | eleq1d 2822 | . . 3 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆)) |
3 | seqf.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → (𝐹‘𝑥) ∈ 𝑆) | |
4 | 3 | ralrimiva 3141 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑍 (𝐹‘𝑥) ∈ 𝑆) |
5 | seqf.2 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
6 | uzid 12736 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
8 | seqf.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
9 | 7, 8 | eleqtrrdi 2849 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
10 | 2, 4, 9 | rspcdva 3580 | . 2 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑆) |
11 | seqf.4 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
12 | peano2uzr 12782 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑥 ∈ (ℤ≥‘𝑀)) | |
13 | 5, 12 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑥 ∈ (ℤ≥‘𝑀)) |
14 | 13, 8 | eleqtrrdi 2849 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑥 ∈ 𝑍) |
15 | 14, 3 | syldan 591 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝑆) |
16 | 10, 11, 8, 5, 15 | seqf2 13881 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 1c1 11010 + caddc 11012 ℤcz 12457 ℤ≥cuz 12721 seqcseq 13860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-seq 13861 |
This theorem is referenced by: serf 13890 serfre 13891 bcval5 14172 prodf 15726 iprodrecl 15839 algrf 16403 pcmptcl 16717 ovolsf 24782 dvnff 25233 elqaalem2 25626 elqaalem3 25627 regamcl 26356 opsqrlem4 30930 sseqf 32820 fsumsermpt 43715 sge0isum 44563 sge0seq 44582 |
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