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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 13985). (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 6882 | . . . 4 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀)) | |
2 | 1 | eleq1d 2810 | . . 3 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆)) |
3 | seqf.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → (𝐹‘𝑥) ∈ 𝑆) | |
4 | 3 | ralrimiva 3138 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑍 (𝐹‘𝑥) ∈ 𝑆) |
5 | seqf.2 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
6 | uzid 12835 | . . . . 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 2836 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
10 | 2, 4, 9 | rspcdva 3605 | . 2 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑆) |
11 | seqf.4 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
12 | peano2uzr 12885 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑥 ∈ (ℤ≥‘𝑀)) | |
13 | 5, 12 | sylan 579 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑥 ∈ (ℤ≥‘𝑀)) |
14 | 13, 8 | eleqtrrdi 2836 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑥 ∈ 𝑍) |
15 | 14, 3 | syldan 590 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝑆) |
16 | 10, 11, 8, 5, 15 | seqf2 13985 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⟶wf 6530 ‘cfv 6534 (class class class)co 7402 1c1 11108 + caddc 11110 ℤcz 12556 ℤ≥cuz 12820 seqcseq 13964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-n0 12471 df-z 12557 df-uz 12821 df-fz 13483 df-seq 13965 |
This theorem is referenced by: serf 13994 serfre 13995 bcval5 14276 prodf 15831 iprodrecl 15944 algrf 16509 pcmptcl 16825 ovolsf 25325 dvnff 25777 elqaalem2 26176 elqaalem3 26177 regamcl 26912 opsqrlem4 31868 sseqf 33883 fsumsermpt 44805 sge0isum 45653 sge0seq 45672 |
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