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Mirrors > Home > MPE Home > Th. List > seqf2 | Structured version Visualization version GIF version |
Description: Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 27-May-2014.) |
Ref | Expression |
---|---|
seqcl2.1 | ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝐶) |
seqcl2.2 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶) |
seqf2.3 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
seqf2.4 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
seqf2.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝐷) |
Ref | Expression |
---|---|
seqf2 | ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqf2.4 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | seqfn 13551 | . . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)) |
4 | seqcl2.1 | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝐶) | |
5 | 4 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑀) ∈ 𝐶) |
6 | seqcl2.2 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶) | |
7 | 6 | adantlr 715 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶) |
8 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
9 | elfzuz 13073 | . . . . . . 7 ⊢ (𝑥 ∈ ((𝑀 + 1)...𝑘) → 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) | |
10 | seqf2.5 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝐷) | |
11 | 9, 10 | sylan2 596 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑀 + 1)...𝑘)) → (𝐹‘𝑥) ∈ 𝐷) |
12 | 11 | adantlr 715 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ ((𝑀 + 1)...𝑘)) → (𝐹‘𝑥) ∈ 𝐷) |
13 | 5, 7, 8, 12 | seqcl2 13559 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹)‘𝑘) ∈ 𝐶) |
14 | 13 | ralrimiva 3095 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑀)(seq𝑀( + , 𝐹)‘𝑘) ∈ 𝐶) |
15 | ffnfv 6913 | . . 3 ⊢ (seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶𝐶 ↔ (seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀) ∧ ∀𝑘 ∈ (ℤ≥‘𝑀)(seq𝑀( + , 𝐹)‘𝑘) ∈ 𝐶)) | |
16 | 3, 14, 15 | sylanbrc 586 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶𝐶) |
17 | seqf2.3 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
18 | 17 | feq2i 6515 | . 2 ⊢ (seq𝑀( + , 𝐹):𝑍⟶𝐶 ↔ seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶𝐶) |
19 | 16, 18 | sylibr 237 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 1c1 10695 + caddc 10697 ℤcz 12141 ℤ≥cuz 12403 ...cfz 13060 seqcseq 13539 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-seq 13540 |
This theorem is referenced by: seqf 13562 ruclem6 15759 sadcf 15975 smupf 16000 sseqfv2 32027 |
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