Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sseqfn | Structured version Visualization version GIF version |
Description: A strong recursive sequence is a function over the nonnegative integers. (Contributed by Thierry Arnoux, 23-Apr-2019.) |
Ref | Expression |
---|---|
sseqval.1 | ⊢ (𝜑 → 𝑆 ∈ V) |
sseqval.2 | ⊢ (𝜑 → 𝑀 ∈ Word 𝑆) |
sseqval.3 | ⊢ 𝑊 = (Word 𝑆 ∩ (◡# “ (ℤ≥‘(#‘𝑀)))) |
sseqval.4 | ⊢ (𝜑 → 𝐹:𝑊⟶𝑆) |
Ref | Expression |
---|---|
sseqfn | ⊢ (𝜑 → (𝑀seqstr𝐹) Fn ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqval.2 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Word 𝑆) | |
2 | wrdfn 13174 | . . . 4 ⊢ (𝑀 ∈ Word 𝑆 → 𝑀 Fn (0..^(#‘𝑀))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 Fn (0..^(#‘𝑀))) |
4 | fvex 6113 | . . . . . 6 ⊢ (𝑥‘((#‘𝑥) − 1)) ∈ V | |
5 | df-lsw 13155 | . . . . . 6 ⊢ lastS = (𝑥 ∈ V ↦ (𝑥‘((#‘𝑥) − 1))) | |
6 | 4, 5 | fnmpti 5935 | . . . . 5 ⊢ lastS Fn V |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → lastS Fn V) |
8 | lencl 13179 | . . . . . 6 ⊢ (𝑀 ∈ Word 𝑆 → (#‘𝑀) ∈ ℕ0) | |
9 | 8 | nn0zd 11356 | . . . . 5 ⊢ (𝑀 ∈ Word 𝑆 → (#‘𝑀) ∈ ℤ) |
10 | seqfn 12675 | . . . . 5 ⊢ ((#‘𝑀) ∈ ℤ → seq(#‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)})) Fn (ℤ≥‘(#‘𝑀))) | |
11 | 1, 9, 10 | 3syl 18 | . . . 4 ⊢ (𝜑 → seq(#‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)})) Fn (ℤ≥‘(#‘𝑀))) |
12 | ssv 3588 | . . . . 5 ⊢ ran seq(#‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)})) ⊆ V | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → ran seq(#‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)})) ⊆ V) |
14 | fnco 5913 | . . . 4 ⊢ (( lastS Fn V ∧ seq(#‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)})) Fn (ℤ≥‘(#‘𝑀)) ∧ ran seq(#‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)})) ⊆ V) → ( lastS ∘ seq(#‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))) Fn (ℤ≥‘(#‘𝑀))) | |
15 | 7, 11, 13, 14 | syl3anc 1318 | . . 3 ⊢ (𝜑 → ( lastS ∘ seq(#‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))) Fn (ℤ≥‘(#‘𝑀))) |
16 | fzouzdisj 12373 | . . . 4 ⊢ ((0..^(#‘𝑀)) ∩ (ℤ≥‘(#‘𝑀))) = ∅ | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → ((0..^(#‘𝑀)) ∩ (ℤ≥‘(#‘𝑀))) = ∅) |
18 | fnun 5911 | . . 3 ⊢ (((𝑀 Fn (0..^(#‘𝑀)) ∧ ( lastS ∘ seq(#‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))) Fn (ℤ≥‘(#‘𝑀))) ∧ ((0..^(#‘𝑀)) ∩ (ℤ≥‘(#‘𝑀))) = ∅) → (𝑀 ∪ ( lastS ∘ seq(#‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)})))) Fn ((0..^(#‘𝑀)) ∪ (ℤ≥‘(#‘𝑀)))) | |
19 | 3, 15, 17, 18 | syl21anc 1317 | . 2 ⊢ (𝜑 → (𝑀 ∪ ( lastS ∘ seq(#‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)})))) Fn ((0..^(#‘𝑀)) ∪ (ℤ≥‘(#‘𝑀)))) |
20 | sseqval.1 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ V) | |
21 | sseqval.3 | . . . 4 ⊢ 𝑊 = (Word 𝑆 ∩ (◡# “ (ℤ≥‘(#‘𝑀)))) | |
22 | sseqval.4 | . . . 4 ⊢ (𝜑 → 𝐹:𝑊⟶𝑆) | |
23 | 20, 1, 21, 22 | sseqval 29777 | . . 3 ⊢ (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ ( lastS ∘ seq(#‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))))) |
24 | nn0uz 11598 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
25 | elnn0uz 11601 | . . . . . 6 ⊢ ((#‘𝑀) ∈ ℕ0 ↔ (#‘𝑀) ∈ (ℤ≥‘0)) | |
26 | fzouzsplit 12372 | . . . . . 6 ⊢ ((#‘𝑀) ∈ (ℤ≥‘0) → (ℤ≥‘0) = ((0..^(#‘𝑀)) ∪ (ℤ≥‘(#‘𝑀)))) | |
27 | 25, 26 | sylbi 206 | . . . . 5 ⊢ ((#‘𝑀) ∈ ℕ0 → (ℤ≥‘0) = ((0..^(#‘𝑀)) ∪ (ℤ≥‘(#‘𝑀)))) |
28 | 1, 8, 27 | 3syl 18 | . . . 4 ⊢ (𝜑 → (ℤ≥‘0) = ((0..^(#‘𝑀)) ∪ (ℤ≥‘(#‘𝑀)))) |
29 | 24, 28 | syl5eq 2656 | . . 3 ⊢ (𝜑 → ℕ0 = ((0..^(#‘𝑀)) ∪ (ℤ≥‘(#‘𝑀)))) |
30 | 23, 29 | fneq12d 5897 | . 2 ⊢ (𝜑 → ((𝑀seqstr𝐹) Fn ℕ0 ↔ (𝑀 ∪ ( lastS ∘ seq(#‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)})))) Fn ((0..^(#‘𝑀)) ∪ (ℤ≥‘(#‘𝑀))))) |
31 | 19, 30 | mpbird 246 | 1 ⊢ (𝜑 → (𝑀seqstr𝐹) Fn ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 × cxp 5036 ◡ccnv 5037 ran crn 5039 “ cima 5041 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 0cc0 9815 1c1 9816 − cmin 10145 ℕ0cn0 11169 ℤcz 11254 ℤ≥cuz 11563 ..^cfzo 12334 seqcseq 12663 #chash 12979 Word cword 13146 lastS clsw 13147 ++ cconcat 13148 〈“cs1 13149 seqstrcsseq 29772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-word 13154 df-lsw 13155 df-s1 13157 df-sseq 29773 |
This theorem is referenced by: sseqfres 29782 |
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