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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 14482 | . . . 4 ⊢ (𝑀 ∈ Word 𝑆 → 𝑀 Fn (0..^(♯‘𝑀))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 Fn (0..^(♯‘𝑀))) |
4 | fvex 6904 | . . . . . 6 ⊢ (𝑥‘((♯‘𝑥) − 1)) ∈ V | |
5 | df-lsw 14517 | . . . . . 6 ⊢ lastS = (𝑥 ∈ V ↦ (𝑥‘((♯‘𝑥) − 1))) | |
6 | 4, 5 | fnmpti 6693 | . . . . 5 ⊢ lastS Fn V |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → lastS Fn V) |
8 | lencl 14487 | . . . . . 6 ⊢ (𝑀 ∈ Word 𝑆 → (♯‘𝑀) ∈ ℕ0) | |
9 | 8 | nn0zd 12588 | . . . . 5 ⊢ (𝑀 ∈ Word 𝑆 → (♯‘𝑀) ∈ ℤ) |
10 | seqfn 13982 | . . . . 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 4006 | . . . . 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 6667 | . . . 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 1371 | . . 3 ⊢ (𝜑 → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))) Fn (ℤ≥‘(♯‘𝑀))) |
16 | fzouzdisj 13672 | . . . 4 ⊢ ((0..^(♯‘𝑀)) ∩ (ℤ≥‘(♯‘𝑀))) = ∅ | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → ((0..^(♯‘𝑀)) ∩ (ℤ≥‘(♯‘𝑀))) = ∅) |
18 | 3, 15, 17 | fnund 6664 | . 2 ⊢ (𝜑 → (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))) Fn ((0..^(♯‘𝑀)) ∪ (ℤ≥‘(♯‘𝑀)))) |
19 | sseqval.1 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ V) | |
20 | sseqval.3 | . . . 4 ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀)))) | |
21 | sseqval.4 | . . . 4 ⊢ (𝜑 → 𝐹:𝑊⟶𝑆) | |
22 | 19, 1, 20, 21 | sseqval 33673 | . . 3 ⊢ (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))))) |
23 | nn0uz 12868 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
24 | elnn0uz 12871 | . . . . . 6 ⊢ ((♯‘𝑀) ∈ ℕ0 ↔ (♯‘𝑀) ∈ (ℤ≥‘0)) | |
25 | fzouzsplit 13671 | . . . . . 6 ⊢ ((♯‘𝑀) ∈ (ℤ≥‘0) → (ℤ≥‘0) = ((0..^(♯‘𝑀)) ∪ (ℤ≥‘(♯‘𝑀)))) | |
26 | 24, 25 | sylbi 216 | . . . . 5 ⊢ ((♯‘𝑀) ∈ ℕ0 → (ℤ≥‘0) = ((0..^(♯‘𝑀)) ∪ (ℤ≥‘(♯‘𝑀)))) |
27 | 1, 8, 26 | 3syl 18 | . . . 4 ⊢ (𝜑 → (ℤ≥‘0) = ((0..^(♯‘𝑀)) ∪ (ℤ≥‘(♯‘𝑀)))) |
28 | 23, 27 | eqtrid 2784 | . . 3 ⊢ (𝜑 → ℕ0 = ((0..^(♯‘𝑀)) ∪ (ℤ≥‘(♯‘𝑀)))) |
29 | 22, 28 | fneq12d 6644 | . 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))))) |
30 | 18, 29 | mpbird 256 | 1 ⊢ (𝜑 → (𝑀seqstr𝐹) Fn ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 {csn 4628 × cxp 5674 ◡ccnv 5675 ran crn 5677 “ cima 5679 ∘ ccom 5680 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ∈ cmpo 7413 0cc0 11112 1c1 11113 − cmin 11448 ℕ0cn0 12476 ℤcz 12562 ℤ≥cuz 12826 ..^cfzo 13631 seqcseq 13970 ♯chash 14294 Word cword 14468 lastSclsw 14516 ++ cconcat 14524 ⟨“cs1 14549 seqstrcsseq 33668 |
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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-word 14469 df-lsw 14517 df-s1 14550 df-sseq 33669 |
This theorem is referenced by: sseqfres 33678 |
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