Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 13877 | . . . 4 ⊢ (𝑀 ∈ Word 𝑆 → 𝑀 Fn (0..^(♯‘𝑀))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 Fn (0..^(♯‘𝑀))) |
4 | fvex 6683 | . . . . . 6 ⊢ (𝑥‘((♯‘𝑥) − 1)) ∈ V | |
5 | df-lsw 13915 | . . . . . 6 ⊢ lastS = (𝑥 ∈ V ↦ (𝑥‘((♯‘𝑥) − 1))) | |
6 | 4, 5 | fnmpti 6491 | . . . . 5 ⊢ lastS Fn V |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → lastS Fn V) |
8 | lencl 13883 | . . . . . 6 ⊢ (𝑀 ∈ Word 𝑆 → (♯‘𝑀) ∈ ℕ0) | |
9 | 8 | nn0zd 12086 | . . . . 5 ⊢ (𝑀 ∈ Word 𝑆 → (♯‘𝑀) ∈ ℤ) |
10 | seqfn 13382 | . . . . 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 3991 | . . . . 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 6465 | . . . 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 1367 | . . 3 ⊢ (𝜑 → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))) Fn (ℤ≥‘(♯‘𝑀))) |
16 | fzouzdisj 13074 | . . . 4 ⊢ ((0..^(♯‘𝑀)) ∩ (ℤ≥‘(♯‘𝑀))) = ∅ | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → ((0..^(♯‘𝑀)) ∩ (ℤ≥‘(♯‘𝑀))) = ∅) |
18 | fnun 6463 | . . 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 835 | . 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 31646 | . . 3 ⊢ (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))))) |
24 | nn0uz 12281 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
25 | elnn0uz 12284 | . . . . . 6 ⊢ ((♯‘𝑀) ∈ ℕ0 ↔ (♯‘𝑀) ∈ (ℤ≥‘0)) | |
26 | fzouzsplit 13073 | . . . . . 6 ⊢ ((♯‘𝑀) ∈ (ℤ≥‘0) → (ℤ≥‘0) = ((0..^(♯‘𝑀)) ∪ (ℤ≥‘(♯‘𝑀)))) | |
27 | 25, 26 | sylbi 219 | . . . . 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 2868 | . . 3 ⊢ (𝜑 → ℕ0 = ((0..^(♯‘𝑀)) ∪ (ℤ≥‘(♯‘𝑀)))) |
30 | 23, 29 | fneq12d 6448 | . 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 259 | 1 ⊢ (𝜑 → (𝑀seqstr𝐹) Fn ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 {csn 4567 × cxp 5553 ◡ccnv 5554 ran crn 5556 “ cima 5558 ∘ ccom 5559 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 0cc0 10537 1c1 10538 − cmin 10870 ℕ0cn0 11898 ℤcz 11982 ℤ≥cuz 12244 ..^cfzo 13034 seqcseq 13370 ♯chash 13691 Word cword 13862 lastSclsw 13914 ++ cconcat 13922 〈“cs1 13949 seqstrcsseq 31641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-word 13863 df-lsw 13915 df-s1 13950 df-sseq 31642 |
This theorem is referenced by: sseqfres 31651 |
Copyright terms: Public domain | W3C validator |