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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sseqfres | Structured version Visualization version GIF version |
Description: The first elements in the strong recursive sequence are the sequence initializer. (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 |
---|---|
sseqfres | ⊢ (𝜑 → ((𝑀seqstr𝐹) ↾ (0..^(♯‘𝑀))) = 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqval.1 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ V) | |
2 | 1 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝑀))) → 𝑆 ∈ V) |
3 | sseqval.2 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ Word 𝑆) | |
4 | 3 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝑀))) → 𝑀 ∈ Word 𝑆) |
5 | sseqval.3 | . . . 4 ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀)))) | |
6 | sseqval.4 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑊⟶𝑆) | |
7 | 6 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝑀))) → 𝐹:𝑊⟶𝑆) |
8 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝑀))) → 𝑖 ∈ (0..^(♯‘𝑀))) | |
9 | 2, 4, 5, 7, 8 | sseqfv1 34050 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝑀))) → ((𝑀seqstr𝐹)‘𝑖) = (𝑀‘𝑖)) |
10 | 9 | ralrimiva 3143 | . 2 ⊢ (𝜑 → ∀𝑖 ∈ (0..^(♯‘𝑀))((𝑀seqstr𝐹)‘𝑖) = (𝑀‘𝑖)) |
11 | 1, 3, 5, 6 | sseqfn 34051 | . . 3 ⊢ (𝜑 → (𝑀seqstr𝐹) Fn ℕ0) |
12 | wrdfn 14520 | . . . 4 ⊢ (𝑀 ∈ Word 𝑆 → 𝑀 Fn (0..^(♯‘𝑀))) | |
13 | 3, 12 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 Fn (0..^(♯‘𝑀))) |
14 | fzo0ssnn0 13755 | . . . 4 ⊢ (0..^(♯‘𝑀)) ⊆ ℕ0 | |
15 | 14 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^(♯‘𝑀)) ⊆ ℕ0) |
16 | fvreseq1 7053 | . . 3 ⊢ ((((𝑀seqstr𝐹) Fn ℕ0 ∧ 𝑀 Fn (0..^(♯‘𝑀))) ∧ (0..^(♯‘𝑀)) ⊆ ℕ0) → (((𝑀seqstr𝐹) ↾ (0..^(♯‘𝑀))) = 𝑀 ↔ ∀𝑖 ∈ (0..^(♯‘𝑀))((𝑀seqstr𝐹)‘𝑖) = (𝑀‘𝑖))) | |
17 | 11, 13, 15, 16 | syl21anc 836 | . 2 ⊢ (𝜑 → (((𝑀seqstr𝐹) ↾ (0..^(♯‘𝑀))) = 𝑀 ↔ ∀𝑖 ∈ (0..^(♯‘𝑀))((𝑀seqstr𝐹)‘𝑖) = (𝑀‘𝑖))) |
18 | 10, 17 | mpbird 256 | 1 ⊢ (𝜑 → ((𝑀seqstr𝐹) ↾ (0..^(♯‘𝑀))) = 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 Vcvv 3473 ∩ cin 3948 ⊆ wss 3949 ◡ccnv 5681 ↾ cres 5684 “ cima 5685 Fn wfn 6548 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 0cc0 11148 ℕ0cn0 12512 ℤ≥cuz 12862 ..^cfzo 13669 ♯chash 14331 Word cword 14506 seqstrcsseq 34044 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-n0 12513 df-z 12599 df-uz 12863 df-fz 13527 df-fzo 13670 df-seq 14009 df-hash 14332 df-word 14507 df-lsw 14555 df-s1 14588 df-sseq 34045 |
This theorem is referenced by: sseqp1 34056 |
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