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 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(#‘𝑀))) → 𝑆 ∈ V) |
3 | sseqval.2 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ Word 𝑆) | |
4 | 3 | adantr 481 | . . . 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 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(#‘𝑀))) → 𝐹:𝑊⟶𝑆) |
8 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(#‘𝑀))) → 𝑖 ∈ (0..^(#‘𝑀))) | |
9 | 2, 4, 5, 7, 8 | sseqfv1 30451 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(#‘𝑀))) → ((𝑀seqstr𝐹)‘𝑖) = (𝑀‘𝑖)) |
10 | 9 | ralrimiva 2966 | . 2 ⊢ (𝜑 → ∀𝑖 ∈ (0..^(#‘𝑀))((𝑀seqstr𝐹)‘𝑖) = (𝑀‘𝑖)) |
11 | 1, 3, 5, 6 | sseqfn 30452 | . . 3 ⊢ (𝜑 → (𝑀seqstr𝐹) Fn ℕ0) |
12 | wrdfn 13319 | . . . 4 ⊢ (𝑀 ∈ Word 𝑆 → 𝑀 Fn (0..^(#‘𝑀))) | |
13 | 3, 12 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 Fn (0..^(#‘𝑀))) |
14 | fzo0ssnn0 12548 | . . . 4 ⊢ (0..^(#‘𝑀)) ⊆ ℕ0 | |
15 | 14 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^(#‘𝑀)) ⊆ ℕ0) |
16 | fvreseq1 6318 | . . 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 1325 | . 2 ⊢ (𝜑 → (((𝑀seqstr𝐹) ↾ (0..^(#‘𝑀))) = 𝑀 ↔ ∀𝑖 ∈ (0..^(#‘𝑀))((𝑀seqstr𝐹)‘𝑖) = (𝑀‘𝑖))) |
18 | 10, 17 | mpbird 247 | 1 ⊢ (𝜑 → ((𝑀seqstr𝐹) ↾ (0..^(#‘𝑀))) = 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 ◡ccnv 5113 ↾ cres 5116 “ cima 5117 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 0cc0 9936 ℕ0cn0 11292 ℤ≥cuz 11687 ..^cfzo 12465 #chash 13117 Word cword 13291 seqstrcsseq 30445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-word 13299 df-lsw 13300 df-s1 13302 df-sseq 30446 |
This theorem is referenced by: sseqp1 30457 |
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