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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sseqfv1 | Structured version Visualization version GIF version |
Description: Value of the strong sequence builder function at one of its initial values. (Contributed by Thierry Arnoux, 21-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) |
sseqfv1.4 | ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝑀))) |
Ref | Expression |
---|---|
sseqfv1 | ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (𝑀‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqval.1 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ V) | |
2 | sseqval.2 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Word 𝑆) | |
3 | sseqval.3 | . . . 4 ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀)))) | |
4 | sseqval.4 | . . . 4 ⊢ (𝜑 → 𝐹:𝑊⟶𝑆) | |
5 | 1, 2, 3, 4 | sseqval 33028 | . . 3 ⊢ (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))))) |
6 | 5 | fveq1d 6849 | . 2 ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = ((𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))))‘𝑁)) |
7 | wrdfn 14423 | . . . 4 ⊢ (𝑀 ∈ Word 𝑆 → 𝑀 Fn (0..^(♯‘𝑀))) | |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 Fn (0..^(♯‘𝑀))) |
9 | fvex 6860 | . . . . . 6 ⊢ (𝑥‘((♯‘𝑥) − 1)) ∈ V | |
10 | df-lsw 14458 | . . . . . 6 ⊢ lastS = (𝑥 ∈ V ↦ (𝑥‘((♯‘𝑥) − 1))) | |
11 | 9, 10 | fnmpti 6649 | . . . . 5 ⊢ lastS Fn V |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → lastS Fn V) |
13 | lencl 14428 | . . . . . . 7 ⊢ (𝑀 ∈ Word 𝑆 → (♯‘𝑀) ∈ ℕ0) | |
14 | 2, 13 | syl 17 | . . . . . 6 ⊢ (𝜑 → (♯‘𝑀) ∈ ℕ0) |
15 | 14 | nn0zd 12532 | . . . . 5 ⊢ (𝜑 → (♯‘𝑀) ∈ ℤ) |
16 | seqfn 13925 | . . . . 5 ⊢ ((♯‘𝑀) ∈ ℤ → seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) Fn (ℤ≥‘(♯‘𝑀))) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝜑 → seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) Fn (ℤ≥‘(♯‘𝑀))) |
18 | ssv 3973 | . . . . 5 ⊢ ran seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) ⊆ V | |
19 | 18 | a1i 11 | . . . 4 ⊢ (𝜑 → ran seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) ⊆ V) |
20 | fnco 6623 | . . . 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))) | |
21 | 12, 17, 19, 20 | syl3anc 1372 | . . 3 ⊢ (𝜑 → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))) Fn (ℤ≥‘(♯‘𝑀))) |
22 | fzouzdisj 13615 | . . . 4 ⊢ ((0..^(♯‘𝑀)) ∩ (ℤ≥‘(♯‘𝑀))) = ∅ | |
23 | 22 | a1i 11 | . . 3 ⊢ (𝜑 → ((0..^(♯‘𝑀)) ∩ (ℤ≥‘(♯‘𝑀))) = ∅) |
24 | sseqfv1.4 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝑀))) | |
25 | fvun1 6937 | . . 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))) = ∅ ∧ 𝑁 ∈ (0..^(♯‘𝑀)))) → ((𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))))‘𝑁) = (𝑀‘𝑁)) | |
26 | 8, 21, 23, 24, 25 | syl112anc 1375 | . 2 ⊢ (𝜑 → ((𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))))‘𝑁) = (𝑀‘𝑁)) |
27 | 6, 26 | eqtrd 2777 | 1 ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (𝑀‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3448 ∪ cun 3913 ∩ cin 3914 ⊆ wss 3915 ∅c0 4287 {csn 4591 × cxp 5636 ◡ccnv 5637 ran crn 5639 “ cima 5641 ∘ ccom 5642 Fn wfn 6496 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 ∈ cmpo 7364 0cc0 11058 1c1 11059 − cmin 11392 ℕ0cn0 12420 ℤcz 12506 ℤ≥cuz 12770 ..^cfzo 13574 seqcseq 13913 ♯chash 14237 Word cword 14409 lastSclsw 14457 ++ cconcat 14465 ⟨“cs1 14490 seqstrcsseq 33023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-fzo 13575 df-seq 13914 df-hash 14238 df-word 14410 df-lsw 14458 df-s1 14491 df-sseq 33024 |
This theorem is referenced by: sseqfres 33033 fib0 33039 fib1 33040 |
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