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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 31756 | . . 3 ⊢ (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))))) |
6 | 5 | fveq1d 6647 | . 2 ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = ((𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))))‘𝑁)) |
7 | wrdfn 13871 | . . . 4 ⊢ (𝑀 ∈ Word 𝑆 → 𝑀 Fn (0..^(♯‘𝑀))) | |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 Fn (0..^(♯‘𝑀))) |
9 | fvex 6658 | . . . . . 6 ⊢ (𝑥‘((♯‘𝑥) − 1)) ∈ V | |
10 | df-lsw 13906 | . . . . . 6 ⊢ lastS = (𝑥 ∈ V ↦ (𝑥‘((♯‘𝑥) − 1))) | |
11 | 9, 10 | fnmpti 6463 | . . . . 5 ⊢ lastS Fn V |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → lastS Fn V) |
13 | lencl 13876 | . . . . . . 7 ⊢ (𝑀 ∈ Word 𝑆 → (♯‘𝑀) ∈ ℕ0) | |
14 | 2, 13 | syl 17 | . . . . . 6 ⊢ (𝜑 → (♯‘𝑀) ∈ ℕ0) |
15 | 14 | nn0zd 12073 | . . . . 5 ⊢ (𝜑 → (♯‘𝑀) ∈ ℤ) |
16 | seqfn 13376 | . . . . 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 3939 | . . . . 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 6437 | . . . 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 1368 | . . 3 ⊢ (𝜑 → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))) Fn (ℤ≥‘(♯‘𝑀))) |
22 | fzouzdisj 13068 | . . . 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 6729 | . . 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 1371 | . 2 ⊢ (𝜑 → ((𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))))‘𝑁) = (𝑀‘𝑁)) |
27 | 6, 26 | eqtrd 2833 | 1 ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (𝑀‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 × cxp 5517 ◡ccnv 5518 ran crn 5520 “ cima 5522 ∘ ccom 5523 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 0cc0 10526 1c1 10527 − cmin 10859 ℕ0cn0 11885 ℤcz 11969 ℤ≥cuz 12231 ..^cfzo 13028 seqcseq 13364 ♯chash 13686 Word cword 13857 lastSclsw 13905 ++ cconcat 13913 〈“cs1 13940 seqstrcsseq 31751 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-word 13858 df-lsw 13906 df-s1 13941 df-sseq 31752 |
This theorem is referenced by: sseqfres 31761 fib0 31767 fib1 31768 |
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