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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 34355 | . . 3 ⊢ (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))))) |
6 | 5 | fveq1d 6924 | . 2 ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = ((𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))))‘𝑁)) |
7 | wrdfn 14578 | . . . 4 ⊢ (𝑀 ∈ Word 𝑆 → 𝑀 Fn (0..^(♯‘𝑀))) | |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 Fn (0..^(♯‘𝑀))) |
9 | fvex 6935 | . . . . . 6 ⊢ (𝑥‘((♯‘𝑥) − 1)) ∈ V | |
10 | df-lsw 14613 | . . . . . 6 ⊢ lastS = (𝑥 ∈ V ↦ (𝑥‘((♯‘𝑥) − 1))) | |
11 | 9, 10 | fnmpti 6725 | . . . . 5 ⊢ lastS Fn V |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → lastS Fn V) |
13 | lencl 14583 | . . . . . . 7 ⊢ (𝑀 ∈ Word 𝑆 → (♯‘𝑀) ∈ ℕ0) | |
14 | 2, 13 | syl 17 | . . . . . 6 ⊢ (𝜑 → (♯‘𝑀) ∈ ℕ0) |
15 | 14 | nn0zd 12667 | . . . . 5 ⊢ (𝜑 → (♯‘𝑀) ∈ ℤ) |
16 | seqfn 14066 | . . . . 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 4033 | . . . . 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 6699 | . . . 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 1371 | . . 3 ⊢ (𝜑 → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))) Fn (ℤ≥‘(♯‘𝑀))) |
22 | fzouzdisj 13754 | . . . 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 7015 | . . 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 1374 | . 2 ⊢ (𝜑 → ((𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))))‘𝑁) = (𝑀‘𝑁)) |
27 | 6, 26 | eqtrd 2780 | 1 ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (𝑀‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 Vcvv 3488 ∪ cun 3974 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 {csn 4648 × cxp 5698 ◡ccnv 5699 ran crn 5701 “ cima 5703 ∘ ccom 5704 Fn wfn 6570 ⟶wf 6571 ‘cfv 6575 (class class class)co 7450 ∈ cmpo 7452 0cc0 11186 1c1 11187 − cmin 11522 ℕ0cn0 12555 ℤcz 12641 ℤ≥cuz 12905 ..^cfzo 13713 seqcseq 14054 ♯chash 14381 Word cword 14564 lastSclsw 14612 ++ cconcat 14620 〈“cs1 14645 seqstrcsseq 34350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-inf2 9712 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-er 8765 df-map 8888 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-card 10010 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-n0 12556 df-z 12642 df-uz 12906 df-fz 13570 df-fzo 13714 df-seq 14055 df-hash 14382 df-word 14565 df-lsw 14613 df-s1 14646 df-sseq 34351 |
This theorem is referenced by: sseqfres 34360 fib0 34366 fib1 34367 |
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