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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 34345 | . . 3 ⊢ (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))))) |
6 | 5 | fveq1d 6921 | . 2 ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = ((𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))))‘𝑁)) |
7 | wrdfn 14572 | . . . 4 ⊢ (𝑀 ∈ Word 𝑆 → 𝑀 Fn (0..^(♯‘𝑀))) | |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 Fn (0..^(♯‘𝑀))) |
9 | fvex 6932 | . . . . . 6 ⊢ (𝑥‘((♯‘𝑥) − 1)) ∈ V | |
10 | df-lsw 14607 | . . . . . 6 ⊢ lastS = (𝑥 ∈ V ↦ (𝑥‘((♯‘𝑥) − 1))) | |
11 | 9, 10 | fnmpti 6722 | . . . . 5 ⊢ lastS Fn V |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → lastS Fn V) |
13 | lencl 14577 | . . . . . . 7 ⊢ (𝑀 ∈ Word 𝑆 → (♯‘𝑀) ∈ ℕ0) | |
14 | 2, 13 | syl 17 | . . . . . 6 ⊢ (𝜑 → (♯‘𝑀) ∈ ℕ0) |
15 | 14 | nn0zd 12661 | . . . . 5 ⊢ (𝜑 → (♯‘𝑀) ∈ ℤ) |
16 | seqfn 14060 | . . . . 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 4027 | . . . . 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 6696 | . . . 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 13748 | . . . 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 7011 | . . 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 2774 | 1 ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (𝑀‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2103 Vcvv 3482 ∪ cun 3968 ∩ cin 3969 ⊆ wss 3970 ∅c0 4347 {csn 4648 × cxp 5697 ◡ccnv 5698 ran crn 5700 “ cima 5702 ∘ ccom 5703 Fn wfn 6567 ⟶wf 6568 ‘cfv 6572 (class class class)co 7445 ∈ cmpo 7447 0cc0 11180 1c1 11181 − cmin 11516 ℕ0cn0 12549 ℤcz 12635 ℤ≥cuz 12899 ..^cfzo 13707 seqcseq 14048 ♯chash 14375 Word cword 14558 lastSclsw 14606 ++ cconcat 14614 〈“cs1 14639 seqstrcsseq 34340 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-inf2 9706 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-n0 12550 df-z 12636 df-uz 12900 df-fz 13564 df-fzo 13708 df-seq 14049 df-hash 14376 df-word 14559 df-lsw 14607 df-s1 14640 df-sseq 34341 |
This theorem is referenced by: sseqfres 34350 fib0 34356 fib1 34357 |
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