Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pfxmpt | Structured version Visualization version GIF version |
Description: Value of the prefix extractor as a mapping. (Contributed by AV, 2-May-2020.) |
Ref | Expression |
---|---|
pfxmpt | ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 prefix 𝐿) = (𝑥 ∈ (0..^𝐿) ↦ (𝑆‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfznn0 13001 | . . 3 ⊢ (𝐿 ∈ (0...(♯‘𝑆)) → 𝐿 ∈ ℕ0) | |
2 | pfxval 14035 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr 〈0, 𝐿〉)) | |
3 | 1, 2 | sylan2 594 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 prefix 𝐿) = (𝑆 substr 〈0, 𝐿〉)) |
4 | simpl 485 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → 𝑆 ∈ Word 𝐴) | |
5 | 1 | adantl 484 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → 𝐿 ∈ ℕ0) |
6 | 0elfz 13005 | . . . 4 ⊢ (𝐿 ∈ ℕ0 → 0 ∈ (0...𝐿)) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → 0 ∈ (0...𝐿)) |
8 | simpr 487 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → 𝐿 ∈ (0...(♯‘𝑆))) | |
9 | swrdval2 14008 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 0 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr 〈0, 𝐿〉) = (𝑥 ∈ (0..^(𝐿 − 0)) ↦ (𝑆‘(𝑥 + 0)))) | |
10 | 4, 7, 8, 9 | syl3anc 1367 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr 〈0, 𝐿〉) = (𝑥 ∈ (0..^(𝐿 − 0)) ↦ (𝑆‘(𝑥 + 0)))) |
11 | nn0cn 11908 | . . . . . . 7 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℂ) | |
12 | 11 | subid1d 10986 | . . . . . 6 ⊢ (𝐿 ∈ ℕ0 → (𝐿 − 0) = 𝐿) |
13 | 1, 12 | syl 17 | . . . . 5 ⊢ (𝐿 ∈ (0...(♯‘𝑆)) → (𝐿 − 0) = 𝐿) |
14 | 13 | oveq2d 7172 | . . . 4 ⊢ (𝐿 ∈ (0...(♯‘𝑆)) → (0..^(𝐿 − 0)) = (0..^𝐿)) |
15 | 14 | adantl 484 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (0..^(𝐿 − 0)) = (0..^𝐿)) |
16 | elfzonn0 13083 | . . . . . 6 ⊢ (𝑥 ∈ (0..^(𝐿 − 0)) → 𝑥 ∈ ℕ0) | |
17 | nn0cn 11908 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℂ) | |
18 | 17 | addid1d 10840 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 → (𝑥 + 0) = 𝑥) |
19 | 16, 18 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ (0..^(𝐿 − 0)) → (𝑥 + 0) = 𝑥) |
20 | 19 | fveq2d 6674 | . . . 4 ⊢ (𝑥 ∈ (0..^(𝐿 − 0)) → (𝑆‘(𝑥 + 0)) = (𝑆‘𝑥)) |
21 | 20 | adantl 484 | . . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝐿 − 0))) → (𝑆‘(𝑥 + 0)) = (𝑆‘𝑥)) |
22 | 15, 21 | mpteq12dva 5150 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑥 ∈ (0..^(𝐿 − 0)) ↦ (𝑆‘(𝑥 + 0))) = (𝑥 ∈ (0..^𝐿) ↦ (𝑆‘𝑥))) |
23 | 3, 10, 22 | 3eqtrd 2860 | 1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 prefix 𝐿) = (𝑥 ∈ (0..^𝐿) ↦ (𝑆‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4573 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 0cc0 10537 + caddc 10540 − cmin 10870 ℕ0cn0 11898 ...cfz 12893 ..^cfzo 13034 ♯chash 13691 Word cword 13862 substr csubstr 14002 prefix cpfx 14032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-substr 14003 df-pfx 14033 |
This theorem is referenced by: pfxres 14041 pfxf 14042 psgnunilem5 18622 |
Copyright terms: Public domain | W3C validator |