![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lpadval | Structured version Visualization version GIF version |
Description: Value of the leftpad function. (Contributed by Thierry Arnoux, 7-Aug-2023.) |
Ref | Expression |
---|---|
lpadval.1 | ⊢ (𝜑 → 𝐿 ∈ ℕ0) |
lpadval.2 | ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) |
lpadval.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
Ref | Expression |
---|---|
lpadval | ⊢ (𝜑 → ((𝐶 leftpad 𝑊)‘𝐿) = (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lpad 34215 | . . . 4 ⊢ leftpad = (𝑐 ∈ V, 𝑤 ∈ V ↦ (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) ++ 𝑤))) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → leftpad = (𝑐 ∈ V, 𝑤 ∈ V ↦ (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) ++ 𝑤)))) |
3 | simprr 770 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → 𝑤 = 𝑊) | |
4 | 3 | fveq2d 6888 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (♯‘𝑤) = (♯‘𝑊)) |
5 | 4 | oveq2d 7420 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (𝑙 − (♯‘𝑤)) = (𝑙 − (♯‘𝑊))) |
6 | 5 | oveq2d 7420 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (0..^(𝑙 − (♯‘𝑤))) = (0..^(𝑙 − (♯‘𝑊)))) |
7 | simprl 768 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → 𝑐 = 𝐶) | |
8 | 7 | sneqd 4635 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → {𝑐} = {𝐶}) |
9 | 6, 8 | xpeq12d 5700 | . . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → ((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) = ((0..^(𝑙 − (♯‘𝑊))) × {𝐶})) |
10 | 9, 3 | oveq12d 7422 | . . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) ++ 𝑤) = (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) |
11 | 10 | mpteq2dv 5243 | . . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) ++ 𝑤)) = (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊))) |
12 | lpadval.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
13 | 12 | elexd 3489 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
14 | lpadval.2 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) | |
15 | 14 | elexd 3489 | . . 3 ⊢ (𝜑 → 𝑊 ∈ V) |
16 | nn0ex 12479 | . . . . 5 ⊢ ℕ0 ∈ V | |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
18 | 17 | mptexd 7220 | . . 3 ⊢ (𝜑 → (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) ∈ V) |
19 | 2, 11, 13, 15, 18 | ovmpod 7555 | . 2 ⊢ (𝜑 → (𝐶 leftpad 𝑊) = (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊))) |
20 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → 𝑙 = 𝐿) | |
21 | 20 | oveq1d 7419 | . . . . 5 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → (𝑙 − (♯‘𝑊)) = (𝐿 − (♯‘𝑊))) |
22 | 21 | oveq2d 7420 | . . . 4 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → (0..^(𝑙 − (♯‘𝑊))) = (0..^(𝐿 − (♯‘𝑊)))) |
23 | 22 | xpeq1d 5698 | . . 3 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → ((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) = ((0..^(𝐿 − (♯‘𝑊))) × {𝐶})) |
24 | 23 | oveq1d 7419 | . 2 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊) = (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) |
25 | lpadval.1 | . 2 ⊢ (𝜑 → 𝐿 ∈ ℕ0) | |
26 | ovexd 7439 | . 2 ⊢ (𝜑 → (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊) ∈ V) | |
27 | 19, 24, 25, 26 | fvmptd 6998 | 1 ⊢ (𝜑 → ((𝐶 leftpad 𝑊)‘𝐿) = (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 {csn 4623 ↦ cmpt 5224 × cxp 5667 ‘cfv 6536 (class class class)co 7404 ∈ cmpo 7406 0cc0 11109 − cmin 11445 ℕ0cn0 12473 ..^cfzo 13630 ♯chash 14292 Word cword 14467 ++ cconcat 14523 leftpad clpad 34214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-1cn 11167 ax-addcl 11169 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-nn 12214 df-n0 12474 df-lpad 34215 |
This theorem is referenced by: lpadlen1 34219 lpadlen2 34221 lpadleft 34223 lpadright 34224 |
Copyright terms: Public domain | W3C validator |