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| 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 34691 | . . . 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 772 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → 𝑤 = 𝑊) | |
| 4 | 3 | fveq2d 6909 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (♯‘𝑤) = (♯‘𝑊)) | 
| 5 | 4 | oveq2d 7448 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (𝑙 − (♯‘𝑤)) = (𝑙 − (♯‘𝑊))) | 
| 6 | 5 | oveq2d 7448 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (0..^(𝑙 − (♯‘𝑤))) = (0..^(𝑙 − (♯‘𝑊)))) | 
| 7 | simprl 770 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → 𝑐 = 𝐶) | |
| 8 | 7 | sneqd 4637 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → {𝑐} = {𝐶}) | 
| 9 | 6, 8 | xpeq12d 5715 | . . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → ((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) = ((0..^(𝑙 − (♯‘𝑊))) × {𝐶})) | 
| 10 | 9, 3 | oveq12d 7450 | . . . 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 3503 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) | 
| 14 | lpadval.2 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) | |
| 15 | 14 | elexd 3503 | . . 3 ⊢ (𝜑 → 𝑊 ∈ V) | 
| 16 | nn0ex 12534 | . . . . 5 ⊢ ℕ0 ∈ V | |
| 17 | 16 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) | 
| 18 | 17 | mptexd 7245 | . . 3 ⊢ (𝜑 → (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) ∈ V) | 
| 19 | 2, 11, 13, 15, 18 | ovmpod 7586 | . 2 ⊢ (𝜑 → (𝐶 leftpad 𝑊) = (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊))) | 
| 20 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → 𝑙 = 𝐿) | |
| 21 | 20 | oveq1d 7447 | . . . . 5 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → (𝑙 − (♯‘𝑊)) = (𝐿 − (♯‘𝑊))) | 
| 22 | 21 | oveq2d 7448 | . . . 4 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → (0..^(𝑙 − (♯‘𝑊))) = (0..^(𝐿 − (♯‘𝑊)))) | 
| 23 | 22 | xpeq1d 5713 | . . 3 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → ((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) = ((0..^(𝐿 − (♯‘𝑊))) × {𝐶})) | 
| 24 | 23 | oveq1d 7447 | . 2 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊) = (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) | 
| 25 | lpadval.1 | . 2 ⊢ (𝜑 → 𝐿 ∈ ℕ0) | |
| 26 | ovexd 7467 | . 2 ⊢ (𝜑 → (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊) ∈ V) | |
| 27 | 19, 24, 25, 26 | fvmptd 7022 | 1 ⊢ (𝜑 → ((𝐶 leftpad 𝑊)‘𝐿) = (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 {csn 4625 ↦ cmpt 5224 × cxp 5682 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 0cc0 11156 − cmin 11493 ℕ0cn0 12528 ..^cfzo 13695 ♯chash 14370 Word cword 14553 ++ cconcat 14609 leftpad clpad 34690 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-1cn 11214 ax-addcl 11216 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-nn 12268 df-n0 12529 df-lpad 34691 | 
| This theorem is referenced by: lpadlen1 34695 lpadlen2 34697 lpadleft 34699 lpadright 34700 | 
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