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Mathbox for Thierry Arnoux |
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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 34340 | . . . 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 771 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → 𝑤 = 𝑊) | |
4 | 3 | fveq2d 6906 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (♯‘𝑤) = (♯‘𝑊)) |
5 | 4 | oveq2d 7442 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (𝑙 − (♯‘𝑤)) = (𝑙 − (♯‘𝑊))) |
6 | 5 | oveq2d 7442 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (0..^(𝑙 − (♯‘𝑤))) = (0..^(𝑙 − (♯‘𝑊)))) |
7 | simprl 769 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → 𝑐 = 𝐶) | |
8 | 7 | sneqd 4644 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → {𝑐} = {𝐶}) |
9 | 6, 8 | xpeq12d 5713 | . . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → ((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) = ((0..^(𝑙 − (♯‘𝑊))) × {𝐶})) |
10 | 9, 3 | oveq12d 7444 | . . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) ++ 𝑤) = (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) |
11 | 10 | mpteq2dv 5254 | . . 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 3494 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
14 | lpadval.2 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) | |
15 | 14 | elexd 3494 | . . 3 ⊢ (𝜑 → 𝑊 ∈ V) |
16 | nn0ex 12516 | . . . . 5 ⊢ ℕ0 ∈ V | |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
18 | 17 | mptexd 7242 | . . 3 ⊢ (𝜑 → (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) ∈ V) |
19 | 2, 11, 13, 15, 18 | ovmpod 7579 | . 2 ⊢ (𝜑 → (𝐶 leftpad 𝑊) = (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊))) |
20 | simpr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → 𝑙 = 𝐿) | |
21 | 20 | oveq1d 7441 | . . . . 5 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → (𝑙 − (♯‘𝑊)) = (𝐿 − (♯‘𝑊))) |
22 | 21 | oveq2d 7442 | . . . 4 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → (0..^(𝑙 − (♯‘𝑊))) = (0..^(𝐿 − (♯‘𝑊)))) |
23 | 22 | xpeq1d 5711 | . . 3 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → ((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) = ((0..^(𝐿 − (♯‘𝑊))) × {𝐶})) |
24 | 23 | oveq1d 7441 | . 2 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊) = (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) |
25 | lpadval.1 | . 2 ⊢ (𝜑 → 𝐿 ∈ ℕ0) | |
26 | ovexd 7461 | . 2 ⊢ (𝜑 → (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊) ∈ V) | |
27 | 19, 24, 25, 26 | fvmptd 7017 | 1 ⊢ (𝜑 → ((𝐶 leftpad 𝑊)‘𝐿) = (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 {csn 4632 ↦ cmpt 5235 × cxp 5680 ‘cfv 6553 (class class class)co 7426 ∈ cmpo 7428 0cc0 11146 − cmin 11482 ℕ0cn0 12510 ..^cfzo 13667 ♯chash 14329 Word cword 14504 ++ cconcat 14560 leftpad clpad 34339 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-1cn 11204 ax-addcl 11206 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-nn 12251 df-n0 12511 df-lpad 34340 |
This theorem is referenced by: lpadlen1 34344 lpadlen2 34346 lpadleft 34348 lpadright 34349 |
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