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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 34669 | . . . 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 773 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → 𝑤 = 𝑊) | |
4 | 3 | fveq2d 6911 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (♯‘𝑤) = (♯‘𝑊)) |
5 | 4 | oveq2d 7447 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (𝑙 − (♯‘𝑤)) = (𝑙 − (♯‘𝑊))) |
6 | 5 | oveq2d 7447 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (0..^(𝑙 − (♯‘𝑤))) = (0..^(𝑙 − (♯‘𝑊)))) |
7 | simprl 771 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → 𝑐 = 𝐶) | |
8 | 7 | sneqd 4643 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → {𝑐} = {𝐶}) |
9 | 6, 8 | xpeq12d 5720 | . . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → ((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) = ((0..^(𝑙 − (♯‘𝑊))) × {𝐶})) |
10 | 9, 3 | oveq12d 7449 | . . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) ++ 𝑤) = (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) |
11 | 10 | mpteq2dv 5250 | . . 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 3502 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
14 | lpadval.2 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) | |
15 | 14 | elexd 3502 | . . 3 ⊢ (𝜑 → 𝑊 ∈ V) |
16 | nn0ex 12530 | . . . . 5 ⊢ ℕ0 ∈ V | |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
18 | 17 | mptexd 7244 | . . 3 ⊢ (𝜑 → (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) ∈ V) |
19 | 2, 11, 13, 15, 18 | ovmpod 7585 | . 2 ⊢ (𝜑 → (𝐶 leftpad 𝑊) = (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊))) |
20 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → 𝑙 = 𝐿) | |
21 | 20 | oveq1d 7446 | . . . . 5 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → (𝑙 − (♯‘𝑊)) = (𝐿 − (♯‘𝑊))) |
22 | 21 | oveq2d 7447 | . . . 4 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → (0..^(𝑙 − (♯‘𝑊))) = (0..^(𝐿 − (♯‘𝑊)))) |
23 | 22 | xpeq1d 5718 | . . 3 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → ((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) = ((0..^(𝐿 − (♯‘𝑊))) × {𝐶})) |
24 | 23 | oveq1d 7446 | . 2 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊) = (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) |
25 | lpadval.1 | . 2 ⊢ (𝜑 → 𝐿 ∈ ℕ0) | |
26 | ovexd 7466 | . 2 ⊢ (𝜑 → (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊) ∈ V) | |
27 | 19, 24, 25, 26 | fvmptd 7023 | 1 ⊢ (𝜑 → ((𝐶 leftpad 𝑊)‘𝐿) = (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 ↦ cmpt 5231 × cxp 5687 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 0cc0 11153 − cmin 11490 ℕ0cn0 12524 ..^cfzo 13691 ♯chash 14366 Word cword 14549 ++ cconcat 14605 leftpad clpad 34668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-n0 12525 df-lpad 34669 |
This theorem is referenced by: lpadlen1 34673 lpadlen2 34675 lpadleft 34677 lpadright 34678 |
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