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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 31965 | . . . 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 6667 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (♯‘𝑤) = (♯‘𝑊)) |
5 | 4 | oveq2d 7165 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (𝑙 − (♯‘𝑤)) = (𝑙 − (♯‘𝑊))) |
6 | 5 | oveq2d 7165 | . . . . . 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 4572 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → {𝑐} = {𝐶}) |
9 | 6, 8 | xpeq12d 5579 | . . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → ((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) = ((0..^(𝑙 − (♯‘𝑊))) × {𝐶})) |
10 | 9, 3 | oveq12d 7167 | . . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑤 = 𝑊)) → (((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) ++ 𝑤) = (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) |
11 | 10 | mpteq2dv 5155 | . . 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 3511 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
14 | lpadval.2 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) | |
15 | 14 | elexd 3511 | . . 3 ⊢ (𝜑 → 𝑊 ∈ V) |
16 | nn0ex 11897 | . . . . 5 ⊢ ℕ0 ∈ V | |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
18 | 17 | mptexd 6980 | . . 3 ⊢ (𝜑 → (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) ∈ V) |
19 | 2, 11, 13, 15, 18 | ovmpod 7295 | . 2 ⊢ (𝜑 → (𝐶 leftpad 𝑊) = (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊))) |
20 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → 𝑙 = 𝐿) | |
21 | 20 | oveq1d 7164 | . . . . 5 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → (𝑙 − (♯‘𝑊)) = (𝐿 − (♯‘𝑊))) |
22 | 21 | oveq2d 7165 | . . . 4 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → (0..^(𝑙 − (♯‘𝑊))) = (0..^(𝐿 − (♯‘𝑊)))) |
23 | 22 | xpeq1d 5577 | . . 3 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → ((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) = ((0..^(𝐿 − (♯‘𝑊))) × {𝐶})) |
24 | 23 | oveq1d 7164 | . 2 ⊢ ((𝜑 ∧ 𝑙 = 𝐿) → (((0..^(𝑙 − (♯‘𝑊))) × {𝐶}) ++ 𝑊) = (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) |
25 | lpadval.1 | . 2 ⊢ (𝜑 → 𝐿 ∈ ℕ0) | |
26 | ovexd 7184 | . 2 ⊢ (𝜑 → (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊) ∈ V) | |
27 | 19, 24, 25, 26 | fvmptd 6768 | 1 ⊢ (𝜑 → ((𝐶 leftpad 𝑊)‘𝐿) = (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3491 {csn 4560 ↦ cmpt 5139 × cxp 5546 ‘cfv 6348 (class class class)co 7149 ∈ cmpo 7151 0cc0 10530 − cmin 10863 ℕ0cn0 11891 ..^cfzo 13030 ♯chash 13687 Word cword 13858 ++ cconcat 13915 leftpad clpad 31964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-1cn 10588 ax-addcl 10590 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-nn 11632 df-n0 11892 df-lpad 31965 |
This theorem is referenced by: lpadlen1 31969 lpadlen2 31971 lpadleft 31973 lpadright 31974 |
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