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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lpadmax | Structured version Visualization version GIF version |
Description: Length of a left-padded word, in the general case, expressed with an if statement. (Contributed by Thierry Arnoux, 7-Aug-2023.) |
Ref | Expression |
---|---|
lpadlen.1 | ⊢ (𝜑 → 𝐿 ∈ ℕ0) |
lpadlen.2 | ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) |
lpadlen.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
Ref | Expression |
---|---|
lpadmax | ⊢ (𝜑 → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = if(𝐿 ≤ (♯‘𝑊), (♯‘𝑊), 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2748 | . 2 ⊢ ((♯‘𝑊) = if(𝐿 ≤ (♯‘𝑊), (♯‘𝑊), 𝐿) → ((♯‘((𝐶 leftpad 𝑊)‘𝐿)) = (♯‘𝑊) ↔ (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = if(𝐿 ≤ (♯‘𝑊), (♯‘𝑊), 𝐿))) | |
2 | eqeq2 2748 | . 2 ⊢ (𝐿 = if(𝐿 ≤ (♯‘𝑊), (♯‘𝑊), 𝐿) → ((♯‘((𝐶 leftpad 𝑊)‘𝐿)) = 𝐿 ↔ (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = if(𝐿 ≤ (♯‘𝑊), (♯‘𝑊), 𝐿))) | |
3 | lpadlen.1 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ ℕ0) | |
4 | 3 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐿 ≤ (♯‘𝑊)) → 𝐿 ∈ ℕ0) |
5 | lpadlen.2 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) | |
6 | 5 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐿 ≤ (♯‘𝑊)) → 𝑊 ∈ Word 𝑆) |
7 | lpadlen.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
8 | 7 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐿 ≤ (♯‘𝑊)) → 𝐶 ∈ 𝑆) |
9 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝐿 ≤ (♯‘𝑊)) → 𝐿 ≤ (♯‘𝑊)) | |
10 | 4, 6, 8, 9 | lpadlen1 33292 | . 2 ⊢ ((𝜑 ∧ 𝐿 ≤ (♯‘𝑊)) → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = (♯‘𝑊)) |
11 | 3 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → 𝐿 ∈ ℕ0) |
12 | 5 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → 𝑊 ∈ Word 𝑆) |
13 | 7 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → 𝐶 ∈ 𝑆) |
14 | lencl 14421 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑆 → (♯‘𝑊) ∈ ℕ0) | |
15 | 5, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ0) |
16 | 15 | nn0red 12474 | . . . . 5 ⊢ (𝜑 → (♯‘𝑊) ∈ ℝ) |
17 | 16 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → (♯‘𝑊) ∈ ℝ) |
18 | 11 | nn0red 12474 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → 𝐿 ∈ ℝ) |
19 | 3 | nn0red 12474 | . . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
20 | 16, 19 | ltnled 11302 | . . . . 5 ⊢ (𝜑 → ((♯‘𝑊) < 𝐿 ↔ ¬ 𝐿 ≤ (♯‘𝑊))) |
21 | 20 | biimpar 478 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → (♯‘𝑊) < 𝐿) |
22 | 17, 18, 21 | ltled 11303 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → (♯‘𝑊) ≤ 𝐿) |
23 | 11, 12, 13, 22 | lpadlen2 33294 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = 𝐿) |
24 | 1, 2, 10, 23 | ifbothda 4524 | 1 ⊢ (𝜑 → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = if(𝐿 ≤ (♯‘𝑊), (♯‘𝑊), 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ifcif 4486 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ℝcr 11050 < clt 11189 ≤ cle 11190 ℕ0cn0 12413 ♯chash 14230 Word cword 14402 leftpad clpad 33287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568 df-hash 14231 df-word 14403 df-concat 14459 df-lpad 33288 |
This theorem is referenced by: (None) |
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