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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 2740 | . 2 ⊢ ((♯‘𝑊) = if(𝐿 ≤ (♯‘𝑊), (♯‘𝑊), 𝐿) → ((♯‘((𝐶 leftpad 𝑊)‘𝐿)) = (♯‘𝑊) ↔ (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = if(𝐿 ≤ (♯‘𝑊), (♯‘𝑊), 𝐿))) | |
2 | eqeq2 2740 | . 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 479 | . . 3 ⊢ ((𝜑 ∧ 𝐿 ≤ (♯‘𝑊)) → 𝐿 ∈ ℕ0) |
5 | lpadlen.2 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) | |
6 | 5 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐿 ≤ (♯‘𝑊)) → 𝑊 ∈ Word 𝑆) |
7 | lpadlen.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
8 | 7 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐿 ≤ (♯‘𝑊)) → 𝐶 ∈ 𝑆) |
9 | simpr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐿 ≤ (♯‘𝑊)) → 𝐿 ≤ (♯‘𝑊)) | |
10 | 4, 6, 8, 9 | lpadlen1 34344 | . 2 ⊢ ((𝜑 ∧ 𝐿 ≤ (♯‘𝑊)) → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = (♯‘𝑊)) |
11 | 3 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → 𝐿 ∈ ℕ0) |
12 | 5 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → 𝑊 ∈ Word 𝑆) |
13 | 7 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → 𝐶 ∈ 𝑆) |
14 | lencl 14523 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑆 → (♯‘𝑊) ∈ ℕ0) | |
15 | 5, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ0) |
16 | 15 | nn0red 12571 | . . . . 5 ⊢ (𝜑 → (♯‘𝑊) ∈ ℝ) |
17 | 16 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → (♯‘𝑊) ∈ ℝ) |
18 | 11 | nn0red 12571 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → 𝐿 ∈ ℝ) |
19 | 3 | nn0red 12571 | . . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
20 | 16, 19 | ltnled 11399 | . . . . 5 ⊢ (𝜑 → ((♯‘𝑊) < 𝐿 ↔ ¬ 𝐿 ≤ (♯‘𝑊))) |
21 | 20 | biimpar 476 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → (♯‘𝑊) < 𝐿) |
22 | 17, 18, 21 | ltled 11400 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → (♯‘𝑊) ≤ 𝐿) |
23 | 11, 12, 13, 22 | lpadlen2 34346 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = 𝐿) |
24 | 1, 2, 10, 23 | ifbothda 4570 | 1 ⊢ (𝜑 → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = if(𝐿 ≤ (♯‘𝑊), (♯‘𝑊), 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ifcif 4532 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 ℝcr 11145 < clt 11286 ≤ cle 11287 ℕ0cn0 12510 ♯chash 14329 Word cword 14504 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-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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-nel 3044 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-int 4954 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-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-oadd 8497 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-hash 14330 df-word 14505 df-concat 14561 df-lpad 34340 |
This theorem is referenced by: (None) |
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