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Mirrors > Home > MPE Home > Th. List > Mathboxes > lpadlem2 | Structured version Visualization version GIF version |
Description: Lemma for the leftpad theorems. (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) |
lpadlen2.1 | ⊢ (𝜑 → (♯‘𝑊) ≤ 𝐿) |
Ref | Expression |
---|---|
lpadlem2 | ⊢ (𝜑 → (♯‘((0..^(𝐿 − (♯‘𝑊))) × {𝐶})) = (𝐿 − (♯‘𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzofi 13878 | . . . 4 ⊢ (0..^(𝐿 − (♯‘𝑊))) ∈ Fin | |
2 | snfi 8987 | . . . 4 ⊢ {𝐶} ∈ Fin | |
3 | hashxp 14333 | . . . 4 ⊢ (((0..^(𝐿 − (♯‘𝑊))) ∈ Fin ∧ {𝐶} ∈ Fin) → (♯‘((0..^(𝐿 − (♯‘𝑊))) × {𝐶})) = ((♯‘(0..^(𝐿 − (♯‘𝑊)))) · (♯‘{𝐶}))) | |
4 | 1, 2, 3 | mp2an 690 | . . 3 ⊢ (♯‘((0..^(𝐿 − (♯‘𝑊))) × {𝐶})) = ((♯‘(0..^(𝐿 − (♯‘𝑊)))) · (♯‘{𝐶})) |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → (♯‘((0..^(𝐿 − (♯‘𝑊))) × {𝐶})) = ((♯‘(0..^(𝐿 − (♯‘𝑊)))) · (♯‘{𝐶}))) |
6 | lpadlen.2 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) | |
7 | lencl 14420 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑆 → (♯‘𝑊) ∈ ℕ0) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ0) |
9 | lpadlen.1 | . . . . 5 ⊢ (𝜑 → 𝐿 ∈ ℕ0) | |
10 | lpadlen2.1 | . . . . 5 ⊢ (𝜑 → (♯‘𝑊) ≤ 𝐿) | |
11 | nn0sub2 12563 | . . . . 5 ⊢ (((♯‘𝑊) ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ (♯‘𝑊) ≤ 𝐿) → (𝐿 − (♯‘𝑊)) ∈ ℕ0) | |
12 | 8, 9, 10, 11 | syl3anc 1371 | . . . 4 ⊢ (𝜑 → (𝐿 − (♯‘𝑊)) ∈ ℕ0) |
13 | hashfzo0 14329 | . . . 4 ⊢ ((𝐿 − (♯‘𝑊)) ∈ ℕ0 → (♯‘(0..^(𝐿 − (♯‘𝑊)))) = (𝐿 − (♯‘𝑊))) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → (♯‘(0..^(𝐿 − (♯‘𝑊)))) = (𝐿 − (♯‘𝑊))) |
15 | lpadlen.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
16 | hashsng 14268 | . . . 4 ⊢ (𝐶 ∈ 𝑆 → (♯‘{𝐶}) = 1) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ (𝜑 → (♯‘{𝐶}) = 1) |
18 | 14, 17 | oveq12d 7374 | . 2 ⊢ (𝜑 → ((♯‘(0..^(𝐿 − (♯‘𝑊)))) · (♯‘{𝐶})) = ((𝐿 − (♯‘𝑊)) · 1)) |
19 | 12 | nn0cnd 12474 | . . 3 ⊢ (𝜑 → (𝐿 − (♯‘𝑊)) ∈ ℂ) |
20 | 19 | mulid1d 11171 | . 2 ⊢ (𝜑 → ((𝐿 − (♯‘𝑊)) · 1) = (𝐿 − (♯‘𝑊))) |
21 | 5, 18, 20 | 3eqtrd 2780 | 1 ⊢ (𝜑 → (♯‘((0..^(𝐿 − (♯‘𝑊))) × {𝐶})) = (𝐿 − (♯‘𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {csn 4586 class class class wbr 5105 × cxp 5631 ‘cfv 6496 (class class class)co 7356 Fincfn 8882 0cc0 11050 1c1 11051 · cmul 11055 ≤ cle 11189 − cmin 11384 ℕ0cn0 12412 ..^cfzo 13566 ♯chash 14229 Word cword 14401 |
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 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
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 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-oadd 8415 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-dju 9836 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-n0 12413 df-z 12499 df-uz 12763 df-fz 13424 df-fzo 13567 df-hash 14230 df-word 14402 |
This theorem is referenced by: lpadlen2 33285 lpadleft 33287 lpadright 33288 |
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