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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 13339 | . . . 4 ⊢ (0..^(𝐿 − (♯‘𝑊))) ∈ Fin | |
2 | snfi 8587 | . . . 4 ⊢ {𝐶} ∈ Fin | |
3 | hashxp 13792 | . . . 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 13877 | . . . . . 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 12037 | . . . . 5 ⊢ (((♯‘𝑊) ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ (♯‘𝑊) ≤ 𝐿) → (𝐿 − (♯‘𝑊)) ∈ ℕ0) | |
12 | 8, 9, 10, 11 | syl3anc 1366 | . . . 4 ⊢ (𝜑 → (𝐿 − (♯‘𝑊)) ∈ ℕ0) |
13 | hashfzo0 13788 | . . . 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 13727 | . . . 4 ⊢ (𝐶 ∈ 𝑆 → (♯‘{𝐶}) = 1) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ (𝜑 → (♯‘{𝐶}) = 1) |
18 | 14, 17 | oveq12d 7167 | . 2 ⊢ (𝜑 → ((♯‘(0..^(𝐿 − (♯‘𝑊)))) · (♯‘{𝐶})) = ((𝐿 − (♯‘𝑊)) · 1)) |
19 | 12 | nn0cnd 11951 | . . 3 ⊢ (𝜑 → (𝐿 − (♯‘𝑊)) ∈ ℂ) |
20 | 19 | mulid1d 10651 | . 2 ⊢ (𝜑 → ((𝐿 − (♯‘𝑊)) · 1) = (𝐿 − (♯‘𝑊))) |
21 | 5, 18, 20 | 3eqtrd 2859 | 1 ⊢ (𝜑 → (♯‘((0..^(𝐿 − (♯‘𝑊))) × {𝐶})) = (𝐿 − (♯‘𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 {csn 4560 class class class wbr 5059 × cxp 5546 ‘cfv 6348 (class class class)co 7149 Fincfn 8502 0cc0 10530 1c1 10531 · cmul 10535 ≤ cle 10669 − cmin 10863 ℕ0cn0 11891 ..^cfzo 13030 ♯chash 13687 Word cword 13858 |
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-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
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-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 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-int 4870 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-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-dju 9323 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-fzo 13031 df-hash 13688 df-word 13859 |
This theorem is referenced by: lpadlen2 31971 lpadleft 31973 lpadright 31974 |
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