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| Mirrors > Home > MPE Home > Th. List > ccatlen | Structured version Visualization version GIF version | ||
| Description: The length of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by JJ, 1-Jan-2024.) |
| Ref | Expression |
|---|---|
| ccatlen | ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑆) + (♯‘𝑇))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ccatfval 14611 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆)))))) | |
| 2 | 1 | fveq2d 6910 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → (♯‘(𝑆 ++ 𝑇)) = (♯‘(𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆))))))) |
| 3 | fvex 6919 | . . . . 5 ⊢ (𝑆‘𝑥) ∈ V | |
| 4 | fvex 6919 | . . . . 5 ⊢ (𝑇‘(𝑥 − (♯‘𝑆))) ∈ V | |
| 5 | 3, 4 | ifex 4576 | . . . 4 ⊢ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆)))) ∈ V |
| 6 | eqid 2737 | . . . 4 ⊢ (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆))))) = (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆))))) | |
| 7 | 5, 6 | fnmpti 6711 | . . 3 ⊢ (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆))))) Fn (0..^((♯‘𝑆) + (♯‘𝑇))) |
| 8 | hashfn 14414 | . . 3 ⊢ ((𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆))))) Fn (0..^((♯‘𝑆) + (♯‘𝑇))) → (♯‘(𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆)))))) = (♯‘(0..^((♯‘𝑆) + (♯‘𝑇))))) | |
| 9 | 7, 8 | mp1i 13 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → (♯‘(𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆)))))) = (♯‘(0..^((♯‘𝑆) + (♯‘𝑇))))) |
| 10 | lencl 14571 | . . . 4 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℕ0) | |
| 11 | lencl 14571 | . . . 4 ⊢ (𝑇 ∈ Word 𝐵 → (♯‘𝑇) ∈ ℕ0) | |
| 12 | nn0addcl 12561 | . . . 4 ⊢ (((♯‘𝑆) ∈ ℕ0 ∧ (♯‘𝑇) ∈ ℕ0) → ((♯‘𝑆) + (♯‘𝑇)) ∈ ℕ0) | |
| 13 | 10, 11, 12 | syl2an 596 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → ((♯‘𝑆) + (♯‘𝑇)) ∈ ℕ0) |
| 14 | hashfzo0 14469 | . . 3 ⊢ (((♯‘𝑆) + (♯‘𝑇)) ∈ ℕ0 → (♯‘(0..^((♯‘𝑆) + (♯‘𝑇)))) = ((♯‘𝑆) + (♯‘𝑇))) | |
| 15 | 13, 14 | syl 17 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → (♯‘(0..^((♯‘𝑆) + (♯‘𝑇)))) = ((♯‘𝑆) + (♯‘𝑇))) |
| 16 | 2, 9, 15 | 3eqtrd 2781 | 1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑆) + (♯‘𝑇))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ifcif 4525 ↦ cmpt 5225 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 0cc0 11155 + caddc 11158 − cmin 11492 ℕ0cn0 12526 ..^cfzo 13694 ♯chash 14369 Word cword 14552 ++ cconcat 14608 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 |
| This theorem is referenced by: ccat0 14614 elfzelfzccat 14618 ccatsymb 14620 ccatass 14626 lswccatn0lsw 14629 ccatws1len 14658 ccat2s1len 14661 ccatswrd 14706 swrdccat2 14707 ccatpfx 14739 pfxccat1 14740 lenrevpfxcctswrd 14750 ccatopth 14754 ccatopth2 14755 swrdccatfn 14762 swrdccatin2 14767 pfxccatin12lem2c 14768 spllen 14792 splfv1 14793 splfv2a 14794 splval2 14795 revccat 14804 cshwlen 14837 cats1len 14899 gsumsgrpccat 18853 psgnuni 19517 efginvrel2 19745 efgsval2 19751 efgsp1 19755 efgredleme 19761 efgredlemc 19763 efgcpbllemb 19773 pgpfaclem1 20101 psgnghm 21598 wwlksnext 29913 wwlksnextbi 29914 clwwlkccatlem 30008 clwlkclwwlk2 30022 clwwlkel 30065 clwwlkwwlksb 30073 clwwlknccat 30082 ccatf1 32933 ccatdmss 32934 splfv3 32943 gsumwrd2dccatlem 33069 cycpmco2lem3 33148 cycpmco2lem4 33149 cycpmco2lem5 33150 cycpmco2lem6 33151 cycpmco2 33153 ofcccat 34558 signstfvn 34584 signstfvp 34586 signstfvc 34589 signsvfn 34597 signshf 34603 lpadlen2 34696 elmrsubrn 35525 ccatcan2d 42292 frlmfzoccat 42515 |
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