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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.) |
Ref | Expression |
---|---|
ccatlen | ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → (#‘(𝑆 ++ 𝑇)) = ((#‘𝑆) + (#‘𝑇))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ccatfval 13211 | . . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇))) ↦ if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆)))))) | |
2 | 1 | fveq2d 6107 | . 2 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → (#‘(𝑆 ++ 𝑇)) = (#‘(𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇))) ↦ if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆))))))) |
3 | fvex 6113 | . . . . 5 ⊢ (𝑆‘𝑥) ∈ V | |
4 | fvex 6113 | . . . . 5 ⊢ (𝑇‘(𝑥 − (#‘𝑆))) ∈ V | |
5 | 3, 4 | ifex 4106 | . . . 4 ⊢ if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆)))) ∈ V |
6 | eqid 2610 | . . . 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 5935 | . . 3 ⊢ (𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇))) ↦ if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆))))) Fn (0..^((#‘𝑆) + (#‘𝑇))) |
8 | hashfn 13025 | . . 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 𝐵 ∧ 𝑇 ∈ Word 𝐵) → (#‘(𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇))) ↦ if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆)))))) = (#‘(0..^((#‘𝑆) + (#‘𝑇))))) |
10 | lencl 13179 | . . . 4 ⊢ (𝑆 ∈ Word 𝐵 → (#‘𝑆) ∈ ℕ0) | |
11 | lencl 13179 | . . . 4 ⊢ (𝑇 ∈ Word 𝐵 → (#‘𝑇) ∈ ℕ0) | |
12 | nn0addcl 11205 | . . . 4 ⊢ (((#‘𝑆) ∈ ℕ0 ∧ (#‘𝑇) ∈ ℕ0) → ((#‘𝑆) + (#‘𝑇)) ∈ ℕ0) | |
13 | 10, 11, 12 | syl2an 493 | . . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → ((#‘𝑆) + (#‘𝑇)) ∈ ℕ0) |
14 | hashfzo0 13077 | . . 3 ⊢ (((#‘𝑆) + (#‘𝑇)) ∈ ℕ0 → (#‘(0..^((#‘𝑆) + (#‘𝑇)))) = ((#‘𝑆) + (#‘𝑇))) | |
15 | 13, 14 | syl 17 | . 2 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → (#‘(0..^((#‘𝑆) + (#‘𝑇)))) = ((#‘𝑆) + (#‘𝑇))) |
16 | 2, 9, 15 | 3eqtrd 2648 | 1 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → (#‘(𝑆 ++ 𝑇)) = ((#‘𝑆) + (#‘𝑇))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ifcif 4036 ↦ cmpt 4643 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 0cc0 9815 + caddc 9818 − cmin 10145 ℕ0cn0 11169 ..^cfzo 12334 #chash 12979 Word cword 13146 ++ cconcat 13148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 |
This theorem is referenced by: elfzelfzccat 13217 ccatsymb 13219 ccatass 13224 lswccatn0lsw 13226 ccatws1len 13251 ccat2s1len 13253 ccatswrd 13308 swrdccat1 13309 swrdccat2 13310 lenrevcctswrd 13319 ccatopth 13322 ccatopth2 13323 swrdccatfn 13333 swrdccatin2 13338 swrdccatin12lem2c 13339 swrdccatin12 13342 spllen 13356 splfv1 13357 splfv2a 13358 splval2 13359 revccat 13366 cshwlen 13396 cats1len 13456 gsumccat 17201 psgnuni 17742 efginvrel2 17963 efgsval2 17969 efgsp1 17973 efgredleme 17979 efgredlemc 17981 efgcpbllemb 17991 pgpfaclem1 18303 psgnghm 19745 wwlknext 26252 wwlknextbi 26253 clwlkisclwwlk2 26318 clwwlkel 26321 wlklenvclwlk 26366 numclwlk1lem2fo 26622 ofcccat 29946 signstfvn 29972 signstfvp 29974 signstfvc 29977 signsvfn 29985 signshf 29991 elmrsubrn 30671 ccatpfx 40272 pfxccat1 40273 lenrevpfxcctswrd 40282 pfxccatin12 40288 1wlklenvclwlk 40863 wwlksnext 41099 wwlksnextbi 41100 clwlkclwwlk2 41212 clwwlksel 41221 av-numclwlk1lem2fo 41525 |
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