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Mirrors > Home > MPE Home > Th. List > ccatrid | Structured version Visualization version GIF version |
Description: Concatenation of a word by the empty word on the right. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 1-May-2020.) |
Ref | Expression |
---|---|
ccatrid | ⊢ (𝑆 ∈ Word 𝐵 → (𝑆 ++ ∅) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrd0 13559 | . . . 4 ⊢ ∅ ∈ Word 𝐵 | |
2 | ccatvalfn 13601 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ ∅ ∈ Word 𝐵) → (𝑆 ++ ∅) Fn (0..^((♯‘𝑆) + (♯‘∅)))) | |
3 | 1, 2 | mpan2 683 | . . 3 ⊢ (𝑆 ∈ Word 𝐵 → (𝑆 ++ ∅) Fn (0..^((♯‘𝑆) + (♯‘∅)))) |
4 | hash0 13408 | . . . . . . 7 ⊢ (♯‘∅) = 0 | |
5 | 4 | oveq2i 6889 | . . . . . 6 ⊢ ((♯‘𝑆) + (♯‘∅)) = ((♯‘𝑆) + 0) |
6 | lencl 13553 | . . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐵 → (♯‘𝑆) ∈ ℕ0) | |
7 | 6 | nn0cnd 11642 | . . . . . . 7 ⊢ (𝑆 ∈ Word 𝐵 → (♯‘𝑆) ∈ ℂ) |
8 | 7 | addid1d 10526 | . . . . . 6 ⊢ (𝑆 ∈ Word 𝐵 → ((♯‘𝑆) + 0) = (♯‘𝑆)) |
9 | 5, 8 | syl5req 2846 | . . . . 5 ⊢ (𝑆 ∈ Word 𝐵 → (♯‘𝑆) = ((♯‘𝑆) + (♯‘∅))) |
10 | 9 | oveq2d 6894 | . . . 4 ⊢ (𝑆 ∈ Word 𝐵 → (0..^(♯‘𝑆)) = (0..^((♯‘𝑆) + (♯‘∅)))) |
11 | 10 | fneq2d 6193 | . . 3 ⊢ (𝑆 ∈ Word 𝐵 → ((𝑆 ++ ∅) Fn (0..^(♯‘𝑆)) ↔ (𝑆 ++ ∅) Fn (0..^((♯‘𝑆) + (♯‘∅))))) |
12 | 3, 11 | mpbird 249 | . 2 ⊢ (𝑆 ∈ Word 𝐵 → (𝑆 ++ ∅) Fn (0..^(♯‘𝑆))) |
13 | wrdfn 13548 | . 2 ⊢ (𝑆 ∈ Word 𝐵 → 𝑆 Fn (0..^(♯‘𝑆))) | |
14 | ccatval1 13597 | . . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ ∅ ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ ∅)‘𝑥) = (𝑆‘𝑥)) | |
15 | 1, 14 | mp3an2 1574 | . 2 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ ∅)‘𝑥) = (𝑆‘𝑥)) |
16 | 12, 13, 15 | eqfnfvd 6540 | 1 ⊢ (𝑆 ∈ Word 𝐵 → (𝑆 ++ ∅) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ∅c0 4115 Fn wfn 6096 ‘cfv 6101 (class class class)co 6878 0cc0 10224 + caddc 10227 ..^cfzo 12720 ♯chash 13370 Word cword 13534 ++ cconcat 13590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-fzo 12721 df-hash 13371 df-word 13535 df-concat 13591 |
This theorem is referenced by: lswccat0lsw 13612 swrdccat 13799 swrdccatOLD 13800 swrdccat3blem 13805 cshw0 13879 gsumccat 17693 frmdmnd 17712 frmd0 17713 efginvrel2 18453 efgredleme 18470 efgcpbllemb 18483 efgcpbl2 18485 frgpnabllem1 18591 signstfvc 31170 |
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