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Mirrors > Home > MPE Home > Th. List > ccatws1len | Structured version Visualization version GIF version |
Description: The length of the concatenation of a word with a singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by AV, 4-Mar-2022.) |
Ref | Expression |
---|---|
ccatws1len | ⊢ (𝑊 ∈ Word 𝑉 → (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = ((♯‘𝑊) + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1cli 14585 | . . 3 ⊢ ⟨“𝑋”⟩ ∈ Word V | |
2 | ccatlen 14555 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑋”⟩ ∈ Word V) → (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = ((♯‘𝑊) + (♯‘⟨“𝑋”⟩))) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = ((♯‘𝑊) + (♯‘⟨“𝑋”⟩))) |
4 | s1len 14586 | . . 3 ⊢ (♯‘⟨“𝑋”⟩) = 1 | |
5 | 4 | oveq2i 7426 | . 2 ⊢ ((♯‘𝑊) + (♯‘⟨“𝑋”⟩)) = ((♯‘𝑊) + 1) |
6 | 3, 5 | eqtrdi 2781 | 1 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = ((♯‘𝑊) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ‘cfv 6542 (class class class)co 7415 1c1 11137 + caddc 11139 ♯chash 14319 Word cword 14494 ++ cconcat 14550 ⟨“cs1 14575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-fzo 13658 df-hash 14320 df-word 14495 df-concat 14551 df-s1 14576 |
This theorem is referenced by: ccatws1lenp1b 14601 wrdlenccats1lenm1 14602 ccatw2s1len 14605 ccatws1n0 14612 ccatw2s1p1 14616 ccatw2s1p2 14617 cats1un 14701 gsmsymgrfix 19385 gsmsymgreqlem2 19388 wlklenvclwlk 29511 wwlksext2clwwlk 29909 cycpmco2lem2 32891 cycpmco2lem3 32892 cycpmco2lem4 32893 cycpmco2lem5 32894 cycpmco2lem6 32895 cycpmco2 32897 sseqf 34068 signstfvneq0 34260 signshf 34276 |
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