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Mirrors > Home > MPE Home > Th. List > s3fv1 | Structured version Visualization version GIF version |
Description: Extract the second symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.) |
Ref | Expression |
---|---|
s3fv1 | ⊢ (𝐵 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-s3 13970 | . 2 ⊢ 〈“𝐴𝐵𝐶”〉 = (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) | |
2 | s2cli 14001 | . 2 ⊢ 〈“𝐴𝐵”〉 ∈ Word V | |
3 | s2len 14010 | . 2 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 | |
4 | s2fv1 14009 | . 2 ⊢ (𝐵 ∈ 𝑉 → (〈“𝐴𝐵”〉‘1) = 𝐵) | |
5 | 1nn0 11636 | . 2 ⊢ 1 ∈ ℕ0 | |
6 | 1lt2 11529 | . 2 ⊢ 1 < 2 | |
7 | 1, 2, 3, 4, 5, 6 | cats1fv 13980 | 1 ⊢ (𝐵 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ‘cfv 6123 1c1 10253 2c2 11406 〈“cs2 13962 〈“cs3 13963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-hash 13411 df-word 13575 df-concat 13631 df-s1 13656 df-s2 13969 df-s3 13970 |
This theorem is referenced by: s4fv1 14017 eqwrds3 14083 uncfval 17227 trgcgrg 25827 israg 26009 iscgra 26118 isinag 26147 isleag 26151 iseqlg 26166 2wlkdlem3 27256 umgr2adedgwlk 27274 midwwlks2s3 27281 wwlks2onv 27282 umgrwwlks2on 27286 elwwlks2 27295 elwspths2spth 27296 wlk2v2elem2 27521 3wlkdlem8 27532 fusgr2wsp2nb 27704 circlemethhgt 31259 |
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