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Mirrors > Home > MPE Home > Th. List > cats1fvn | Structured version Visualization version GIF version |
Description: The last symbol of a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
cats1cld.1 | ⊢ 𝑇 = (𝑆 ++ 〈“𝑋”〉) |
cats1cli.2 | ⊢ 𝑆 ∈ Word V |
cats1fvn.3 | ⊢ (♯‘𝑆) = 𝑀 |
Ref | Expression |
---|---|
cats1fvn | ⊢ (𝑋 ∈ 𝑉 → (𝑇‘𝑀) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cats1cld.1 | . . . 4 ⊢ 𝑇 = (𝑆 ++ 〈“𝑋”〉) | |
2 | cats1fvn.3 | . . . . . 6 ⊢ (♯‘𝑆) = 𝑀 | |
3 | 2 | oveq2i 6921 | . . . . 5 ⊢ (0 + (♯‘𝑆)) = (0 + 𝑀) |
4 | cats1cli.2 | . . . . . . . . 9 ⊢ 𝑆 ∈ Word V | |
5 | lencl 13600 | . . . . . . . . 9 ⊢ (𝑆 ∈ Word V → (♯‘𝑆) ∈ ℕ0) | |
6 | 4, 5 | ax-mp 5 | . . . . . . . 8 ⊢ (♯‘𝑆) ∈ ℕ0 |
7 | 2, 6 | eqeltrri 2903 | . . . . . . 7 ⊢ 𝑀 ∈ ℕ0 |
8 | 7 | nn0cni 11638 | . . . . . 6 ⊢ 𝑀 ∈ ℂ |
9 | 8 | addid2i 10550 | . . . . 5 ⊢ (0 + 𝑀) = 𝑀 |
10 | 3, 9 | eqtr2i 2850 | . . . 4 ⊢ 𝑀 = (0 + (♯‘𝑆)) |
11 | 1, 10 | fveq12i 6443 | . . 3 ⊢ (𝑇‘𝑀) = ((𝑆 ++ 〈“𝑋”〉)‘(0 + (♯‘𝑆))) |
12 | s1cli 13672 | . . . 4 ⊢ 〈“𝑋”〉 ∈ Word V | |
13 | s1len 13673 | . . . . . 6 ⊢ (♯‘〈“𝑋”〉) = 1 | |
14 | 1nn 11370 | . . . . . 6 ⊢ 1 ∈ ℕ | |
15 | 13, 14 | eqeltri 2902 | . . . . 5 ⊢ (♯‘〈“𝑋”〉) ∈ ℕ |
16 | lbfzo0 12810 | . . . . 5 ⊢ (0 ∈ (0..^(♯‘〈“𝑋”〉)) ↔ (♯‘〈“𝑋”〉) ∈ ℕ) | |
17 | 15, 16 | mpbir 223 | . . . 4 ⊢ 0 ∈ (0..^(♯‘〈“𝑋”〉)) |
18 | ccatval3 13646 | . . . 4 ⊢ ((𝑆 ∈ Word V ∧ 〈“𝑋”〉 ∈ Word V ∧ 0 ∈ (0..^(♯‘〈“𝑋”〉))) → ((𝑆 ++ 〈“𝑋”〉)‘(0 + (♯‘𝑆))) = (〈“𝑋”〉‘0)) | |
19 | 4, 12, 17, 18 | mp3an 1589 | . . 3 ⊢ ((𝑆 ++ 〈“𝑋”〉)‘(0 + (♯‘𝑆))) = (〈“𝑋”〉‘0) |
20 | 11, 19 | eqtri 2849 | . 2 ⊢ (𝑇‘𝑀) = (〈“𝑋”〉‘0) |
21 | s1fv 13677 | . 2 ⊢ (𝑋 ∈ 𝑉 → (〈“𝑋”〉‘0) = 𝑋) | |
22 | 20, 21 | syl5eq 2873 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝑇‘𝑀) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ‘cfv 6127 (class class class)co 6910 0cc0 10259 1c1 10260 + caddc 10262 ℕcn 11357 ℕ0cn0 11625 ..^cfzo 12767 ♯chash 13417 Word cword 13581 ++ cconcat 13637 〈“cs1 13662 |
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 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
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 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-fzo 12768 df-hash 13418 df-word 13582 df-concat 13638 df-s1 13663 |
This theorem is referenced by: s2fv1 14016 s3fv2 14021 s4fv3 14026 |
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