![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6825 | . . . . 5 ⊢ (0 + (♯‘𝑆)) = (0 + 𝑀) |
4 | cats1cli.2 | . . . . . . . . 9 ⊢ 𝑆 ∈ Word V | |
5 | lencl 13530 | . . . . . . . . 9 ⊢ (𝑆 ∈ Word V → (♯‘𝑆) ∈ ℕ0) | |
6 | 4, 5 | ax-mp 5 | . . . . . . . 8 ⊢ (♯‘𝑆) ∈ ℕ0 |
7 | 2, 6 | eqeltrri 2836 | . . . . . . 7 ⊢ 𝑀 ∈ ℕ0 |
8 | 7 | nn0cni 11516 | . . . . . 6 ⊢ 𝑀 ∈ ℂ |
9 | 8 | addid2i 10436 | . . . . 5 ⊢ (0 + 𝑀) = 𝑀 |
10 | 3, 9 | eqtr2i 2783 | . . . 4 ⊢ 𝑀 = (0 + (♯‘𝑆)) |
11 | 1, 10 | fveq12i 6358 | . . 3 ⊢ (𝑇‘𝑀) = ((𝑆 ++ 〈“𝑋”〉)‘(0 + (♯‘𝑆))) |
12 | s1cli 13595 | . . . 4 ⊢ 〈“𝑋”〉 ∈ Word V | |
13 | s1len 13596 | . . . . . 6 ⊢ (♯‘〈“𝑋”〉) = 1 | |
14 | 1nn 11243 | . . . . . 6 ⊢ 1 ∈ ℕ | |
15 | 13, 14 | eqeltri 2835 | . . . . 5 ⊢ (♯‘〈“𝑋”〉) ∈ ℕ |
16 | lbfzo0 12722 | . . . . 5 ⊢ (0 ∈ (0..^(♯‘〈“𝑋”〉)) ↔ (♯‘〈“𝑋”〉) ∈ ℕ) | |
17 | 15, 16 | mpbir 221 | . . . 4 ⊢ 0 ∈ (0..^(♯‘〈“𝑋”〉)) |
18 | ccatval3 13571 | . . . 4 ⊢ ((𝑆 ∈ Word V ∧ 〈“𝑋”〉 ∈ Word V ∧ 0 ∈ (0..^(♯‘〈“𝑋”〉))) → ((𝑆 ++ 〈“𝑋”〉)‘(0 + (♯‘𝑆))) = (〈“𝑋”〉‘0)) | |
19 | 4, 12, 17, 18 | mp3an 1573 | . . 3 ⊢ ((𝑆 ++ 〈“𝑋”〉)‘(0 + (♯‘𝑆))) = (〈“𝑋”〉‘0) |
20 | 11, 19 | eqtri 2782 | . 2 ⊢ (𝑇‘𝑀) = (〈“𝑋”〉‘0) |
21 | s1fv 13601 | . 2 ⊢ (𝑋 ∈ 𝑉 → (〈“𝑋”〉‘0) = 𝑋) | |
22 | 20, 21 | syl5eq 2806 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝑇‘𝑀) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ‘cfv 6049 (class class class)co 6814 0cc0 10148 1c1 10149 + caddc 10151 ℕcn 11232 ℕ0cn0 11504 ..^cfzo 12679 ♯chash 13331 Word cword 13497 ++ cconcat 13499 〈“cs1 13500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-card 8975 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-n0 11505 df-z 11590 df-uz 11900 df-fz 12540 df-fzo 12680 df-hash 13332 df-word 13505 df-concat 13507 df-s1 13508 |
This theorem is referenced by: s2fv1 13853 s3fv2 13858 s4fv3 13863 |
Copyright terms: Public domain | W3C validator |