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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 7373 | . . . . 5 ⊢ (0 + (♯‘𝑆)) = (0 + 𝑀) |
4 | cats1cli.2 | . . . . . . . . 9 ⊢ 𝑆 ∈ Word V | |
5 | lencl 14428 | . . . . . . . . 9 ⊢ (𝑆 ∈ Word V → (♯‘𝑆) ∈ ℕ0) | |
6 | 4, 5 | ax-mp 5 | . . . . . . . 8 ⊢ (♯‘𝑆) ∈ ℕ0 |
7 | 2, 6 | eqeltrri 2835 | . . . . . . 7 ⊢ 𝑀 ∈ ℕ0 |
8 | 7 | nn0cni 12432 | . . . . . 6 ⊢ 𝑀 ∈ ℂ |
9 | 8 | addid2i 11350 | . . . . 5 ⊢ (0 + 𝑀) = 𝑀 |
10 | 3, 9 | eqtr2i 2766 | . . . 4 ⊢ 𝑀 = (0 + (♯‘𝑆)) |
11 | 1, 10 | fveq12i 6853 | . . 3 ⊢ (𝑇‘𝑀) = ((𝑆 ++ ⟨“𝑋”⟩)‘(0 + (♯‘𝑆))) |
12 | s1cli 14500 | . . . 4 ⊢ ⟨“𝑋”⟩ ∈ Word V | |
13 | s1len 14501 | . . . . . 6 ⊢ (♯‘⟨“𝑋”⟩) = 1 | |
14 | 1nn 12171 | . . . . . 6 ⊢ 1 ∈ ℕ | |
15 | 13, 14 | eqeltri 2834 | . . . . 5 ⊢ (♯‘⟨“𝑋”⟩) ∈ ℕ |
16 | lbfzo0 13619 | . . . . 5 ⊢ (0 ∈ (0..^(♯‘⟨“𝑋”⟩)) ↔ (♯‘⟨“𝑋”⟩) ∈ ℕ) | |
17 | 15, 16 | mpbir 230 | . . . 4 ⊢ 0 ∈ (0..^(♯‘⟨“𝑋”⟩)) |
18 | ccatval3 14474 | . . . 4 ⊢ ((𝑆 ∈ Word V ∧ ⟨“𝑋”⟩ ∈ Word V ∧ 0 ∈ (0..^(♯‘⟨“𝑋”⟩))) → ((𝑆 ++ ⟨“𝑋”⟩)‘(0 + (♯‘𝑆))) = (⟨“𝑋”⟩‘0)) | |
19 | 4, 12, 17, 18 | mp3an 1462 | . . 3 ⊢ ((𝑆 ++ ⟨“𝑋”⟩)‘(0 + (♯‘𝑆))) = (⟨“𝑋”⟩‘0) |
20 | 11, 19 | eqtri 2765 | . 2 ⊢ (𝑇‘𝑀) = (⟨“𝑋”⟩‘0) |
21 | s1fv 14505 | . 2 ⊢ (𝑋 ∈ 𝑉 → (⟨“𝑋”⟩‘0) = 𝑋) | |
22 | 20, 21 | eqtrid 2789 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝑇‘𝑀) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3448 ‘cfv 6501 (class class class)co 7362 0cc0 11058 1c1 11059 + caddc 11061 ℕcn 12160 ℕ0cn0 12420 ..^cfzo 13574 ♯chash 14237 Word cword 14409 ++ cconcat 14465 ⟨“cs1 14490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-fzo 13575 df-hash 14238 df-word 14410 df-concat 14466 df-s1 14491 |
This theorem is referenced by: s2fv1 14784 s3fv2 14789 s4fv3 14794 |
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