Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ccat1st1st | Structured version Visualization version GIF version |
Description: The first symbol of a word concatenated with its first symbol is the first symbol of the word. This theorem holds even if 𝑊 is the empty word. (Contributed by AV, 26-Mar-2022.) |
Ref | Expression |
---|---|
ccat1st1st | ⊢ (𝑊 ∈ Word 𝑉 → ((𝑊 ++ 〈“(𝑊‘0)”〉)‘0) = (𝑊‘0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hasheq0 13718 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → ((♯‘𝑊) = 0 ↔ 𝑊 = ∅)) | |
2 | 1 | biimpa 479 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 0) → 𝑊 = ∅) |
3 | s1cli 13953 | . . . . . . 7 ⊢ 〈“∅”〉 ∈ Word V | |
4 | ccatlid 13934 | . . . . . . 7 ⊢ (〈“∅”〉 ∈ Word V → (∅ ++ 〈“∅”〉) = 〈“∅”〉) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ (∅ ++ 〈“∅”〉) = 〈“∅”〉 |
6 | 5 | fveq1i 6666 | . . . . 5 ⊢ ((∅ ++ 〈“∅”〉)‘0) = (〈“∅”〉‘0) |
7 | 0ex 5204 | . . . . . 6 ⊢ ∅ ∈ V | |
8 | s1fv 13958 | . . . . . 6 ⊢ (∅ ∈ V → (〈“∅”〉‘0) = ∅) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ (〈“∅”〉‘0) = ∅ |
10 | 6, 9 | eqtri 2844 | . . . 4 ⊢ ((∅ ++ 〈“∅”〉)‘0) = ∅ |
11 | id 22 | . . . . . 6 ⊢ (𝑊 = ∅ → 𝑊 = ∅) | |
12 | fveq1 6664 | . . . . . . . 8 ⊢ (𝑊 = ∅ → (𝑊‘0) = (∅‘0)) | |
13 | 0fv 6704 | . . . . . . . 8 ⊢ (∅‘0) = ∅ | |
14 | 12, 13 | syl6eq 2872 | . . . . . . 7 ⊢ (𝑊 = ∅ → (𝑊‘0) = ∅) |
15 | 14 | s1eqd 13949 | . . . . . 6 ⊢ (𝑊 = ∅ → 〈“(𝑊‘0)”〉 = 〈“∅”〉) |
16 | 11, 15 | oveq12d 7168 | . . . . 5 ⊢ (𝑊 = ∅ → (𝑊 ++ 〈“(𝑊‘0)”〉) = (∅ ++ 〈“∅”〉)) |
17 | 16 | fveq1d 6667 | . . . 4 ⊢ (𝑊 = ∅ → ((𝑊 ++ 〈“(𝑊‘0)”〉)‘0) = ((∅ ++ 〈“∅”〉)‘0)) |
18 | 10, 17, 14 | 3eqtr4a 2882 | . . 3 ⊢ (𝑊 = ∅ → ((𝑊 ++ 〈“(𝑊‘0)”〉)‘0) = (𝑊‘0)) |
19 | 2, 18 | syl 17 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 0) → ((𝑊 ++ 〈“(𝑊‘0)”〉)‘0) = (𝑊‘0)) |
20 | 1 | necon3bid 3060 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → ((♯‘𝑊) ≠ 0 ↔ 𝑊 ≠ ∅)) |
21 | 20 | biimpa 479 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ≠ 0) → 𝑊 ≠ ∅) |
22 | lennncl 13878 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ) | |
23 | 21, 22 | syldan 593 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ≠ 0) → (♯‘𝑊) ∈ ℕ) |
24 | lbfzo0 13071 | . . . 4 ⊢ (0 ∈ (0..^(♯‘𝑊)) ↔ (♯‘𝑊) ∈ ℕ) | |
25 | 23, 24 | sylibr 236 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ≠ 0) → 0 ∈ (0..^(♯‘𝑊))) |
26 | ccats1val1 13975 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0..^(♯‘𝑊))) → ((𝑊 ++ 〈“(𝑊‘0)”〉)‘0) = (𝑊‘0)) | |
27 | 25, 26 | syldan 593 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ≠ 0) → ((𝑊 ++ 〈“(𝑊‘0)”〉)‘0) = (𝑊‘0)) |
28 | 19, 27 | pm2.61dane 3104 | 1 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑊 ++ 〈“(𝑊‘0)”〉)‘0) = (𝑊‘0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3495 ∅c0 4291 ‘cfv 6350 (class class class)co 7150 0cc0 10531 ℕcn 11632 ..^cfzo 13027 ♯chash 13684 Word cword 13855 ++ cconcat 13916 〈“cs1 13943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 |
This theorem is referenced by: clwwlknonwwlknonb 27879 |
Copyright terms: Public domain | W3C validator |