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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 13533 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → ((♯‘𝑊) = 0 ↔ 𝑊 = ∅)) | |
2 | 1 | biimpac 471 | . . 3 ⊢ (((♯‘𝑊) = 0 ∧ 𝑊 ∈ Word 𝑉) → 𝑊 = ∅) |
3 | s1cli 13762 | . . . . . . 7 ⊢ 〈“∅”〉 ∈ Word V | |
4 | ccatlid 13743 | . . . . . . 7 ⊢ (〈“∅”〉 ∈ Word V → (∅ ++ 〈“∅”〉) = 〈“∅”〉) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ (∅ ++ 〈“∅”〉) = 〈“∅”〉 |
6 | 5 | fveq1i 6494 | . . . . 5 ⊢ ((∅ ++ 〈“∅”〉)‘0) = (〈“∅”〉‘0) |
7 | 0ex 5062 | . . . . . 6 ⊢ ∅ ∈ V | |
8 | s1fv 13767 | . . . . . 6 ⊢ (∅ ∈ V → (〈“∅”〉‘0) = ∅) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ (〈“∅”〉‘0) = ∅ |
10 | 6, 9 | eqtri 2796 | . . . 4 ⊢ ((∅ ++ 〈“∅”〉)‘0) = ∅ |
11 | id 22 | . . . . . 6 ⊢ (𝑊 = ∅ → 𝑊 = ∅) | |
12 | fveq1 6492 | . . . . . . . 8 ⊢ (𝑊 = ∅ → (𝑊‘0) = (∅‘0)) | |
13 | 0fv 6533 | . . . . . . . 8 ⊢ (∅‘0) = ∅ | |
14 | 12, 13 | syl6eq 2824 | . . . . . . 7 ⊢ (𝑊 = ∅ → (𝑊‘0) = ∅) |
15 | 14 | s1eqd 13758 | . . . . . 6 ⊢ (𝑊 = ∅ → 〈“(𝑊‘0)”〉 = 〈“∅”〉) |
16 | 11, 15 | oveq12d 6988 | . . . . 5 ⊢ (𝑊 = ∅ → (𝑊 ++ 〈“(𝑊‘0)”〉) = (∅ ++ 〈“∅”〉)) |
17 | 16 | fveq1d 6495 | . . . 4 ⊢ (𝑊 = ∅ → ((𝑊 ++ 〈“(𝑊‘0)”〉)‘0) = ((∅ ++ 〈“∅”〉)‘0)) |
18 | 10, 17, 14 | 3eqtr4a 2834 | . . 3 ⊢ (𝑊 = ∅ → ((𝑊 ++ 〈“(𝑊‘0)”〉)‘0) = (𝑊‘0)) |
19 | 2, 18 | syl 17 | . 2 ⊢ (((♯‘𝑊) = 0 ∧ 𝑊 ∈ Word 𝑉) → ((𝑊 ++ 〈“(𝑊‘0)”〉)‘0) = (𝑊‘0)) |
20 | wrdv 13682 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → 𝑊 ∈ Word V) | |
21 | 20 | adantl 474 | . . 3 ⊢ ((¬ (♯‘𝑊) = 0 ∧ 𝑊 ∈ Word 𝑉) → 𝑊 ∈ Word V) |
22 | fvexd 6508 | . . 3 ⊢ ((¬ (♯‘𝑊) = 0 ∧ 𝑊 ∈ Word 𝑉) → (𝑊‘0) ∈ V) | |
23 | lencl 13688 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ0) | |
24 | df-ne 2962 | . . . . . . 7 ⊢ ((♯‘𝑊) ≠ 0 ↔ ¬ (♯‘𝑊) = 0) | |
25 | elnnne0 11717 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℕ ↔ ((♯‘𝑊) ∈ ℕ0 ∧ (♯‘𝑊) ≠ 0)) | |
26 | 25 | simplbi2 493 | . . . . . . 7 ⊢ ((♯‘𝑊) ∈ ℕ0 → ((♯‘𝑊) ≠ 0 → (♯‘𝑊) ∈ ℕ)) |
27 | 24, 26 | syl5bir 235 | . . . . . 6 ⊢ ((♯‘𝑊) ∈ ℕ0 → (¬ (♯‘𝑊) = 0 → (♯‘𝑊) ∈ ℕ)) |
28 | 23, 27 | syl 17 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (¬ (♯‘𝑊) = 0 → (♯‘𝑊) ∈ ℕ)) |
29 | 28 | impcom 399 | . . . 4 ⊢ ((¬ (♯‘𝑊) = 0 ∧ 𝑊 ∈ Word 𝑉) → (♯‘𝑊) ∈ ℕ) |
30 | lbfzo0 12886 | . . . 4 ⊢ (0 ∈ (0..^(♯‘𝑊)) ↔ (♯‘𝑊) ∈ ℕ) | |
31 | 29, 30 | sylibr 226 | . . 3 ⊢ ((¬ (♯‘𝑊) = 0 ∧ 𝑊 ∈ Word 𝑉) → 0 ∈ (0..^(♯‘𝑊))) |
32 | ccats1val1 13783 | . . 3 ⊢ ((𝑊 ∈ Word V ∧ (𝑊‘0) ∈ V ∧ 0 ∈ (0..^(♯‘𝑊))) → ((𝑊 ++ 〈“(𝑊‘0)”〉)‘0) = (𝑊‘0)) | |
33 | 21, 22, 31, 32 | syl3anc 1351 | . 2 ⊢ ((¬ (♯‘𝑊) = 0 ∧ 𝑊 ∈ Word 𝑉) → ((𝑊 ++ 〈“(𝑊‘0)”〉)‘0) = (𝑊‘0)) |
34 | 19, 33 | pm2.61ian 799 | 1 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑊 ++ 〈“(𝑊‘0)”〉)‘0) = (𝑊‘0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ≠ wne 2961 Vcvv 3409 ∅c0 4172 ‘cfv 6182 (class class class)co 6970 0cc0 10329 ℕcn 11433 ℕ0cn0 11701 ..^cfzo 12843 ♯chash 13499 Word cword 13666 ++ cconcat 13727 〈“cs1 13752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-oadd 7903 df-er 8083 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-card 9156 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-nn 11434 df-n0 11702 df-z 11788 df-uz 12053 df-fz 12703 df-fzo 12844 df-hash 13500 df-word 13667 df-concat 13728 df-s1 13753 |
This theorem is referenced by: clwwlknonwwlknonb 27628 |
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