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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 14319 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → ((♯‘𝑊) = 0 ↔ 𝑊 = ∅)) | |
2 | 1 | biimpa 477 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 0) → 𝑊 = ∅) |
3 | s1cli 14551 | . . . . . . 7 ⊢ ⟨“∅”⟩ ∈ Word V | |
4 | ccatlid 14532 | . . . . . . 7 ⊢ (⟨“∅”⟩ ∈ Word V → (∅ ++ ⟨“∅”⟩) = ⟨“∅”⟩) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ (∅ ++ ⟨“∅”⟩) = ⟨“∅”⟩ |
6 | 5 | fveq1i 6889 | . . . . 5 ⊢ ((∅ ++ ⟨“∅”⟩)‘0) = (⟨“∅”⟩‘0) |
7 | 0ex 5306 | . . . . . 6 ⊢ ∅ ∈ V | |
8 | s1fv 14556 | . . . . . 6 ⊢ (∅ ∈ V → (⟨“∅”⟩‘0) = ∅) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ (⟨“∅”⟩‘0) = ∅ |
10 | 6, 9 | eqtri 2760 | . . . 4 ⊢ ((∅ ++ ⟨“∅”⟩)‘0) = ∅ |
11 | id 22 | . . . . . 6 ⊢ (𝑊 = ∅ → 𝑊 = ∅) | |
12 | fveq1 6887 | . . . . . . . 8 ⊢ (𝑊 = ∅ → (𝑊‘0) = (∅‘0)) | |
13 | 0fv 6932 | . . . . . . . 8 ⊢ (∅‘0) = ∅ | |
14 | 12, 13 | eqtrdi 2788 | . . . . . . 7 ⊢ (𝑊 = ∅ → (𝑊‘0) = ∅) |
15 | 14 | s1eqd 14547 | . . . . . 6 ⊢ (𝑊 = ∅ → ⟨“(𝑊‘0)”⟩ = ⟨“∅”⟩) |
16 | 11, 15 | oveq12d 7423 | . . . . 5 ⊢ (𝑊 = ∅ → (𝑊 ++ ⟨“(𝑊‘0)”⟩) = (∅ ++ ⟨“∅”⟩)) |
17 | 16 | fveq1d 6890 | . . . 4 ⊢ (𝑊 = ∅ → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = ((∅ ++ ⟨“∅”⟩)‘0)) |
18 | 10, 17, 14 | 3eqtr4a 2798 | . . 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 2985 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → ((♯‘𝑊) ≠ 0 ↔ 𝑊 ≠ ∅)) |
21 | 20 | biimpa 477 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ≠ 0) → 𝑊 ≠ ∅) |
22 | lennncl 14480 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ) | |
23 | 21, 22 | syldan 591 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ≠ 0) → (♯‘𝑊) ∈ ℕ) |
24 | lbfzo0 13668 | . . . 4 ⊢ (0 ∈ (0..^(♯‘𝑊)) ↔ (♯‘𝑊) ∈ ℕ) | |
25 | 23, 24 | sylibr 233 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ≠ 0) → 0 ∈ (0..^(♯‘𝑊))) |
26 | ccats1val1 14572 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0..^(♯‘𝑊))) → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0)) | |
27 | 25, 26 | syldan 591 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ≠ 0) → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0)) |
28 | 19, 27 | pm2.61dane 3029 | 1 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 ∅c0 4321 ‘cfv 6540 (class class class)co 7405 0cc0 11106 ℕcn 12208 ..^cfzo 13623 ♯chash 14286 Word cword 14460 ++ cconcat 14516 ⟨“cs1 14541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-concat 14517 df-s1 14542 |
This theorem is referenced by: clwwlknonwwlknonb 29348 |
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