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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 13725 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → ((♯‘𝑊) = 0 ↔ 𝑊 = ∅)) | |
2 | 1 | biimpa 479 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 0) → 𝑊 = ∅) |
3 | s1cli 13959 | . . . . . . 7 ⊢ 〈“∅”〉 ∈ Word V | |
4 | ccatlid 13940 | . . . . . . 7 ⊢ (〈“∅”〉 ∈ Word V → (∅ ++ 〈“∅”〉) = 〈“∅”〉) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ (∅ ++ 〈“∅”〉) = 〈“∅”〉 |
6 | 5 | fveq1i 6671 | . . . . 5 ⊢ ((∅ ++ 〈“∅”〉)‘0) = (〈“∅”〉‘0) |
7 | 0ex 5211 | . . . . . 6 ⊢ ∅ ∈ V | |
8 | s1fv 13964 | . . . . . 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 6669 | . . . . . . . 8 ⊢ (𝑊 = ∅ → (𝑊‘0) = (∅‘0)) | |
13 | 0fv 6709 | . . . . . . . 8 ⊢ (∅‘0) = ∅ | |
14 | 12, 13 | syl6eq 2872 | . . . . . . 7 ⊢ (𝑊 = ∅ → (𝑊‘0) = ∅) |
15 | 14 | s1eqd 13955 | . . . . . 6 ⊢ (𝑊 = ∅ → 〈“(𝑊‘0)”〉 = 〈“∅”〉) |
16 | 11, 15 | oveq12d 7174 | . . . . 5 ⊢ (𝑊 = ∅ → (𝑊 ++ 〈“(𝑊‘0)”〉) = (∅ ++ 〈“∅”〉)) |
17 | 16 | fveq1d 6672 | . . . 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 13884 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ) | |
23 | 21, 22 | syldan 593 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ≠ 0) → (♯‘𝑊) ∈ ℕ) |
24 | lbfzo0 13078 | . . . 4 ⊢ (0 ∈ (0..^(♯‘𝑊)) ↔ (♯‘𝑊) ∈ ℕ) | |
25 | 23, 24 | sylibr 236 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ≠ 0) → 0 ∈ (0..^(♯‘𝑊))) |
26 | ccats1val1 13981 | . . 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 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ∅c0 4291 ‘cfv 6355 (class class class)co 7156 0cc0 10537 ℕcn 11638 ..^cfzo 13034 ♯chash 13691 Word cword 13862 ++ cconcat 13922 〈“cs1 13949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 |
This theorem is referenced by: clwwlknonwwlknonb 27885 |
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