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Mirrors > Home > MPE Home > Th. List > ccatw2s1p1 | Structured version Visualization version GIF version |
Description: Extract the symbol of the first singleton word of a word concatenated with this singleton word and another singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 1-May-2020.) (Revised by AV, 1-May-2020.) (Revised by AV, 29-Jan-2024.) |
Ref | Expression |
---|---|
ccatw2s1p1 | ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘𝑁) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ccatws1cl 13970 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑊 ++ 〈“𝑋”〉) ∈ Word 𝑉) | |
2 | 1 | 3adant2 1127 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → (𝑊 ++ 〈“𝑋”〉) ∈ Word 𝑉) |
3 | lencl 13883 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ0) | |
4 | fzonn0p1 13115 | . . . . . . 7 ⊢ ((♯‘𝑊) ∈ ℕ0 → (♯‘𝑊) ∈ (0..^((♯‘𝑊) + 1))) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ (0..^((♯‘𝑊) + 1))) |
6 | 5 | adantr 483 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) → (♯‘𝑊) ∈ (0..^((♯‘𝑊) + 1))) |
7 | simpr 487 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) → (♯‘𝑊) = 𝑁) | |
8 | 7 | eqcomd 2827 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) → 𝑁 = (♯‘𝑊)) |
9 | ccatws1len 13974 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘(𝑊 ++ 〈“𝑋”〉)) = ((♯‘𝑊) + 1)) | |
10 | 9 | adantr 483 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) → (♯‘(𝑊 ++ 〈“𝑋”〉)) = ((♯‘𝑊) + 1)) |
11 | 10 | oveq2d 7172 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) → (0..^(♯‘(𝑊 ++ 〈“𝑋”〉))) = (0..^((♯‘𝑊) + 1))) |
12 | 6, 8, 11 | 3eltr4d 2928 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) → 𝑁 ∈ (0..^(♯‘(𝑊 ++ 〈“𝑋”〉)))) |
13 | 12 | 3adant3 1128 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → 𝑁 ∈ (0..^(♯‘(𝑊 ++ 〈“𝑋”〉)))) |
14 | ccats1val1 13981 | . . 3 ⊢ (((𝑊 ++ 〈“𝑋”〉) ∈ Word 𝑉 ∧ 𝑁 ∈ (0..^(♯‘(𝑊 ++ 〈“𝑋”〉)))) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘𝑁) = ((𝑊 ++ 〈“𝑋”〉)‘𝑁)) | |
15 | 2, 13, 14 | syl2anc 586 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘𝑁) = ((𝑊 ++ 〈“𝑋”〉)‘𝑁)) |
16 | simp1 1132 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ Word 𝑉) | |
17 | simp3 1134 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
18 | eqcom 2828 | . . . . 5 ⊢ ((♯‘𝑊) = 𝑁 ↔ 𝑁 = (♯‘𝑊)) | |
19 | 18 | biimpi 218 | . . . 4 ⊢ ((♯‘𝑊) = 𝑁 → 𝑁 = (♯‘𝑊)) |
20 | 19 | 3ad2ant2 1130 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → 𝑁 = (♯‘𝑊)) |
21 | ccats1val2 13983 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊)) → ((𝑊 ++ 〈“𝑋”〉)‘𝑁) = 𝑋) | |
22 | 16, 17, 20, 21 | syl3anc 1367 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → ((𝑊 ++ 〈“𝑋”〉)‘𝑁) = 𝑋) |
23 | 15, 22 | eqtrd 2856 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘𝑁) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 ℕ0cn0 11898 ..^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: clwwlknonex2lem2 27887 numclwwlk1lem2foalem 28130 |
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