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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 14624 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑊 ++ 〈“𝑋”〉) ∈ Word 𝑉) | |
2 | 1 | 3adant2 1128 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → (𝑊 ++ 〈“𝑋”〉) ∈ Word 𝑉) |
3 | lencl 14541 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ0) | |
4 | fzonn0p1 13763 | . . . . . . 7 ⊢ ((♯‘𝑊) ∈ ℕ0 → (♯‘𝑊) ∈ (0..^((♯‘𝑊) + 1))) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ (0..^((♯‘𝑊) + 1))) |
6 | 5 | adantr 479 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) → (♯‘𝑊) ∈ (0..^((♯‘𝑊) + 1))) |
7 | simpr 483 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) → (♯‘𝑊) = 𝑁) | |
8 | 7 | eqcomd 2732 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) → 𝑁 = (♯‘𝑊)) |
9 | ccatws1len 14628 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘(𝑊 ++ 〈“𝑋”〉)) = ((♯‘𝑊) + 1)) | |
10 | 9 | adantr 479 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) → (♯‘(𝑊 ++ 〈“𝑋”〉)) = ((♯‘𝑊) + 1)) |
11 | 10 | oveq2d 7440 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) → (0..^(♯‘(𝑊 ++ 〈“𝑋”〉))) = (0..^((♯‘𝑊) + 1))) |
12 | 6, 8, 11 | 3eltr4d 2841 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) → 𝑁 ∈ (0..^(♯‘(𝑊 ++ 〈“𝑋”〉)))) |
13 | 12 | 3adant3 1129 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → 𝑁 ∈ (0..^(♯‘(𝑊 ++ 〈“𝑋”〉)))) |
14 | ccats1val1 14634 | . . 3 ⊢ (((𝑊 ++ 〈“𝑋”〉) ∈ Word 𝑉 ∧ 𝑁 ∈ (0..^(♯‘(𝑊 ++ 〈“𝑋”〉)))) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘𝑁) = ((𝑊 ++ 〈“𝑋”〉)‘𝑁)) | |
15 | 2, 13, 14 | syl2anc 582 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘𝑁) = ((𝑊 ++ 〈“𝑋”〉)‘𝑁)) |
16 | simp1 1133 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ Word 𝑉) | |
17 | simp3 1135 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
18 | eqcom 2733 | . . . . 5 ⊢ ((♯‘𝑊) = 𝑁 ↔ 𝑁 = (♯‘𝑊)) | |
19 | 18 | biimpi 215 | . . . 4 ⊢ ((♯‘𝑊) = 𝑁 → 𝑁 = (♯‘𝑊)) |
20 | 19 | 3ad2ant2 1131 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → 𝑁 = (♯‘𝑊)) |
21 | ccats1val2 14635 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊)) → ((𝑊 ++ 〈“𝑋”〉)‘𝑁) = 𝑋) | |
22 | 16, 17, 20, 21 | syl3anc 1368 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → ((𝑊 ++ 〈“𝑋”〉)‘𝑁) = 𝑋) |
23 | 15, 22 | eqtrd 2766 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘𝑁) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ‘cfv 6554 (class class class)co 7424 0cc0 11158 1c1 11159 + caddc 11161 ℕ0cn0 12524 ..^cfzo 13681 ♯chash 14347 Word cword 14522 ++ cconcat 14578 〈“cs1 14603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-fzo 13682 df-hash 14348 df-word 14523 df-concat 14579 df-s1 14604 |
This theorem is referenced by: clwwlknonex2lem2 30041 numclwwlk1lem2foalem 30284 |
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