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Mirrors > Home > MPE Home > Th. List > ccats1alpha | Structured version Visualization version GIF version |
Description: A concatenation of a word with a singleton word is a word over an alphabet 𝑆 iff the symbols of both words belong to the alphabet 𝑆. (Contributed by AV, 27-Mar-2022.) |
Ref | Expression |
---|---|
ccats1alpha | ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑈) → ((𝐴 ++ 〈“𝑋”〉) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 𝑋 ∈ 𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrdv 13872 | . . 3 ⊢ (𝐴 ∈ Word 𝑉 → 𝐴 ∈ Word V) | |
2 | s1cli 13950 | . . . 4 ⊢ 〈“𝑋”〉 ∈ Word V | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑋 ∈ 𝑈 → 〈“𝑋”〉 ∈ Word V) |
4 | ccatalpha 13938 | . . 3 ⊢ ((𝐴 ∈ Word V ∧ 〈“𝑋”〉 ∈ Word V) → ((𝐴 ++ 〈“𝑋”〉) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 〈“𝑋”〉 ∈ Word 𝑆))) | |
5 | 1, 3, 4 | syl2an 598 | . 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑈) → ((𝐴 ++ 〈“𝑋”〉) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 〈“𝑋”〉 ∈ Word 𝑆))) |
6 | simpr 488 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) → 〈“𝑋”〉 ∈ Word 𝑆) | |
7 | s1len 13951 | . . . . . . . 8 ⊢ (♯‘〈“𝑋”〉) = 1 | |
8 | wrdl1exs1 13958 | . . . . . . . 8 ⊢ ((〈“𝑋”〉 ∈ Word 𝑆 ∧ (♯‘〈“𝑋”〉) = 1) → ∃𝑤 ∈ 𝑆 〈“𝑋”〉 = 〈“𝑤”〉) | |
9 | 6, 7, 8 | sylancl 589 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) → ∃𝑤 ∈ 𝑆 〈“𝑋”〉 = 〈“𝑤”〉) |
10 | elex 3459 | . . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ V) | |
11 | 10 | adantr 484 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) → 𝑋 ∈ V) |
12 | elex 3459 | . . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑆 → 𝑤 ∈ V) | |
13 | s111 13960 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝑤 ∈ V) → (〈“𝑋”〉 = 〈“𝑤”〉 ↔ 𝑋 = 𝑤)) | |
14 | 11, 12, 13 | syl2an 598 | . . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ∧ 𝑤 ∈ 𝑆) → (〈“𝑋”〉 = 〈“𝑤”〉 ↔ 𝑋 = 𝑤)) |
15 | simpr 488 | . . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ∧ 𝑤 ∈ 𝑆) → 𝑤 ∈ 𝑆) | |
16 | eleq1 2877 | . . . . . . . . . 10 ⊢ (𝑋 = 𝑤 → (𝑋 ∈ 𝑆 ↔ 𝑤 ∈ 𝑆)) | |
17 | 15, 16 | syl5ibrcom 250 | . . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ∧ 𝑤 ∈ 𝑆) → (𝑋 = 𝑤 → 𝑋 ∈ 𝑆)) |
18 | 14, 17 | sylbid 243 | . . . . . . . 8 ⊢ (((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ∧ 𝑤 ∈ 𝑆) → (〈“𝑋”〉 = 〈“𝑤”〉 → 𝑋 ∈ 𝑆)) |
19 | 18 | rexlimdva 3243 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) → (∃𝑤 ∈ 𝑆 〈“𝑋”〉 = 〈“𝑤”〉 → 𝑋 ∈ 𝑆)) |
20 | 9, 19 | mpd 15 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) → 𝑋 ∈ 𝑆) |
21 | 20 | ex 416 | . . . . 5 ⊢ (𝑋 ∈ 𝑈 → (〈“𝑋”〉 ∈ Word 𝑆 → 𝑋 ∈ 𝑆)) |
22 | s1cl 13947 | . . . . 5 ⊢ (𝑋 ∈ 𝑆 → 〈“𝑋”〉 ∈ Word 𝑆) | |
23 | 21, 22 | impbid1 228 | . . . 4 ⊢ (𝑋 ∈ 𝑈 → (〈“𝑋”〉 ∈ Word 𝑆 ↔ 𝑋 ∈ 𝑆)) |
24 | 23 | anbi2d 631 | . . 3 ⊢ (𝑋 ∈ 𝑈 → ((𝐴 ∈ Word 𝑆 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ↔ (𝐴 ∈ Word 𝑆 ∧ 𝑋 ∈ 𝑆))) |
25 | 24 | adantl 485 | . 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑈) → ((𝐴 ∈ Word 𝑆 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ↔ (𝐴 ∈ Word 𝑆 ∧ 𝑋 ∈ 𝑆))) |
26 | 5, 25 | bitrd 282 | 1 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑈) → ((𝐴 ++ 〈“𝑋”〉) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 𝑋 ∈ 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 ‘cfv 6324 (class class class)co 7135 1c1 10527 ♯chash 13686 Word cword 13857 ++ cconcat 13913 〈“cs1 13940 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 |
This theorem is referenced by: clwwlknonwwlknonb 27891 |
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