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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 14532 | . . 3 ⊢ (𝐴 ∈ Word 𝑉 → 𝐴 ∈ Word V) | |
2 | s1cli 14608 | . . . 4 ⊢ 〈“𝑋”〉 ∈ Word V | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑋 ∈ 𝑈 → 〈“𝑋”〉 ∈ Word V) |
4 | ccatalpha 14596 | . . 3 ⊢ ((𝐴 ∈ Word V ∧ 〈“𝑋”〉 ∈ Word V) → ((𝐴 ++ 〈“𝑋”〉) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 〈“𝑋”〉 ∈ Word 𝑆))) | |
5 | 1, 3, 4 | syl2an 594 | . 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑈) → ((𝐴 ++ 〈“𝑋”〉) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 〈“𝑋”〉 ∈ Word 𝑆))) |
6 | simpr 483 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) → 〈“𝑋”〉 ∈ Word 𝑆) | |
7 | s1len 14609 | . . . . . . . 8 ⊢ (♯‘〈“𝑋”〉) = 1 | |
8 | wrdl1exs1 14616 | . . . . . . . 8 ⊢ ((〈“𝑋”〉 ∈ Word 𝑆 ∧ (♯‘〈“𝑋”〉) = 1) → ∃𝑤 ∈ 𝑆 〈“𝑋”〉 = 〈“𝑤”〉) | |
9 | 6, 7, 8 | sylancl 584 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) → ∃𝑤 ∈ 𝑆 〈“𝑋”〉 = 〈“𝑤”〉) |
10 | elex 3482 | . . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ V) | |
11 | 10 | adantr 479 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) → 𝑋 ∈ V) |
12 | elex 3482 | . . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑆 → 𝑤 ∈ V) | |
13 | s111 14618 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝑤 ∈ V) → (〈“𝑋”〉 = 〈“𝑤”〉 ↔ 𝑋 = 𝑤)) | |
14 | 11, 12, 13 | syl2an 594 | . . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ∧ 𝑤 ∈ 𝑆) → (〈“𝑋”〉 = 〈“𝑤”〉 ↔ 𝑋 = 𝑤)) |
15 | simpr 483 | . . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ∧ 𝑤 ∈ 𝑆) → 𝑤 ∈ 𝑆) | |
16 | eleq1 2814 | . . . . . . . . . 10 ⊢ (𝑋 = 𝑤 → (𝑋 ∈ 𝑆 ↔ 𝑤 ∈ 𝑆)) | |
17 | 15, 16 | syl5ibrcom 246 | . . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ∧ 𝑤 ∈ 𝑆) → (𝑋 = 𝑤 → 𝑋 ∈ 𝑆)) |
18 | 14, 17 | sylbid 239 | . . . . . . . 8 ⊢ (((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ∧ 𝑤 ∈ 𝑆) → (〈“𝑋”〉 = 〈“𝑤”〉 → 𝑋 ∈ 𝑆)) |
19 | 18 | rexlimdva 3145 | . . . . . . 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 411 | . . . . 5 ⊢ (𝑋 ∈ 𝑈 → (〈“𝑋”〉 ∈ Word 𝑆 → 𝑋 ∈ 𝑆)) |
22 | s1cl 14605 | . . . . 5 ⊢ (𝑋 ∈ 𝑆 → 〈“𝑋”〉 ∈ Word 𝑆) | |
23 | 21, 22 | impbid1 224 | . . . 4 ⊢ (𝑋 ∈ 𝑈 → (〈“𝑋”〉 ∈ Word 𝑆 ↔ 𝑋 ∈ 𝑆)) |
24 | 23 | anbi2d 628 | . . 3 ⊢ (𝑋 ∈ 𝑈 → ((𝐴 ∈ Word 𝑆 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ↔ (𝐴 ∈ Word 𝑆 ∧ 𝑋 ∈ 𝑆))) |
25 | 24 | adantl 480 | . 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑈) → ((𝐴 ∈ Word 𝑆 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ↔ (𝐴 ∈ Word 𝑆 ∧ 𝑋 ∈ 𝑆))) |
26 | 5, 25 | bitrd 278 | 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 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∃wrex 3060 Vcvv 3462 ‘cfv 6546 (class class class)co 7416 1c1 11150 ♯chash 14342 Word cword 14517 ++ cconcat 14573 〈“cs1 14598 |
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 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-oadd 8492 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-dju 9937 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-n0 12519 df-xnn0 12591 df-z 12605 df-uz 12869 df-fz 13533 df-fzo 13676 df-hash 14343 df-word 14518 df-concat 14574 df-s1 14599 |
This theorem is referenced by: clwwlknonwwlknonb 30036 |
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