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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 14428 | . . 3 ⊢ (𝐴 ∈ Word 𝑉 → 𝐴 ∈ Word V) | |
| 2 | s1cli 14505 | . . . 4 ⊢ 〈“𝑋”〉 ∈ Word V | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝑋 ∈ 𝑈 → 〈“𝑋”〉 ∈ Word V) |
| 4 | ccatalpha 14493 | . . 3 ⊢ ((𝐴 ∈ Word V ∧ 〈“𝑋”〉 ∈ Word V) → ((𝐴 ++ 〈“𝑋”〉) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 〈“𝑋”〉 ∈ Word 𝑆))) | |
| 5 | 1, 3, 4 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑈) → ((𝐴 ++ 〈“𝑋”〉) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 〈“𝑋”〉 ∈ Word 𝑆))) |
| 6 | simpr 484 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) → 〈“𝑋”〉 ∈ Word 𝑆) | |
| 7 | s1len 14506 | . . . . . . . 8 ⊢ (♯‘〈“𝑋”〉) = 1 | |
| 8 | wrdl1exs1 14513 | . . . . . . . 8 ⊢ ((〈“𝑋”〉 ∈ Word 𝑆 ∧ (♯‘〈“𝑋”〉) = 1) → ∃𝑤 ∈ 𝑆 〈“𝑋”〉 = 〈“𝑤”〉) | |
| 9 | 6, 7, 8 | sylancl 586 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) → ∃𝑤 ∈ 𝑆 〈“𝑋”〉 = 〈“𝑤”〉) |
| 10 | elex 3455 | . . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ V) | |
| 11 | 10 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) → 𝑋 ∈ V) |
| 12 | elex 3455 | . . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑆 → 𝑤 ∈ V) | |
| 13 | s111 14515 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝑤 ∈ V) → (〈“𝑋”〉 = 〈“𝑤”〉 ↔ 𝑋 = 𝑤)) | |
| 14 | 11, 12, 13 | syl2an 596 | . . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ∧ 𝑤 ∈ 𝑆) → (〈“𝑋”〉 = 〈“𝑤”〉 ↔ 𝑋 = 𝑤)) |
| 15 | simpr 484 | . . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ∧ 𝑤 ∈ 𝑆) → 𝑤 ∈ 𝑆) | |
| 16 | eleq1 2817 | . . . . . . . . . 10 ⊢ (𝑋 = 𝑤 → (𝑋 ∈ 𝑆 ↔ 𝑤 ∈ 𝑆)) | |
| 17 | 15, 16 | syl5ibrcom 247 | . . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ∧ 𝑤 ∈ 𝑆) → (𝑋 = 𝑤 → 𝑋 ∈ 𝑆)) |
| 18 | 14, 17 | sylbid 240 | . . . . . . . 8 ⊢ (((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ∧ 𝑤 ∈ 𝑆) → (〈“𝑋”〉 = 〈“𝑤”〉 → 𝑋 ∈ 𝑆)) |
| 19 | 18 | rexlimdva 3131 | . . . . . . 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 412 | . . . . 5 ⊢ (𝑋 ∈ 𝑈 → (〈“𝑋”〉 ∈ Word 𝑆 → 𝑋 ∈ 𝑆)) |
| 22 | s1cl 14502 | . . . . 5 ⊢ (𝑋 ∈ 𝑆 → 〈“𝑋”〉 ∈ Word 𝑆) | |
| 23 | 21, 22 | impbid1 225 | . . . 4 ⊢ (𝑋 ∈ 𝑈 → (〈“𝑋”〉 ∈ Word 𝑆 ↔ 𝑋 ∈ 𝑆)) |
| 24 | 23 | anbi2d 630 | . . 3 ⊢ (𝑋 ∈ 𝑈 → ((𝐴 ∈ Word 𝑆 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ↔ (𝐴 ∈ Word 𝑆 ∧ 𝑋 ∈ 𝑆))) |
| 25 | 24 | adantl 481 | . 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑈) → ((𝐴 ∈ Word 𝑆 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ↔ (𝐴 ∈ Word 𝑆 ∧ 𝑋 ∈ 𝑆))) |
| 26 | 5, 25 | bitrd 279 | 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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2110 ∃wrex 3054 Vcvv 3434 ‘cfv 6477 (class class class)co 7341 1c1 10999 ♯chash 14229 Word cword 14412 ++ cconcat 14469 〈“cs1 14495 |
| 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 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-dju 9786 df-card 9824 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-nn 12118 df-2 12180 df-n0 12374 df-xnn0 12447 df-z 12461 df-uz 12725 df-fz 13400 df-fzo 13547 df-hash 14230 df-word 14413 df-concat 14470 df-s1 14496 |
| This theorem is referenced by: clwwlknonwwlknonb 30076 |
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