![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 13913 | . . 3 ⊢ (𝐴 ∈ Word 𝑉 → 𝐴 ∈ Word V) | |
2 | s1cli 13991 | . . . 4 ⊢ 〈“𝑋”〉 ∈ Word V | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑋 ∈ 𝑈 → 〈“𝑋”〉 ∈ Word V) |
4 | ccatalpha 13979 | . . 3 ⊢ ((𝐴 ∈ Word V ∧ 〈“𝑋”〉 ∈ Word V) → ((𝐴 ++ 〈“𝑋”〉) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 〈“𝑋”〉 ∈ Word 𝑆))) | |
5 | 1, 3, 4 | syl2an 599 | . 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑈) → ((𝐴 ++ 〈“𝑋”〉) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 〈“𝑋”〉 ∈ Word 𝑆))) |
6 | simpr 489 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) → 〈“𝑋”〉 ∈ Word 𝑆) | |
7 | s1len 13992 | . . . . . . . 8 ⊢ (♯‘〈“𝑋”〉) = 1 | |
8 | wrdl1exs1 13999 | . . . . . . . 8 ⊢ ((〈“𝑋”〉 ∈ Word 𝑆 ∧ (♯‘〈“𝑋”〉) = 1) → ∃𝑤 ∈ 𝑆 〈“𝑋”〉 = 〈“𝑤”〉) | |
9 | 6, 7, 8 | sylancl 590 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) → ∃𝑤 ∈ 𝑆 〈“𝑋”〉 = 〈“𝑤”〉) |
10 | elex 3427 | . . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ V) | |
11 | 10 | adantr 485 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) → 𝑋 ∈ V) |
12 | elex 3427 | . . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑆 → 𝑤 ∈ V) | |
13 | s111 14001 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝑤 ∈ V) → (〈“𝑋”〉 = 〈“𝑤”〉 ↔ 𝑋 = 𝑤)) | |
14 | 11, 12, 13 | syl2an 599 | . . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ∧ 𝑤 ∈ 𝑆) → (〈“𝑋”〉 = 〈“𝑤”〉 ↔ 𝑋 = 𝑤)) |
15 | simpr 489 | . . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑈 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ∧ 𝑤 ∈ 𝑆) → 𝑤 ∈ 𝑆) | |
16 | eleq1 2838 | . . . . . . . . . 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 3206 | . . . . . . 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 417 | . . . . 5 ⊢ (𝑋 ∈ 𝑈 → (〈“𝑋”〉 ∈ Word 𝑆 → 𝑋 ∈ 𝑆)) |
22 | s1cl 13988 | . . . . 5 ⊢ (𝑋 ∈ 𝑆 → 〈“𝑋”〉 ∈ Word 𝑆) | |
23 | 21, 22 | impbid1 228 | . . . 4 ⊢ (𝑋 ∈ 𝑈 → (〈“𝑋”〉 ∈ Word 𝑆 ↔ 𝑋 ∈ 𝑆)) |
24 | 23 | anbi2d 632 | . . 3 ⊢ (𝑋 ∈ 𝑈 → ((𝐴 ∈ Word 𝑆 ∧ 〈“𝑋”〉 ∈ Word 𝑆) ↔ (𝐴 ∈ Word 𝑆 ∧ 𝑋 ∈ 𝑆))) |
25 | 24 | adantl 486 | . 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 400 = wceq 1539 ∈ wcel 2112 ∃wrex 3069 Vcvv 3407 ‘cfv 6328 (class class class)co 7143 1c1 10561 ♯chash 13725 Word cword 13898 ++ cconcat 13954 〈“cs1 13981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-dju 9348 df-card 9386 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-n0 11920 df-xnn0 11992 df-z 12006 df-uz 12268 df-fz 12925 df-fzo 13068 df-hash 13726 df-word 13899 df-concat 13955 df-s1 13982 |
This theorem is referenced by: clwwlknonwwlknonb 27975 |
Copyright terms: Public domain | W3C validator |