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Mirrors > Home > MPE Home > Th. List > wrdl1s1 | Structured version Visualization version GIF version |
Description: A word of length 1 is a singleton word consisting of the first symbol of the word. (Contributed by AV, 22-Jul-2018.) (Proof shortened by AV, 14-Oct-2018.) |
Ref | Expression |
---|---|
wrdl1s1 | ⊢ (𝑆 ∈ 𝑉 → (𝑊 = 〈“𝑆”〉 ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1cl 13955 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → 〈“𝑆”〉 ∈ Word 𝑉) | |
2 | s1len 13959 | . . . . 5 ⊢ (♯‘〈“𝑆”〉) = 1 | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → (♯‘〈“𝑆”〉) = 1) |
4 | s1fv 13963 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → (〈“𝑆”〉‘0) = 𝑆) | |
5 | 1, 3, 4 | 3jca 1124 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (〈“𝑆”〉 ∈ Word 𝑉 ∧ (♯‘〈“𝑆”〉) = 1 ∧ (〈“𝑆”〉‘0) = 𝑆)) |
6 | eleq1 2900 | . . . 4 ⊢ (𝑊 = 〈“𝑆”〉 → (𝑊 ∈ Word 𝑉 ↔ 〈“𝑆”〉 ∈ Word 𝑉)) | |
7 | fveqeq2 6678 | . . . 4 ⊢ (𝑊 = 〈“𝑆”〉 → ((♯‘𝑊) = 1 ↔ (♯‘〈“𝑆”〉) = 1)) | |
8 | fveq1 6668 | . . . . 5 ⊢ (𝑊 = 〈“𝑆”〉 → (𝑊‘0) = (〈“𝑆”〉‘0)) | |
9 | 8 | eqeq1d 2823 | . . . 4 ⊢ (𝑊 = 〈“𝑆”〉 → ((𝑊‘0) = 𝑆 ↔ (〈“𝑆”〉‘0) = 𝑆)) |
10 | 6, 7, 9 | 3anbi123d 1432 | . . 3 ⊢ (𝑊 = 〈“𝑆”〉 → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆) ↔ (〈“𝑆”〉 ∈ Word 𝑉 ∧ (♯‘〈“𝑆”〉) = 1 ∧ (〈“𝑆”〉‘0) = 𝑆))) |
11 | 5, 10 | syl5ibrcom 249 | . 2 ⊢ (𝑆 ∈ 𝑉 → (𝑊 = 〈“𝑆”〉 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆))) |
12 | eqs1 13965 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1) → 𝑊 = 〈“(𝑊‘0)”〉) | |
13 | s1eq 13953 | . . . . 5 ⊢ ((𝑊‘0) = 𝑆 → 〈“(𝑊‘0)”〉 = 〈“𝑆”〉) | |
14 | 13 | eqeq2d 2832 | . . . 4 ⊢ ((𝑊‘0) = 𝑆 → (𝑊 = 〈“(𝑊‘0)”〉 ↔ 𝑊 = 〈“𝑆”〉)) |
15 | 12, 14 | syl5ibcom 247 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1) → ((𝑊‘0) = 𝑆 → 𝑊 = 〈“𝑆”〉)) |
16 | 15 | 3impia 1113 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆) → 𝑊 = 〈“𝑆”〉) |
17 | 11, 16 | impbid1 227 | 1 ⊢ (𝑆 ∈ 𝑉 → (𝑊 = 〈“𝑆”〉 ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 0cc0 10536 1c1 10537 ♯chash 13689 Word cword 13860 〈“cs1 13948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-s1 13949 |
This theorem is referenced by: rusgrnumwwlkb0 27749 clwwlknon1 27875 |
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