![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 13805 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → 〈“𝑆”〉 ∈ Word 𝑉) | |
2 | s1len 13809 | . . . . 5 ⊢ (♯‘〈“𝑆”〉) = 1 | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → (♯‘〈“𝑆”〉) = 1) |
4 | s1fv 13813 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → (〈“𝑆”〉‘0) = 𝑆) | |
5 | 1, 3, 4 | 3jca 1121 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (〈“𝑆”〉 ∈ Word 𝑉 ∧ (♯‘〈“𝑆”〉) = 1 ∧ (〈“𝑆”〉‘0) = 𝑆)) |
6 | eleq1 2870 | . . . 4 ⊢ (𝑊 = 〈“𝑆”〉 → (𝑊 ∈ Word 𝑉 ↔ 〈“𝑆”〉 ∈ Word 𝑉)) | |
7 | fveqeq2 6552 | . . . 4 ⊢ (𝑊 = 〈“𝑆”〉 → ((♯‘𝑊) = 1 ↔ (♯‘〈“𝑆”〉) = 1)) | |
8 | fveq1 6542 | . . . . 5 ⊢ (𝑊 = 〈“𝑆”〉 → (𝑊‘0) = (〈“𝑆”〉‘0)) | |
9 | 8 | eqeq1d 2797 | . . . 4 ⊢ (𝑊 = 〈“𝑆”〉 → ((𝑊‘0) = 𝑆 ↔ (〈“𝑆”〉‘0) = 𝑆)) |
10 | 6, 7, 9 | 3anbi123d 1428 | . . 3 ⊢ (𝑊 = 〈“𝑆”〉 → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆) ↔ (〈“𝑆”〉 ∈ Word 𝑉 ∧ (♯‘〈“𝑆”〉) = 1 ∧ (〈“𝑆”〉‘0) = 𝑆))) |
11 | 5, 10 | syl5ibrcom 248 | . 2 ⊢ (𝑆 ∈ 𝑉 → (𝑊 = 〈“𝑆”〉 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆))) |
12 | eqs1 13815 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1) → 𝑊 = 〈“(𝑊‘0)”〉) | |
13 | s1eq 13803 | . . . . 5 ⊢ ((𝑊‘0) = 𝑆 → 〈“(𝑊‘0)”〉 = 〈“𝑆”〉) | |
14 | 13 | eqeq2d 2805 | . . . 4 ⊢ ((𝑊‘0) = 𝑆 → (𝑊 = 〈“(𝑊‘0)”〉 ↔ 𝑊 = 〈“𝑆”〉)) |
15 | 12, 14 | syl5ibcom 246 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1) → ((𝑊‘0) = 𝑆 → 𝑊 = 〈“𝑆”〉)) |
16 | 15 | 3impia 1110 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆) → 𝑊 = 〈“𝑆”〉) |
17 | 11, 16 | impbid1 226 | 1 ⊢ (𝑆 ∈ 𝑉 → (𝑊 = 〈“𝑆”〉 ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ‘cfv 6230 0cc0 10388 1c1 10389 ♯chash 13545 Word cword 13712 〈“cs1 13798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-oadd 7962 df-er 8144 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-card 9219 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-n0 11751 df-z 11835 df-uz 12099 df-fz 12748 df-fzo 12889 df-hash 13546 df-word 13713 df-s1 13799 |
This theorem is referenced by: rusgrnumwwlkb0 27442 clwwlknon1 27568 |
Copyright terms: Public domain | W3C validator |