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Mirrors > Home > MPE Home > Th. List > eqs1 | Structured version Visualization version GIF version |
Description: A word of length 1 is a singleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 1-May-2020.) |
Ref | Expression |
---|---|
eqs1 | ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → 𝑊 = 〈“(𝑊‘0)”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . 5 ⊢ ((♯‘𝑊) = 1 → (♯‘𝑊) = 1) | |
2 | s1len 13948 | . . . . 5 ⊢ (♯‘〈“(𝑊‘0)”〉) = 1 | |
3 | 1, 2 | syl6eqr 2871 | . . . 4 ⊢ ((♯‘𝑊) = 1 → (♯‘𝑊) = (♯‘〈“(𝑊‘0)”〉)) |
4 | fvex 6676 | . . . . . . . 8 ⊢ (𝑊‘0) ∈ V | |
5 | s1fv 13952 | . . . . . . . 8 ⊢ ((𝑊‘0) ∈ V → (〈“(𝑊‘0)”〉‘0) = (𝑊‘0)) | |
6 | 4, 5 | ax-mp 5 | . . . . . . 7 ⊢ (〈“(𝑊‘0)”〉‘0) = (𝑊‘0) |
7 | 6 | eqcomi 2827 | . . . . . 6 ⊢ (𝑊‘0) = (〈“(𝑊‘0)”〉‘0) |
8 | c0ex 10623 | . . . . . . 7 ⊢ 0 ∈ V | |
9 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑥 = 0 → (𝑊‘𝑥) = (𝑊‘0)) | |
10 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑥 = 0 → (〈“(𝑊‘0)”〉‘𝑥) = (〈“(𝑊‘0)”〉‘0)) | |
11 | 9, 10 | eqeq12d 2834 | . . . . . . 7 ⊢ (𝑥 = 0 → ((𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥) ↔ (𝑊‘0) = (〈“(𝑊‘0)”〉‘0))) |
12 | 8, 11 | ralsn 4611 | . . . . . 6 ⊢ (∀𝑥 ∈ {0} (𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥) ↔ (𝑊‘0) = (〈“(𝑊‘0)”〉‘0)) |
13 | 7, 12 | mpbir 232 | . . . . 5 ⊢ ∀𝑥 ∈ {0} (𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥) |
14 | oveq2 7153 | . . . . . . 7 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = (0..^1)) | |
15 | fzo01 13107 | . . . . . . 7 ⊢ (0..^1) = {0} | |
16 | 14, 15 | syl6eq 2869 | . . . . . 6 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = {0}) |
17 | 16 | raleqdv 3413 | . . . . 5 ⊢ ((♯‘𝑊) = 1 → (∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥) ↔ ∀𝑥 ∈ {0} (𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥))) |
18 | 13, 17 | mpbiri 259 | . . . 4 ⊢ ((♯‘𝑊) = 1 → ∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥)) |
19 | 3, 18 | jca 512 | . . 3 ⊢ ((♯‘𝑊) = 1 → ((♯‘𝑊) = (♯‘〈“(𝑊‘0)”〉) ∧ ∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥))) |
20 | s1cli 13947 | . . . 4 ⊢ 〈“(𝑊‘0)”〉 ∈ Word V | |
21 | eqwrd 13897 | . . . 4 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 〈“(𝑊‘0)”〉 ∈ Word V) → (𝑊 = 〈“(𝑊‘0)”〉 ↔ ((♯‘𝑊) = (♯‘〈“(𝑊‘0)”〉) ∧ ∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥)))) | |
22 | 20, 21 | mpan2 687 | . . 3 ⊢ (𝑊 ∈ Word 𝐴 → (𝑊 = 〈“(𝑊‘0)”〉 ↔ ((♯‘𝑊) = (♯‘〈“(𝑊‘0)”〉) ∧ ∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥)))) |
23 | 19, 22 | syl5ibr 247 | . 2 ⊢ (𝑊 ∈ Word 𝐴 → ((♯‘𝑊) = 1 → 𝑊 = 〈“(𝑊‘0)”〉)) |
24 | 23 | imp 407 | 1 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → 𝑊 = 〈“(𝑊‘0)”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 {csn 4557 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 ..^cfzo 13021 ♯chash 13678 Word cword 13849 〈“cs1 13937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-s1 13938 |
This theorem is referenced by: wrdl1exs1 13955 wrdl1s1 13956 swrds1 14016 revs1 14115 signsvtn0 31739 |
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