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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 14554 | . . . . 5 ⊢ (♯‘⟨“(𝑊‘0)”⟩) = 1 | |
3 | 1, 2 | eqtr4di 2782 | . . . 4 ⊢ ((♯‘𝑊) = 1 → (♯‘𝑊) = (♯‘⟨“(𝑊‘0)”⟩)) |
4 | fvex 6895 | . . . . . . . 8 ⊢ (𝑊‘0) ∈ V | |
5 | s1fv 14558 | . . . . . . . 8 ⊢ ((𝑊‘0) ∈ V → (⟨“(𝑊‘0)”⟩‘0) = (𝑊‘0)) | |
6 | 4, 5 | ax-mp 5 | . . . . . . 7 ⊢ (⟨“(𝑊‘0)”⟩‘0) = (𝑊‘0) |
7 | 6 | eqcomi 2733 | . . . . . 6 ⊢ (𝑊‘0) = (⟨“(𝑊‘0)”⟩‘0) |
8 | c0ex 11206 | . . . . . . 7 ⊢ 0 ∈ V | |
9 | fveq2 6882 | . . . . . . . 8 ⊢ (𝑥 = 0 → (𝑊‘𝑥) = (𝑊‘0)) | |
10 | fveq2 6882 | . . . . . . . 8 ⊢ (𝑥 = 0 → (⟨“(𝑊‘0)”⟩‘𝑥) = (⟨“(𝑊‘0)”⟩‘0)) | |
11 | 9, 10 | eqeq12d 2740 | . . . . . . 7 ⊢ (𝑥 = 0 → ((𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥) ↔ (𝑊‘0) = (⟨“(𝑊‘0)”⟩‘0))) |
12 | 8, 11 | ralsn 4678 | . . . . . 6 ⊢ (∀𝑥 ∈ {0} (𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥) ↔ (𝑊‘0) = (⟨“(𝑊‘0)”⟩‘0)) |
13 | 7, 12 | mpbir 230 | . . . . 5 ⊢ ∀𝑥 ∈ {0} (𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥) |
14 | oveq2 7410 | . . . . . . 7 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = (0..^1)) | |
15 | fzo01 13712 | . . . . . . 7 ⊢ (0..^1) = {0} | |
16 | 14, 15 | eqtrdi 2780 | . . . . . 6 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = {0}) |
17 | 16 | raleqdv 3317 | . . . . 5 ⊢ ((♯‘𝑊) = 1 → (∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥) ↔ ∀𝑥 ∈ {0} (𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥))) |
18 | 13, 17 | mpbiri 258 | . . . 4 ⊢ ((♯‘𝑊) = 1 → ∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥)) |
19 | 3, 18 | jca 511 | . . 3 ⊢ ((♯‘𝑊) = 1 → ((♯‘𝑊) = (♯‘⟨“(𝑊‘0)”⟩) ∧ ∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥))) |
20 | s1cli 14553 | . . . 4 ⊢ ⟨“(𝑊‘0)”⟩ ∈ Word V | |
21 | eqwrd 14505 | . . . 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 688 | . . 3 ⊢ (𝑊 ∈ Word 𝐴 → (𝑊 = ⟨“(𝑊‘0)”⟩ ↔ ((♯‘𝑊) = (♯‘⟨“(𝑊‘0)”⟩) ∧ ∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥)))) |
23 | 19, 22 | imbitrrid 245 | . 2 ⊢ (𝑊 ∈ Word 𝐴 → ((♯‘𝑊) = 1 → 𝑊 = ⟨“(𝑊‘0)”⟩)) |
24 | 23 | imp 406 | 1 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → 𝑊 = ⟨“(𝑊‘0)”⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 Vcvv 3466 {csn 4621 ‘cfv 6534 (class class class)co 7402 0cc0 11107 1c1 11108 ..^cfzo 13625 ♯chash 14288 Word cword 14462 ⟨“cs1 14543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-n0 12471 df-z 12557 df-uz 12821 df-fz 13483 df-fzo 13626 df-hash 14289 df-word 14463 df-s1 14544 |
This theorem is referenced by: wrdl1exs1 14561 wrdl1s1 14562 swrds1 14614 revs1 14713 signsvtn0 34073 |
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