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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 | simpr 477 | . . 3 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → (♯‘𝑊) = 1) | |
2 | s1len 13767 | . . 3 ⊢ (♯‘〈“(𝑊‘0)”〉) = 1 | |
3 | 1, 2 | syl6eqr 2825 | . 2 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → (♯‘𝑊) = (♯‘〈“(𝑊‘0)”〉)) |
4 | fvex 6509 | . . . . 5 ⊢ (𝑊‘0) ∈ V | |
5 | s1fv 13771 | . . . . . 6 ⊢ ((𝑊‘0) ∈ V → (〈“(𝑊‘0)”〉‘0) = (𝑊‘0)) | |
6 | 5 | eqcomd 2777 | . . . . 5 ⊢ ((𝑊‘0) ∈ V → (𝑊‘0) = (〈“(𝑊‘0)”〉‘0)) |
7 | 4, 6 | mp1i 13 | . . . 4 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → (𝑊‘0) = (〈“(𝑊‘0)”〉‘0)) |
8 | c0ex 10431 | . . . . 5 ⊢ 0 ∈ V | |
9 | fveq2 6496 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑊‘𝑥) = (𝑊‘0)) | |
10 | fveq2 6496 | . . . . . 6 ⊢ (𝑥 = 0 → (〈“(𝑊‘0)”〉‘𝑥) = (〈“(𝑊‘0)”〉‘0)) | |
11 | 9, 10 | eqeq12d 2786 | . . . . 5 ⊢ (𝑥 = 0 → ((𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥) ↔ (𝑊‘0) = (〈“(𝑊‘0)”〉‘0))) |
12 | 8, 11 | ralsn 4489 | . . . 4 ⊢ (∀𝑥 ∈ {0} (𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥) ↔ (𝑊‘0) = (〈“(𝑊‘0)”〉‘0)) |
13 | 7, 12 | sylibr 226 | . . 3 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → ∀𝑥 ∈ {0} (𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥)) |
14 | oveq2 6982 | . . . . . 6 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = (0..^1)) | |
15 | 14 | adantl 474 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → (0..^(♯‘𝑊)) = (0..^1)) |
16 | fzo01 12932 | . . . . 5 ⊢ (0..^1) = {0} | |
17 | 15, 16 | syl6eq 2823 | . . . 4 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → (0..^(♯‘𝑊)) = {0}) |
18 | 17 | raleqdv 3348 | . . 3 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → (∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥) ↔ ∀𝑥 ∈ {0} (𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥))) |
19 | 13, 18 | mpbird 249 | . 2 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → ∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥)) |
20 | 1nn 11450 | . . . . 5 ⊢ 1 ∈ ℕ | |
21 | fstwrdne0 13717 | . . . . 5 ⊢ ((1 ∈ ℕ ∧ (𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1)) → (𝑊‘0) ∈ 𝐴) | |
22 | 20, 21 | mpan 678 | . . . 4 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → (𝑊‘0) ∈ 𝐴) |
23 | 22 | s1cld 13764 | . . 3 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → 〈“(𝑊‘0)”〉 ∈ Word 𝐴) |
24 | eqwrd 13718 | . . 3 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 〈“(𝑊‘0)”〉 ∈ Word 𝐴) → (𝑊 = 〈“(𝑊‘0)”〉 ↔ ((♯‘𝑊) = (♯‘〈“(𝑊‘0)”〉) ∧ ∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥)))) | |
25 | 23, 24 | syldan 583 | . 2 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → (𝑊 = 〈“(𝑊‘0)”〉 ↔ ((♯‘𝑊) = (♯‘〈“(𝑊‘0)”〉) ∧ ∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥)))) |
26 | 3, 19, 25 | mpbir2and 701 | 1 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → 𝑊 = 〈“(𝑊‘0)”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∀wral 3081 Vcvv 3408 {csn 4435 ‘cfv 6185 (class class class)co 6974 0cc0 10333 1c1 10334 ℕcn 11437 ..^cfzo 12847 ♯chash 13503 Word cword 13670 〈“cs1 13756 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-n0 11706 df-z 11792 df-uz 12057 df-fz 12707 df-fzo 12848 df-hash 13504 df-word 13671 df-s1 13757 |
This theorem is referenced by: wrdl1exs1 13774 wrdl1s1 13775 swrds1 13842 revs1 13982 signsvtn0 31518 signsvtn0OLD 31519 |
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