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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 476 | . . 3 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (#‘𝑊) = 1) → (#‘𝑊) = 1) | |
2 | s1len 13238 | . . 3 ⊢ (#‘〈“(𝑊‘0)”〉) = 1 | |
3 | 1, 2 | syl6eqr 2662 | . 2 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (#‘𝑊) = 1) → (#‘𝑊) = (#‘〈“(𝑊‘0)”〉)) |
4 | fvex 6113 | . . . . 5 ⊢ (𝑊‘0) ∈ V | |
5 | s1fv 13243 | . . . . . 6 ⊢ ((𝑊‘0) ∈ V → (〈“(𝑊‘0)”〉‘0) = (𝑊‘0)) | |
6 | 5 | eqcomd 2616 | . . . . 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 9913 | . . . . 5 ⊢ 0 ∈ V | |
9 | fveq2 6103 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑊‘𝑥) = (𝑊‘0)) | |
10 | fveq2 6103 | . . . . . 6 ⊢ (𝑥 = 0 → (〈“(𝑊‘0)”〉‘𝑥) = (〈“(𝑊‘0)”〉‘0)) | |
11 | 9, 10 | eqeq12d 2625 | . . . . 5 ⊢ (𝑥 = 0 → ((𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥) ↔ (𝑊‘0) = (〈“(𝑊‘0)”〉‘0))) |
12 | 8, 11 | ralsn 4169 | . . . 4 ⊢ (∀𝑥 ∈ {0} (𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥) ↔ (𝑊‘0) = (〈“(𝑊‘0)”〉‘0)) |
13 | 7, 12 | sylibr 223 | . . 3 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (#‘𝑊) = 1) → ∀𝑥 ∈ {0} (𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥)) |
14 | oveq2 6557 | . . . . . 6 ⊢ ((#‘𝑊) = 1 → (0..^(#‘𝑊)) = (0..^1)) | |
15 | 14 | adantl 481 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (#‘𝑊) = 1) → (0..^(#‘𝑊)) = (0..^1)) |
16 | fzo01 12417 | . . . . 5 ⊢ (0..^1) = {0} | |
17 | 15, 16 | syl6eq 2660 | . . . 4 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (#‘𝑊) = 1) → (0..^(#‘𝑊)) = {0}) |
18 | 17 | raleqdv 3121 | . . 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 246 | . 2 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (#‘𝑊) = 1) → ∀𝑥 ∈ (0..^(#‘𝑊))(𝑊‘𝑥) = (〈“(𝑊‘0)”〉‘𝑥)) |
20 | 1nn 10908 | . . . . 5 ⊢ 1 ∈ ℕ | |
21 | fstwrdne0 13200 | . . . . 5 ⊢ ((1 ∈ ℕ ∧ (𝑊 ∈ Word 𝐴 ∧ (#‘𝑊) = 1)) → (𝑊‘0) ∈ 𝐴) | |
22 | 20, 21 | mpan 702 | . . . 4 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (#‘𝑊) = 1) → (𝑊‘0) ∈ 𝐴) |
23 | 22 | s1cld 13236 | . . 3 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (#‘𝑊) = 1) → 〈“(𝑊‘0)”〉 ∈ Word 𝐴) |
24 | eqwrd 13201 | . . 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 486 | . 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 959 | 1 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (#‘𝑊) = 1) → 𝑊 = 〈“(𝑊‘0)”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 {csn 4125 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 ℕcn 10897 ..^cfzo 12334 #chash 12979 Word cword 13146 〈“cs1 13149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-s1 13157 |
This theorem is referenced by: wrdl1exs1 13246 wrdl1s1 13247 swrds1 13303 revs1 13365 wwlkn0 26217 signsvtn0 29973 |
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