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Mirrors > Home > MPE Home > Th. List > Mathboxes > ccats1pfxeqrex | Structured version Visualization version GIF version |
Description: There exists a symbol such that its concatenation with the prefix obtained by deleting the last symbol of a nonempty word results in the word itself. Could replace ccats1swrdeqrex 13330. (Contributed by AV, 9-May-2020.) |
Ref | Expression |
---|---|
ccats1pfxeqrex | ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 prefix (#‘𝑊)) → ∃𝑠 ∈ 𝑉 𝑈 = (𝑊 ++ 〈“𝑠”〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1055 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → 𝑈 ∈ Word 𝑉) | |
2 | lencl 13179 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0) | |
3 | 2 | 3ad2ant1 1075 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (#‘𝑊) ∈ ℕ0) |
4 | nn0p1nn 11209 | . . . . . 6 ⊢ ((#‘𝑊) ∈ ℕ0 → ((#‘𝑊) + 1) ∈ ℕ) | |
5 | nngt0 10926 | . . . . . 6 ⊢ (((#‘𝑊) + 1) ∈ ℕ → 0 < ((#‘𝑊) + 1)) | |
6 | 3, 4, 5 | 3syl 18 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → 0 < ((#‘𝑊) + 1)) |
7 | breq2 4587 | . . . . . 6 ⊢ ((#‘𝑈) = ((#‘𝑊) + 1) → (0 < (#‘𝑈) ↔ 0 < ((#‘𝑊) + 1))) | |
8 | 7 | 3ad2ant3 1077 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (0 < (#‘𝑈) ↔ 0 < ((#‘𝑊) + 1))) |
9 | 6, 8 | mpbird 246 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → 0 < (#‘𝑈)) |
10 | hashgt0n0 13017 | . . . 4 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → 𝑈 ≠ ∅) | |
11 | 1, 9, 10 | syl2anc 691 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → 𝑈 ≠ ∅) |
12 | lswcl 13208 | . . 3 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 𝑈 ≠ ∅) → ( lastS ‘𝑈) ∈ 𝑉) | |
13 | 1, 11, 12 | syl2anc 691 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → ( lastS ‘𝑈) ∈ 𝑉) |
14 | ccats1pfxeq 40284 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 prefix (#‘𝑊)) → 𝑈 = (𝑊 ++ 〈“( lastS ‘𝑈)”〉))) | |
15 | s1eq 13233 | . . . . 5 ⊢ (𝑠 = ( lastS ‘𝑈) → 〈“𝑠”〉 = 〈“( lastS ‘𝑈)”〉) | |
16 | 15 | oveq2d 6565 | . . . 4 ⊢ (𝑠 = ( lastS ‘𝑈) → (𝑊 ++ 〈“𝑠”〉) = (𝑊 ++ 〈“( lastS ‘𝑈)”〉)) |
17 | 16 | eqeq2d 2620 | . . 3 ⊢ (𝑠 = ( lastS ‘𝑈) → (𝑈 = (𝑊 ++ 〈“𝑠”〉) ↔ 𝑈 = (𝑊 ++ 〈“( lastS ‘𝑈)”〉))) |
18 | 17 | rspcev 3282 | . 2 ⊢ ((( lastS ‘𝑈) ∈ 𝑉 ∧ 𝑈 = (𝑊 ++ 〈“( lastS ‘𝑈)”〉)) → ∃𝑠 ∈ 𝑉 𝑈 = (𝑊 ++ 〈“𝑠”〉)) |
19 | 13, 14, 18 | syl6an 566 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 prefix (#‘𝑊)) → ∃𝑠 ∈ 𝑉 𝑈 = (𝑊 ++ 〈“𝑠”〉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ∅c0 3874 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ℕcn 10897 ℕ0cn0 11169 #chash 12979 Word cword 13146 lastS clsw 13147 ++ cconcat 13148 〈“cs1 13149 prefix cpfx 40244 |
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-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-pfx 40245 |
This theorem is referenced by: reuccatpfxs1lem 40296 |
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