Theorem ccats1swrdeqbi 13349

Description: A word is a prefix of a word with length greater by 1 than the first word iff the second word is the first word concatenated with the last symbol of the second word. (Contributed by AV, 24-Oct-2018.)

Assertion

Ref	Expression
ccats1swrdeqbi	⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩) ↔ 𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩)))

Proof of Theorem ccats1swrdeqbi

Step	Hyp	Ref	Expression
1		ccats1swrdeq 13321	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩) → 𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩)))
2		simp1 1054	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → 𝑊 ∈ Word 𝑉)
3		lencl 13179	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
4		nn0p1nn 11209	. . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℕ0 → ((#‘𝑊) + 1) ∈ ℕ)
5	3, 4	syl 17	. . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → ((#‘𝑊) + 1) ∈ ℕ)
6	5	3ad2ant1 1075	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → ((#‘𝑊) + 1) ∈ ℕ)
7		3simpc 1053	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)))
8		lswlgt0cl 13209	. . . . . . 7 ⊢ ((((#‘𝑊) + 1) ∈ ℕ ∧ (𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1))) → ( lastS ‘𝑈) ∈ 𝑉)
9	6, 7, 8	syl2anc 691	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → ( lastS ‘𝑈) ∈ 𝑉)
10	9	s1cld 13236	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → ⟨“( lastS ‘𝑈)”⟩ ∈ Word 𝑉)
11		eqidd 2611	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (#‘𝑊) = (#‘𝑊))
12		swrdccatid 13348	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“( lastS ‘𝑈)”⟩ ∈ Word 𝑉 ∧ (#‘𝑊) = (#‘𝑊)) → ((𝑊 ++ ⟨“( lastS ‘𝑈)”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊)
13	12	eqcomd 2616	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“( lastS ‘𝑈)”⟩ ∈ Word 𝑉 ∧ (#‘𝑊) = (#‘𝑊)) → 𝑊 = ((𝑊 ++ ⟨“( lastS ‘𝑈)”⟩) substr ⟨0, (#‘𝑊)⟩))
14	2, 10, 11, 13	syl3anc 1318	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → 𝑊 = ((𝑊 ++ ⟨“( lastS ‘𝑈)”⟩) substr ⟨0, (#‘𝑊)⟩))
15		oveq1 6556	. . . . 5 ⊢ (𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩) → (𝑈 substr ⟨0, (#‘𝑊)⟩) = ((𝑊 ++ ⟨“( lastS ‘𝑈)”⟩) substr ⟨0, (#‘𝑊)⟩))
16	15	eqcomd 2616	. . . 4 ⊢ (𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩) → ((𝑊 ++ ⟨“( lastS ‘𝑈)”⟩) substr ⟨0, (#‘𝑊)⟩) = (𝑈 substr ⟨0, (#‘𝑊)⟩))
17	14, 16	sylan9eq 2664	. . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) ∧ 𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩)) → 𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩))
18	17	ex 449	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩) → 𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩)))
19	1, 18	impbid 201	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩) ↔ 𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ℕcn 10897  ℕ₀cn0 11169  #chash 12979  Word cword 13146   lastS clsw 13147   ++ cconcat 13148  ⟨“cs1 13149   substr csubstr 13150

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

This theorem is referenced by: (None)