Theorem swrdccat3b 13347

Description: A suffix of a concatenation is either a suffix of the second concatenated word or a concatenation of a suffix of the first word with the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by Alexander van der Vekens, 30-May-2018.)

Hypothesis

Ref	Expression
swrdccatin12.l	⊢ 𝐿 = (#‘𝐴)

Assertion

Ref	Expression
swrdccat3b	⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵))))

Proof of Theorem swrdccat3b

Step	Hyp	Ref	Expression
1		simpl 472	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
2		simpr 476	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → 𝑀 ∈ (0...(𝐿 + (#‘𝐵))))
3		elfzubelfz 12224	. . . . 5 ⊢ (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) → (𝐿 + (#‘𝐵)) ∈ (0...(𝐿 + (#‘𝐵))))
4	3	adantl 481	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝐿 + (#‘𝐵)) ∈ (0...(𝐿 + (#‘𝐵))))
5		swrdccatin12.l	. . . . . 6 ⊢ 𝐿 = (#‘𝐴)
6	5	swrdccat3 13343	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (𝐿 + (#‘𝐵)) ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = if((𝐿 + (#‘𝐵)) ≤ 𝐿, (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩), if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), ((𝐿 + (#‘𝐵)) − 𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩))))))
7	6	imp 444	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (𝐿 + (#‘𝐵)) ∈ (0...(𝐿 + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = if((𝐿 + (#‘𝐵)) ≤ 𝐿, (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩), if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), ((𝐿 + (#‘𝐵)) − 𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩)))))
8	1, 2, 4, 7	syl12anc 1316	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = if((𝐿 + (#‘𝐵)) ≤ 𝐿, (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩), if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), ((𝐿 + (#‘𝐵)) − 𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩)))))
9	5	swrdccat3blem 13346	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ (𝐿 + (#‘𝐵)) ≤ 𝐿) → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩))
10		iftrue 4042	. . . . . 6 ⊢ (𝐿 ≤ 𝑀 → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩))
11	10	3ad2ant3 1077	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩))
12		lencl 13179	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
13	12	nn0cnd 11230	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℂ)
14		lencl 13179	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℕ0)
15	14	nn0cnd 11230	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℂ)
16	5	eqcomi 2619	. . . . . . . . . . . . 13 ⊢ (#‘𝐴) = 𝐿
17	16	eleq1i 2679	. . . . . . . . . . . 12 ⊢ ((#‘𝐴) ∈ ℂ ↔ 𝐿 ∈ ℂ)
18		pncan2 10167	. . . . . . . . . . . 12 ⊢ ((𝐿 ∈ ℂ ∧ (#‘𝐵) ∈ ℂ) → ((𝐿 + (#‘𝐵)) − 𝐿) = (#‘𝐵))
19	17, 18	sylanb 488	. . . . . . . . . . 11 ⊢ (((#‘𝐴) ∈ ℂ ∧ (#‘𝐵) ∈ ℂ) → ((𝐿 + (#‘𝐵)) − 𝐿) = (#‘𝐵))
20	13, 15, 19	syl2an 493	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝐿 + (#‘𝐵)) − 𝐿) = (#‘𝐵))
21	20	eqcomd 2616	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (#‘𝐵) = ((𝐿 + (#‘𝐵)) − 𝐿))
22	21	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → (#‘𝐵) = ((𝐿 + (#‘𝐵)) − 𝐿))
23	22	3ad2ant1 1075	. . . . . . 7 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → (#‘𝐵) = ((𝐿 + (#‘𝐵)) − 𝐿))
24	23	opeq2d 4347	. . . . . 6 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → ⟨(𝑀 − 𝐿), (#‘𝐵)⟩ = ⟨(𝑀 − 𝐿), ((𝐿 + (#‘𝐵)) − 𝐿)⟩)
25	24	oveq2d 6565	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩) = (𝐵 substr ⟨(𝑀 − 𝐿), ((𝐿 + (#‘𝐵)) − 𝐿)⟩))
26	11, 25	eqtrd 2644	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐵 substr ⟨(𝑀 − 𝐿), ((𝐿 + (#‘𝐵)) − 𝐿)⟩))
27		iffalse 4045	. . . . . 6 ⊢ (¬ 𝐿 ≤ 𝑀 → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵))
28	27	3ad2ant3 1077	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵))
29	20	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐿 + (#‘𝐵)) − 𝐿) = (#‘𝐵))
30	29	3ad2ant1 1075	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → ((𝐿 + (#‘𝐵)) − 𝐿) = (#‘𝐵))
31	30	opeq2d 4347	. . . . . . . 8 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩ = ⟨0, (#‘𝐵)⟩)
32	31	oveq2d 6565	. . . . . . 7 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩) = (𝐵 substr ⟨0, (#‘𝐵)⟩))
33		swrdid 13280	. . . . . . . . . 10 ⊢ (𝐵 ∈ Word 𝑉 → (𝐵 substr ⟨0, (#‘𝐵)⟩) = 𝐵)
34	33	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐵 substr ⟨0, (#‘𝐵)⟩) = 𝐵)
35	34	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝐵 substr ⟨0, (#‘𝐵)⟩) = 𝐵)
36	35	3ad2ant1 1075	. . . . . . 7 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → (𝐵 substr ⟨0, (#‘𝐵)⟩) = 𝐵)
37	32, 36	eqtr2d 2645	. . . . . 6 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → 𝐵 = (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩))
38	37	oveq2d 6565	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩)))
39	28, 38	eqtrd 2644	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩)))
40	9, 26, 39	2if2 4086	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = if((𝐿 + (#‘𝐵)) ≤ 𝐿, (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩), if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), ((𝐿 + (#‘𝐵)) − 𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩)))))
41	8, 40	eqtr4d 2647	. 2 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)))
42	41	ex 449	1 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ifcif 4036  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815   + caddc 9818   ≤ cle 9954   − cmin 10145  ...cfz 12197  #chash 12979  Word cword 13146   ++ cconcat 13148   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-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158

This theorem is referenced by: (None)