Theorem swrdccatid 13348

Description: A prefix of a concatenation of length of the first concatenated word is the first word itself. (Contributed by Alexander van der Vekens, 20-Sep-2018.)

Assertion

Ref	Expression
swrdccatid	⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = 𝐴)

Proof of Theorem swrdccatid

Step	Hyp	Ref	Expression
1		3simpa 1051	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
2		lencl 13179	. . . . 5 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
3		lencl 13179	. . . . . 6 ⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℕ0)
4		simplr 788	. . . . . . . . 9 ⊢ ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → (#‘𝐴) ∈ ℕ0)
5		eleq1 2676	. . . . . . . . . 10 ⊢ (𝑁 = (#‘𝐴) → (𝑁 ∈ ℕ0 ↔ (#‘𝐴) ∈ ℕ0))
6	5	adantl 481	. . . . . . . . 9 ⊢ ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → (𝑁 ∈ ℕ0 ↔ (#‘𝐴) ∈ ℕ0))
7	4, 6	mpbird 246	. . . . . . . 8 ⊢ ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → 𝑁 ∈ ℕ0)
8		nn0addcl 11205	. . . . . . . . . 10 ⊢ (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0) → ((#‘𝐴) + (#‘𝐵)) ∈ ℕ0)
9	8	ancoms 468	. . . . . . . . 9 ⊢ (((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → ((#‘𝐴) + (#‘𝐵)) ∈ ℕ0)
10	9	adantr 480	. . . . . . . 8 ⊢ ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → ((#‘𝐴) + (#‘𝐵)) ∈ ℕ0)
11		nn0re 11178	. . . . . . . . . . . . 13 ⊢ ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ ℝ)
12	11	anim1i 590	. . . . . . . . . . . 12 ⊢ (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0) → ((#‘𝐴) ∈ ℝ ∧ (#‘𝐵) ∈ ℕ0))
13	12	ancoms 468	. . . . . . . . . . 11 ⊢ (((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → ((#‘𝐴) ∈ ℝ ∧ (#‘𝐵) ∈ ℕ0))
14		nn0addge1 11216	. . . . . . . . . . 11 ⊢ (((#‘𝐴) ∈ ℝ ∧ (#‘𝐵) ∈ ℕ0) → (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝐵)))
15	13, 14	syl 17	. . . . . . . . . 10 ⊢ (((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝐵)))
16	15	adantr 480	. . . . . . . . 9 ⊢ ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝐵)))
17		breq1 4586	. . . . . . . . . 10 ⊢ (𝑁 = (#‘𝐴) → (𝑁 ≤ ((#‘𝐴) + (#‘𝐵)) ↔ (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝐵))))
18	17	adantl 481	. . . . . . . . 9 ⊢ ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → (𝑁 ≤ ((#‘𝐴) + (#‘𝐵)) ↔ (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝐵))))
19	16, 18	mpbird 246	. . . . . . . 8 ⊢ ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → 𝑁 ≤ ((#‘𝐴) + (#‘𝐵)))
20		elfz2nn0 12300	. . . . . . . 8 ⊢ (𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))) ↔ (𝑁 ∈ ℕ0 ∧ ((#‘𝐴) + (#‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ ((#‘𝐴) + (#‘𝐵))))
21	7, 10, 19, 20	syl3anbrc 1239	. . . . . . 7 ⊢ ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))
22	21	exp31 628	. . . . . 6 ⊢ ((#‘𝐵) ∈ ℕ0 → ((#‘𝐴) ∈ ℕ0 → (𝑁 = (#‘𝐴) → 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))))
23	3, 22	syl 17	. . . . 5 ⊢ (𝐵 ∈ Word 𝑉 → ((#‘𝐴) ∈ ℕ0 → (𝑁 = (#‘𝐴) → 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))))
24	2, 23	syl5com 31	. . . 4 ⊢ (𝐴 ∈ Word 𝑉 → (𝐵 ∈ Word 𝑉 → (𝑁 = (#‘𝐴) → 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))))
25	24	3imp 1249	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))
26		eqid 2610	. . . 4 ⊢ (#‘𝐴) = (#‘𝐴)
27	26	swrdccat3a 13345	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = if(𝑁 ≤ (#‘𝐴), (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − (#‘𝐴))⟩)))))
28	1, 25, 27	sylc 63	. 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = if(𝑁 ≤ (#‘𝐴), (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − (#‘𝐴))⟩))))
29	2, 11	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℝ)
30	29	leidd 10473	. . . . 5 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ≤ (#‘𝐴))
31	30	3ad2ant1 1075	. . . 4 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → (#‘𝐴) ≤ (#‘𝐴))
32		breq1 4586	. . . . 5 ⊢ (𝑁 = (#‘𝐴) → (𝑁 ≤ (#‘𝐴) ↔ (#‘𝐴) ≤ (#‘𝐴)))
33	32	3ad2ant3 1077	. . . 4 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → (𝑁 ≤ (#‘𝐴) ↔ (#‘𝐴) ≤ (#‘𝐴)))
34	31, 33	mpbird 246	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → 𝑁 ≤ (#‘𝐴))
35	34	iftrued 4044	. 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → if(𝑁 ≤ (#‘𝐴), (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − (#‘𝐴))⟩))) = (𝐴 substr ⟨0, 𝑁⟩))
36		swrdid 13280	. . . 4 ⊢ (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨0, (#‘𝐴)⟩) = 𝐴)
37	36	3ad2ant1 1075	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → (𝐴 substr ⟨0, (#‘𝐴)⟩) = 𝐴)
38		opeq2 4341	. . . . . 6 ⊢ (𝑁 = (#‘𝐴) → ⟨0, 𝑁⟩ = ⟨0, (#‘𝐴)⟩)
39	38	oveq2d 6565	. . . . 5 ⊢ (𝑁 = (#‘𝐴) → (𝐴 substr ⟨0, 𝑁⟩) = (𝐴 substr ⟨0, (#‘𝐴)⟩))
40	39	eqeq1d 2612	. . . 4 ⊢ (𝑁 = (#‘𝐴) → ((𝐴 substr ⟨0, 𝑁⟩) = 𝐴 ↔ (𝐴 substr ⟨0, (#‘𝐴)⟩) = 𝐴))
41	40	3ad2ant3 1077	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → ((𝐴 substr ⟨0, 𝑁⟩) = 𝐴 ↔ (𝐴 substr ⟨0, (#‘𝐴)⟩) = 𝐴))
42	37, 41	mpbird 246	. 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → (𝐴 substr ⟨0, 𝑁⟩) = 𝐴)
43	28, 35, 42	3eqtrd 2648	1 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ifcif 4036  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815   + caddc 9818   ≤ cle 9954   − cmin 10145  ℕ₀cn0 11169  ...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:  ccats1swrdeqbi  13349  clwlkisclwwlk2  26318  clwlkfoclwwlk  26372  numclwlk1lem2foa  26618  numclwlk1lem2fo  26622  clwlkclwwlk2  41212  clwlksfoclwwlk  41270  av-numclwlk1lem2foa  41521  av-numclwlk1lem2fo  41525