Theorem swrdccatid 13497

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 1058	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
2		lencl 13324	. . . . 5 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
3		lencl 13324	. . . . . 6 ⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℕ0)
4		simplr 792	. . . . . . . . 9 ⊢ ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → (#‘𝐴) ∈ ℕ0)
5		eleq1 2689	. . . . . . . . . 10 ⊢ (𝑁 = (#‘𝐴) → (𝑁 ∈ ℕ0 ↔ (#‘𝐴) ∈ ℕ0))
6	5	adantl 482	. . . . . . . . 9 ⊢ ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → (𝑁 ∈ ℕ0 ↔ (#‘𝐴) ∈ ℕ0))
7	4, 6	mpbird 247	. . . . . . . 8 ⊢ ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → 𝑁 ∈ ℕ0)
8		nn0addcl 11328	. . . . . . . . . 10 ⊢ (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0) → ((#‘𝐴) + (#‘𝐵)) ∈ ℕ0)
9	8	ancoms 469	. . . . . . . . 9 ⊢ (((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → ((#‘𝐴) + (#‘𝐵)) ∈ ℕ0)
10	9	adantr 481	. . . . . . . 8 ⊢ ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → ((#‘𝐴) + (#‘𝐵)) ∈ ℕ0)
11		nn0re 11301	. . . . . . . . . . . . 13 ⊢ ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ ℝ)
12	11	anim1i 592	. . . . . . . . . . . 12 ⊢ (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0) → ((#‘𝐴) ∈ ℝ ∧ (#‘𝐵) ∈ ℕ0))
13	12	ancoms 469	. . . . . . . . . . 11 ⊢ (((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → ((#‘𝐴) ∈ ℝ ∧ (#‘𝐵) ∈ ℕ0))
14		nn0addge1 11339	. . . . . . . . . . 11 ⊢ (((#‘𝐴) ∈ ℝ ∧ (#‘𝐵) ∈ ℕ0) → (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝐵)))
15	13, 14	syl 17	. . . . . . . . . 10 ⊢ (((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝐵)))
16	15	adantr 481	. . . . . . . . 9 ⊢ ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝐵)))
17		breq1 4656	. . . . . . . . . 10 ⊢ (𝑁 = (#‘𝐴) → (𝑁 ≤ ((#‘𝐴) + (#‘𝐵)) ↔ (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝐵))))
18	17	adantl 482	. . . . . . . . 9 ⊢ ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → (𝑁 ≤ ((#‘𝐴) + (#‘𝐵)) ↔ (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝐵))))
19	16, 18	mpbird 247	. . . . . . . 8 ⊢ ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → 𝑁 ≤ ((#‘𝐴) + (#‘𝐵)))
20		elfz2nn0 12431	. . . . . . . 8 ⊢ (𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))) ↔ (𝑁 ∈ ℕ0 ∧ ((#‘𝐴) + (#‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ ((#‘𝐴) + (#‘𝐵))))
21	7, 10, 19, 20	syl3anbrc 1246	. . . . . . 7 ⊢ ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))
22	21	exp31 630	. . . . . 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 1256	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))
26		eqid 2622	. . . 4 ⊢ (#‘𝐴) = (#‘𝐴)
27	26	swrdccat3a 13494	. . 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 65	. 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 10594	. . . . 5 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ≤ (#‘𝐴))
31	30	3ad2ant1 1082	. . . 4 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → (#‘𝐴) ≤ (#‘𝐴))
32		breq1 4656	. . . . 5 ⊢ (𝑁 = (#‘𝐴) → (𝑁 ≤ (#‘𝐴) ↔ (#‘𝐴) ≤ (#‘𝐴)))
33	32	3ad2ant3 1084	. . . 4 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → (𝑁 ≤ (#‘𝐴) ↔ (#‘𝐴) ≤ (#‘𝐴)))
34	31, 33	mpbird 247	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → 𝑁 ≤ (#‘𝐴))
35	34	iftrued 4094	. 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 13428	. . . 4 ⊢ (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨0, (#‘𝐴)⟩) = 𝐴)
37	36	3ad2ant1 1082	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → (𝐴 substr ⟨0, (#‘𝐴)⟩) = 𝐴)
38		opeq2 4403	. . . . . 6 ⊢ (𝑁 = (#‘𝐴) → ⟨0, 𝑁⟩ = ⟨0, (#‘𝐴)⟩)
39	38	oveq2d 6666	. . . . 5 ⊢ (𝑁 = (#‘𝐴) → (𝐴 substr ⟨0, 𝑁⟩) = (𝐴 substr ⟨0, (#‘𝐴)⟩))
40	39	eqeq1d 2624	. . . 4 ⊢ (𝑁 = (#‘𝐴) → ((𝐴 substr ⟨0, 𝑁⟩) = 𝐴 ↔ (𝐴 substr ⟨0, (#‘𝐴)⟩) = 𝐴))
41	40	3ad2ant3 1084	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → ((𝐴 substr ⟨0, 𝑁⟩) = 𝐴 ↔ (𝐴 substr ⟨0, (#‘𝐴)⟩) = 𝐴))
42	37, 41	mpbird 247	. 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → (𝐴 substr ⟨0, 𝑁⟩) = 𝐴)
43	28, 35, 42	3eqtrd 2660	1 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝐴)) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ifcif 4086  ⟨cop 4183   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  0cc0 9936   + caddc 9939   ≤ cle 10075   − cmin 10266  ℕ₀cn0 11292  ...cfz 12326  #chash 13117  Word cword 13291   ++ cconcat 13293   substr csubstr 13295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303

This theorem is referenced by:  ccats1swrdeqbi  13498  clwlkclwwlk2  26904  clwlksfoclwwlk  26963  numclwlk1lem2foalem  27222  numclwlk1lem2fo  27228