Theorem repswpfx 40299

Description: A prefix of a repeated symbol word is a repeated symbol word. (Contributed by AV, 11-May-2020.)

Assertion

Ref	Expression
repswpfx	⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → ((𝑆 repeatS 𝑁) prefix 𝐿) = (𝑆 repeatS 𝐿))

Proof of Theorem repswpfx

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		repsw 13373	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) ∈ Word 𝑉)
2	1	3adant3 1074	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → (𝑆 repeatS 𝑁) ∈ Word 𝑉)
3		repswlen 13374	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (#‘(𝑆 repeatS 𝑁)) = 𝑁)
4	3	eqcomd 2616	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑁 = (#‘(𝑆 repeatS 𝑁)))
5	4	oveq2d 6565	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (0...𝑁) = (0...(#‘(𝑆 repeatS 𝑁))))
6	5	eleq2d 2673	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐿 ∈ (0...𝑁) ↔ 𝐿 ∈ (0...(#‘(𝑆 repeatS 𝑁)))))
7	6	biimp3a 1424	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → 𝐿 ∈ (0...(#‘(𝑆 repeatS 𝑁))))
8		pfxlen 40254	. . . 4 ⊢ (((𝑆 repeatS 𝑁) ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(#‘(𝑆 repeatS 𝑁)))) → (#‘((𝑆 repeatS 𝑁) prefix 𝐿)) = 𝐿)
9	2, 7, 8	syl2anc 691	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → (#‘((𝑆 repeatS 𝑁) prefix 𝐿)) = 𝐿)
10		elfznn0 12302	. . . . . 6 ⊢ (𝐿 ∈ (0...𝑁) → 𝐿 ∈ ℕ0)
11	10	anim2i 591	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ (0...𝑁)) → (𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0))
12	11	3adant2 1073	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → (𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0))
13		repswlen 13374	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → (#‘(𝑆 repeatS 𝐿)) = 𝐿)
14	12, 13	syl 17	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → (#‘(𝑆 repeatS 𝐿)) = 𝐿)
15	9, 14	eqtr4d 2647	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → (#‘((𝑆 repeatS 𝑁) prefix 𝐿)) = (#‘(𝑆 repeatS 𝐿)))
16		simpl1 1057	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) ∧ 𝑖 ∈ (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿)))) → 𝑆 ∈ 𝑉)
17		simpl2 1058	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) ∧ 𝑖 ∈ (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿)))) → 𝑁 ∈ ℕ0)
18		elfzuz3 12210	. . . . . . . . 9 ⊢ (𝐿 ∈ (0...𝑁) → 𝑁 ∈ (ℤ≥‘𝐿))
19	18	3ad2ant3 1077	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ≥‘𝐿))
20	9	fveq2d 6107	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → (ℤ≥‘(#‘((𝑆 repeatS 𝑁) prefix 𝐿))) = (ℤ≥‘𝐿))
21	19, 20	eleqtrrd 2691	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ≥‘(#‘((𝑆 repeatS 𝑁) prefix 𝐿))))
22		fzoss2 12365	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘(#‘((𝑆 repeatS 𝑁) prefix 𝐿))) → (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿))) ⊆ (0..^𝑁))
23	21, 22	syl 17	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿))) ⊆ (0..^𝑁))
24	23	sselda 3568	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) ∧ 𝑖 ∈ (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿)))) → 𝑖 ∈ (0..^𝑁))
25		repswsymb 13372	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝑖) = 𝑆)
26	16, 17, 24, 25	syl3anc 1318	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) ∧ 𝑖 ∈ (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿)))) → ((𝑆 repeatS 𝑁)‘𝑖) = 𝑆)
27	2	adantr 480	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) ∧ 𝑖 ∈ (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿)))) → (𝑆 repeatS 𝑁) ∈ Word 𝑉)
28	7	adantr 480	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) ∧ 𝑖 ∈ (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿)))) → 𝐿 ∈ (0...(#‘(𝑆 repeatS 𝑁))))
29	9	oveq2d 6565	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿))) = (0..^𝐿))
30	29	eleq2d 2673	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → (𝑖 ∈ (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿))) ↔ 𝑖 ∈ (0..^𝐿)))
31	30	biimpa 500	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) ∧ 𝑖 ∈ (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿)))) → 𝑖 ∈ (0..^𝐿))
32		pfxfv 40262	. . . . 5 ⊢ (((𝑆 repeatS 𝑁) ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(#‘(𝑆 repeatS 𝑁))) ∧ 𝑖 ∈ (0..^𝐿)) → (((𝑆 repeatS 𝑁) prefix 𝐿)‘𝑖) = ((𝑆 repeatS 𝑁)‘𝑖))
33	27, 28, 31, 32	syl3anc 1318	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) ∧ 𝑖 ∈ (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿)))) → (((𝑆 repeatS 𝑁) prefix 𝐿)‘𝑖) = ((𝑆 repeatS 𝑁)‘𝑖))
34	10	3ad2ant3 1077	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → 𝐿 ∈ ℕ0)
35	34	adantr 480	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) ∧ 𝑖 ∈ (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿)))) → 𝐿 ∈ ℕ0)
36		repswsymb 13372	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0 ∧ 𝑖 ∈ (0..^𝐿)) → ((𝑆 repeatS 𝐿)‘𝑖) = 𝑆)
37	16, 35, 31, 36	syl3anc 1318	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) ∧ 𝑖 ∈ (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿)))) → ((𝑆 repeatS 𝐿)‘𝑖) = 𝑆)
38	26, 33, 37	3eqtr4d 2654	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) ∧ 𝑖 ∈ (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿)))) → (((𝑆 repeatS 𝑁) prefix 𝐿)‘𝑖) = ((𝑆 repeatS 𝐿)‘𝑖))
39	38	ralrimiva 2949	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → ∀𝑖 ∈ (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿)))(((𝑆 repeatS 𝑁) prefix 𝐿)‘𝑖) = ((𝑆 repeatS 𝐿)‘𝑖))
40		pfxcl 40249	. . . 4 ⊢ ((𝑆 repeatS 𝑁) ∈ Word 𝑉 → ((𝑆 repeatS 𝑁) prefix 𝐿) ∈ Word 𝑉)
41	2, 40	syl 17	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → ((𝑆 repeatS 𝑁) prefix 𝐿) ∈ Word 𝑉)
42		repsw 13373	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑆 repeatS 𝐿) ∈ Word 𝑉)
43	12, 42	syl 17	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → (𝑆 repeatS 𝐿) ∈ Word 𝑉)
44		eqwrd 13201	. . 3 ⊢ ((((𝑆 repeatS 𝑁) prefix 𝐿) ∈ Word 𝑉 ∧ (𝑆 repeatS 𝐿) ∈ Word 𝑉) → (((𝑆 repeatS 𝑁) prefix 𝐿) = (𝑆 repeatS 𝐿) ↔ ((#‘((𝑆 repeatS 𝑁) prefix 𝐿)) = (#‘(𝑆 repeatS 𝐿)) ∧ ∀𝑖 ∈ (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿)))(((𝑆 repeatS 𝑁) prefix 𝐿)‘𝑖) = ((𝑆 repeatS 𝐿)‘𝑖))))
45	41, 43, 44	syl2anc 691	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → (((𝑆 repeatS 𝑁) prefix 𝐿) = (𝑆 repeatS 𝐿) ↔ ((#‘((𝑆 repeatS 𝑁) prefix 𝐿)) = (#‘(𝑆 repeatS 𝐿)) ∧ ∀𝑖 ∈ (0..^(#‘((𝑆 repeatS 𝑁) prefix 𝐿)))(((𝑆 repeatS 𝑁) prefix 𝐿)‘𝑖) = ((𝑆 repeatS 𝐿)‘𝑖))))
46	15, 39, 45	mpbir2and 959	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → ((𝑆 repeatS 𝑁) prefix 𝐿) = (𝑆 repeatS 𝐿))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℕ₀cn0 11169  ℤ_≥cuz 11563  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   repeatS creps 13153   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-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-substr 13158  df-reps 13161  df-pfx 40245

This theorem is referenced by: (None)