Theorem repswsymballbi 13527

Description: A word is a "repeated symbol word" iff each of its symbols equals the first symbol of the word. (Contributed by AV, 10-Nov-2018.)

Assertion

Ref	Expression
repswsymballbi	⊢ (𝑊 ∈ Word 𝑉 → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))

Distinct variable group:   𝑖,𝑊

Allowed substitution hint:   𝑉(𝑖)

Proof of Theorem repswsymballbi

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 ⊢ (𝑊 = ∅ → (#‘𝑊) = (#‘∅))
2		hash0 13158	. . . . 5 ⊢ (#‘∅) = 0
3	1, 2	syl6eq 2672	. . . 4 ⊢ (𝑊 = ∅ → (#‘𝑊) = 0)
4		fvex 6201	. . . . . . . 8 ⊢ (𝑊‘0) ∈ V
5		repsw0 13524	. . . . . . . 8 ⊢ ((𝑊‘0) ∈ V → ((𝑊‘0) repeatS 0) = ∅)
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ ((𝑊‘0) repeatS 0) = ∅
7	6	eqcomi 2631	. . . . . 6 ⊢ ∅ = ((𝑊‘0) repeatS 0)
8		simpr 477	. . . . . 6 ⊢ (((#‘𝑊) = 0 ∧ 𝑊 = ∅) → 𝑊 = ∅)
9		oveq2 6658	. . . . . . 7 ⊢ ((#‘𝑊) = 0 → ((𝑊‘0) repeatS (#‘𝑊)) = ((𝑊‘0) repeatS 0))
10	9	adantr 481	. . . . . 6 ⊢ (((#‘𝑊) = 0 ∧ 𝑊 = ∅) → ((𝑊‘0) repeatS (#‘𝑊)) = ((𝑊‘0) repeatS 0))
11	7, 8, 10	3eqtr4a 2682	. . . . 5 ⊢ (((#‘𝑊) = 0 ∧ 𝑊 = ∅) → 𝑊 = ((𝑊‘0) repeatS (#‘𝑊)))
12		ral0 4076	. . . . . . 7 ⊢ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)
13		oveq2 6658	. . . . . . . . 9 ⊢ ((#‘𝑊) = 0 → (0..^(#‘𝑊)) = (0..^0))
14		fzo0 12492	. . . . . . . . 9 ⊢ (0..^0) = ∅
15	13, 14	syl6eq 2672	. . . . . . . 8 ⊢ ((#‘𝑊) = 0 → (0..^(#‘𝑊)) = ∅)
16	15	raleqdv 3144	. . . . . . 7 ⊢ ((#‘𝑊) = 0 → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) ↔ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)))
17	12, 16	mpbiri 248	. . . . . 6 ⊢ ((#‘𝑊) = 0 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
18	17	adantr 481	. . . . 5 ⊢ (((#‘𝑊) = 0 ∧ 𝑊 = ∅) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
19	11, 18	2thd 255	. . . 4 ⊢ (((#‘𝑊) = 0 ∧ 𝑊 = ∅) → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
20	3, 19	mpancom 703	. . 3 ⊢ (𝑊 = ∅ → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
21	20	a1d 25	. 2 ⊢ (𝑊 = ∅ → (𝑊 ∈ Word 𝑉 → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))))
22		df-3an 1039	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (#‘𝑊) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)) ↔ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (#‘𝑊)) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
23	22	a1i 11	. . . 4 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (#‘𝑊) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)) ↔ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (#‘𝑊)) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))))
24		fstwrdne 13344	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (𝑊‘0) ∈ 𝑉)
25	24	ancoms 469	. . . . 5 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) → (𝑊‘0) ∈ 𝑉)
26		lencl 13324	. . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
27	26	adantl 482	. . . . 5 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) → (#‘𝑊) ∈ ℕ0)
28		repsdf2 13525	. . . . 5 ⊢ (((𝑊‘0) ∈ 𝑉 ∧ (#‘𝑊) ∈ ℕ0) → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (#‘𝑊) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))))
29	25, 27, 28	syl2anc 693	. . . 4 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (#‘𝑊) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))))
30		simpr 477	. . . . . 6 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) → 𝑊 ∈ Word 𝑉)
31		eqidd 2623	. . . . . 6 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) → (#‘𝑊) = (#‘𝑊))
32	30, 31	jca 554	. . . . 5 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (#‘𝑊)))
33	32	biantrurd 529	. . . 4 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) ↔ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (#‘𝑊)) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))))
34	23, 29, 33	3bitr4d 300	. . 3 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
35	34	ex 450	. 2 ⊢ (𝑊 ≠ ∅ → (𝑊 ∈ Word 𝑉 → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))))
36	21, 35	pm2.61ine 2877	1 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200  ∅c0 3915  ‘cfv 5888  (class class class)co 6650  0cc0 9936  ℕ₀cn0 11292  ..^cfzo 12465  #chash 13117  Word cword 13291   repeatS creps 13298

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-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-reps 13306

This theorem is referenced by:  cshw1repsw  13569  cshwsidrepsw  15800  cshwshashlem1  15802  cshwshash  15811