Theorem repswsymballbi 13378

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 6103	. . . . 5 ⊢ (𝑊 = ∅ → (#‘𝑊) = (#‘∅))
2		hash0 13019	. . . . 5 ⊢ (#‘∅) = 0
3	1, 2	syl6eq 2660	. . . 4 ⊢ (𝑊 = ∅ → (#‘𝑊) = 0)
4		fvex 6113	. . . . . . . 8 ⊢ (𝑊‘0) ∈ V
5		repsw0 13375	. . . . . . . 8 ⊢ ((𝑊‘0) ∈ V → ((𝑊‘0) repeatS 0) = ∅)
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ ((𝑊‘0) repeatS 0) = ∅
7	6	eqcomi 2619	. . . . . 6 ⊢ ∅ = ((𝑊‘0) repeatS 0)
8		simpr 476	. . . . . 6 ⊢ (((#‘𝑊) = 0 ∧ 𝑊 = ∅) → 𝑊 = ∅)
9		oveq2 6557	. . . . . . 7 ⊢ ((#‘𝑊) = 0 → ((𝑊‘0) repeatS (#‘𝑊)) = ((𝑊‘0) repeatS 0))
10	9	adantr 480	. . . . . 6 ⊢ (((#‘𝑊) = 0 ∧ 𝑊 = ∅) → ((𝑊‘0) repeatS (#‘𝑊)) = ((𝑊‘0) repeatS 0))
11	7, 8, 10	3eqtr4a 2670	. . . . 5 ⊢ (((#‘𝑊) = 0 ∧ 𝑊 = ∅) → 𝑊 = ((𝑊‘0) repeatS (#‘𝑊)))
12		ral0 4028	. . . . . . 7 ⊢ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)
13		oveq2 6557	. . . . . . . . 9 ⊢ ((#‘𝑊) = 0 → (0..^(#‘𝑊)) = (0..^0))
14		fzo0 12361	. . . . . . . . 9 ⊢ (0..^0) = ∅
15	13, 14	syl6eq 2660	. . . . . . . 8 ⊢ ((#‘𝑊) = 0 → (0..^(#‘𝑊)) = ∅)
16	15	raleqdv 3121	. . . . . . 7 ⊢ ((#‘𝑊) = 0 → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) ↔ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)))
17	12, 16	mpbiri 247	. . . . . 6 ⊢ ((#‘𝑊) = 0 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
18	17	adantr 480	. . . . 5 ⊢ (((#‘𝑊) = 0 ∧ 𝑊 = ∅) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
19	11, 18	2thd 254	. . . 4 ⊢ (((#‘𝑊) = 0 ∧ 𝑊 = ∅) → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
20	3, 19	mpancom 700	. . 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 1033	. . . . 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 13199	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (𝑊‘0) ∈ 𝑉)
25	24	ancoms 468	. . . . 5 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) → (𝑊‘0) ∈ 𝑉)
26		lencl 13179	. . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
27	26	adantl 481	. . . . 5 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) → (#‘𝑊) ∈ ℕ0)
28		repsdf2 13376	. . . . 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 691	. . . 4 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (#‘𝑊) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))))
30		simpr 476	. . . . . 6 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) → 𝑊 ∈ Word 𝑉)
31		eqidd 2611	. . . . . 6 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) → (#‘𝑊) = (#‘𝑊))
32	30, 31	jca 553	. . . . 5 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (#‘𝑊)))
33	32	biantrurd 528	. . . 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 299	. . 3 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
35	34	ex 449	. 2 ⊢ (𝑊 ≠ ∅ → (𝑊 ∈ Word 𝑉 → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))))
36	21, 35	pm2.61ine 2865	1 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  ∅c0 3874  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℕ₀cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   repeatS creps 13153

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-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-reps 13161

This theorem is referenced by:  cshw1repsw  13420  cshwsidrepsw  15638  cshwshashlem1  15640  cshwshash  15649