Theorem cshwsexaOLD 14780

Description: Obsolete version of cshwsexa 14779 as of 15-Jan-2025. (Contributed by AV, 8-Jun-2018.) (Revised by Mario Carneiro/AV, 25-Oct-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cshwsexaOLD	⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V

Distinct variable groups:   𝑛,𝑊,𝑤   𝑛,𝑉

Allowed substitution hint:   𝑉(𝑤)

Proof of Theorem cshwsexaOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 3432	. . 3 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)}
2		r19.42v 3189	. . . . 5 ⊢ (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) ↔ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
3	2	bicomi 223	. . . 4 ⊢ ((𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤) ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤))
4	3	abbii 2801	. . 3 ⊢ {𝑤 ∣ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)} = {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤)}
5		df-rex 3070	. . . 4 ⊢ (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) ↔ ∃𝑛(𝑛 ∈ (0..^(♯‘𝑊)) ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤)))
6	5	abbii 2801	. . 3 ⊢ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤)} = {𝑤 ∣ ∃𝑛(𝑛 ∈ (0..^(♯‘𝑊)) ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤))}
7	1, 4, 6	3eqtri 2763	. 2 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ ∃𝑛(𝑛 ∈ (0..^(♯‘𝑊)) ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤))}
8		abid2 2870	. . . 4 ⊢ {𝑛 ∣ 𝑛 ∈ (0..^(♯‘𝑊))} = (0..^(♯‘𝑊))
9	8	ovexi 7446	. . 3 ⊢ {𝑛 ∣ 𝑛 ∈ (0..^(♯‘𝑊))} ∈ V
10		tru 1544	. . . . 5 ⊢ ⊤
11	10, 10	pm3.2i 470	. . . 4 ⊢ (⊤ ∧ ⊤)
12		ovexd 7447	. . . . . 6 ⊢ (⊤ → (𝑊 cyclShift 𝑛) ∈ V)
13		eqtr3 2757	. . . . . . . . . . . . 13 ⊢ ((𝑤 = (𝑊 cyclShift 𝑛) ∧ 𝑦 = (𝑊 cyclShift 𝑛)) → 𝑤 = 𝑦)
14	13	ex 412	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑊 cyclShift 𝑛) → (𝑦 = (𝑊 cyclShift 𝑛) → 𝑤 = 𝑦))
15	14	eqcoms 2739	. . . . . . . . . . 11 ⊢ ((𝑊 cyclShift 𝑛) = 𝑤 → (𝑦 = (𝑊 cyclShift 𝑛) → 𝑤 = 𝑦))
16	15	adantl 481	. . . . . . . . . 10 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → (𝑦 = (𝑊 cyclShift 𝑛) → 𝑤 = 𝑦))
17	16	com12 32	. . . . . . . . 9 ⊢ (𝑦 = (𝑊 cyclShift 𝑛) → ((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦))
18	17	ad2antlr 724	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 = (𝑊 cyclShift 𝑛)) ∧ ⊤) → ((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦))
19	18	alrimiv 1929	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 = (𝑊 cyclShift 𝑛)) ∧ ⊤) → ∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦))
20	19	ex 412	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 = (𝑊 cyclShift 𝑛)) → (⊤ → ∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦)))
21	12, 20	spcimedv 3585	. . . . 5 ⊢ (⊤ → (⊤ → ∃𝑦∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦)))
22	21	imp 406	. . . 4 ⊢ ((⊤ ∧ ⊤) → ∃𝑦∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦))
23	11, 22	mp1i 13	. . 3 ⊢ (𝑛 ∈ (0..^(♯‘𝑊)) → ∃𝑦∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦))
24	9, 23	zfrep4 5296	. 2 ⊢ {𝑤 ∣ ∃𝑛(𝑛 ∈ (0..^(♯‘𝑊)) ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤))} ∈ V
25	7, 24	eqeltri 2828	1 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  ⊤wtru 1541  ∃wex 1780   ∈ wcel 2105  {cab 2708  ∃wrex 3069  {crab 3431  Vcvv 3473  ‘cfv 6543  (class class class)co 7412  0cc0 11114  ..^cfzo 13632  ♯chash 14295  Word cword 14469   cyclShift ccsh 14743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-uni 4909  df-iota 6495  df-fv 6551  df-ov 7415

This theorem is referenced by: (None)