repsdf2 - Metamath Proof Explorer

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Theorem repsdf2 13725

Description: Alternative definition of a "repeated symbol word". (Contributed by AV, 7-Nov-2018.)

**Assertion**
Ref	Expression
repsdf2	⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑊 = (𝑆 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆)))

Distinct variable groups: 𝑖,𝑁 𝑆,𝑖 𝑖,𝑊 Allowed substitution hint: 𝑉(𝑖)

**Proof of Theorem repsdf2**
Step	Hyp	Ref	Expression
1		repsconst 13719	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑆 repeatS 𝑁) = ((0..^𝑁) × {𝑆}))
2	1	eqeq2d 2770	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑊 = (𝑆 repeatS 𝑁) ↔ 𝑊 = ((0..^𝑁) × {𝑆})))
3		fconst2g 6632	. . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑊:(0..^𝑁)⟶{𝑆} ↔ 𝑊 = ((0..^𝑁) × {𝑆})))
4	3	adantr 472	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑊:(0..^𝑁)⟶{𝑆} ↔ 𝑊 = ((0..^𝑁) × {𝑆})))
5		fconstfv 6640	. . . . . . . . 9 ⊢ (𝑊:(0..^𝑁)⟶{𝑆} ↔ (𝑊 Fn (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆))
6		simpr 479	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ 𝑊:(0..^𝑁)⟶{𝑆}) → 𝑊:(0..^𝑁)⟶{𝑆})
7		snssi 4484	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ 𝑉 → {𝑆} ⊆ 𝑉)
8	7	adantr 472	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → {𝑆} ⊆ 𝑉)
9	8	adantr 472	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ 𝑊:(0..^𝑁)⟶{𝑆}) → {𝑆} ⊆ 𝑉)
10	6, 9	jca 555	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ 𝑊:(0..^𝑁)⟶{𝑆}) → (𝑊:(0..^𝑁)⟶{𝑆} ∧ {𝑆} ⊆ 𝑉))
11		fss 6217	. . . . . . . . . . . 12 ⊢ ((𝑊:(0..^𝑁)⟶{𝑆} ∧ {𝑆} ⊆ 𝑉) → 𝑊:(0..^𝑁)⟶𝑉)
12		iswrdi 13495	. . . . . . . . . . . 12 ⊢ (𝑊:(0..^𝑁)⟶𝑉 → 𝑊 ∈ Word 𝑉)
13	10, 11, 12	3syl 18	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ 𝑊:(0..^𝑁)⟶{𝑆}) → 𝑊 ∈ Word 𝑉)
14		ffzo0hash 13425	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑊 Fn (0..^𝑁)) → (♯‘𝑊) = 𝑁)
15	14	expcom 450	. . . . . . . . . . . . . 14 ⊢ (𝑊 Fn (0..^𝑁) → (𝑁 ∈ ℕ₀ → (♯‘𝑊) = 𝑁))
16		ffn 6206	. . . . . . . . . . . . . 14 ⊢ (𝑊:(0..^𝑁)⟶{𝑆} → 𝑊 Fn (0..^𝑁))
17	15, 16	syl11 33	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → (𝑊:(0..^𝑁)⟶{𝑆} → (♯‘𝑊) = 𝑁))
18	17	adantl 473	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑊:(0..^𝑁)⟶{𝑆} → (♯‘𝑊) = 𝑁))
19	18	imp 444	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ 𝑊:(0..^𝑁)⟶{𝑆}) → (♯‘𝑊) = 𝑁)
20	13, 19	jca 555	. . . . . . . . . 10 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ 𝑊:(0..^𝑁)⟶{𝑆}) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁))
21	20	ex 449	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑊:(0..^𝑁)⟶{𝑆} → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁)))
22	5, 21	syl5bir 233	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((𝑊 Fn (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁)))
23	22	expcomd 453	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆 → (𝑊 Fn (0..^𝑁) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁))))
24	23	imp 444	. . . . . 6 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆) → (𝑊 Fn (0..^𝑁) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁)))
25		wrdf 13496	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝑉 → 𝑊:(0..^(♯‘𝑊))⟶𝑉)
26		ffn 6206	. . . . . . . . . 10 ⊢ (𝑊:(0..^(♯‘𝑊))⟶𝑉 → 𝑊 Fn (0..^(♯‘𝑊)))
27		oveq2 6821	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑊) = 𝑁 → (0..^(♯‘𝑊)) = (0..^𝑁))
28	27	fneq2d 6143	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) = 𝑁 → (𝑊 Fn (0..^(♯‘𝑊)) ↔ 𝑊 Fn (0..^𝑁)))
29	28	biimpd 219	. . . . . . . . . . . 12 ⊢ ((♯‘𝑊) = 𝑁 → (𝑊 Fn (0..^(♯‘𝑊)) → 𝑊 Fn (0..^𝑁)))
30	29	a1d 25	. . . . . . . . . . 11 ⊢ ((♯‘𝑊) = 𝑁 → ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑊 Fn (0..^(♯‘𝑊)) → 𝑊 Fn (0..^𝑁))))
31	30	com13 88	. . . . . . . . . 10 ⊢ (𝑊 Fn (0..^(♯‘𝑊)) → ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((♯‘𝑊) = 𝑁 → 𝑊 Fn (0..^𝑁))))
32	25, 26, 31	3syl 18	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((♯‘𝑊) = 𝑁 → 𝑊 Fn (0..^𝑁))))
33	32	com12 32	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑊 ∈ Word 𝑉 → ((♯‘𝑊) = 𝑁 → 𝑊 Fn (0..^𝑁))))
34	33	impd 446	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) → 𝑊 Fn (0..^𝑁)))
35	34	adantr 472	. . . . . 6 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) → 𝑊 Fn (0..^𝑁)))
36	24, 35	impbid 202	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆) → (𝑊 Fn (0..^𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁)))
37	36	ex 449	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆 → (𝑊 Fn (0..^𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁))))
38	37	pm5.32rd 675	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((𝑊 Fn (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆) ↔ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆)))
39		df-3an 1074	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆) ↔ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆))
40	38, 5, 39	3bitr4g 303	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑊:(0..^𝑁)⟶{𝑆} ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆)))
41	2, 4, 40	3bitr2d 296	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑊 = (𝑆 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ∀wral 3050 ⊆ wss 3715 {csn 4321 × cxp 5264 Fn wfn 6044 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 0cc0 10128 ℕ₀cn0 11484 ..^cfzo 12659 ♯chash 13311 Word cword 13477 repeatS creps 13484

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-card 8955 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 df-fzo 12660 df-hash 13312 df-word 13485 df-reps 13492

This theorem is referenced by: repswsymball 13726 repswsymballbi 13727 cshwrepswhash1 16011

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