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Theorem wrdeqs1cat 14703

Description: Decompose a nonempty word by separating off the first symbol. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 12-Oct-2022.)

**Assertion**
Ref	Expression
wrdeqs1cat	⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑊 ≠ ∅) → 𝑊 = (⟨“(𝑊‘0)”⟩ ++ (𝑊 substr ⟨1, (♯‘𝑊)⟩)))

**Proof of Theorem wrdeqs1cat**
Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑊 ≠ ∅) → 𝑊 ∈ Word 𝐴)
2		wrdfin 14515	. . . 4 ⊢ (𝑊 ∈ Word 𝐴 → 𝑊 ∈ Fin)
3		1elfz0hash 14382	. . . 4 ⊢ ((𝑊 ∈ Fin ∧ 𝑊 ≠ ∅) → 1 ∈ (0...(♯‘𝑊)))
4	2, 3	sylan 579	. . 3 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑊 ≠ ∅) → 1 ∈ (0...(♯‘𝑊)))
5		lennncl 14517	. . . . 5 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ)
6	5	nnnn0d 12563	. . . 4 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ₀)
7		eluzfz2 13542	. . . . 5 ⊢ ((♯‘𝑊) ∈ (ℤ_≥‘0) → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
8		nn0uz 12895	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
9	7, 8	eleq2s 2847	. . . 4 ⊢ ((♯‘𝑊) ∈ ℕ₀ → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
10	6, 9	syl 17	. . 3 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
11		ccatpfx 14684	. . 3 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 1 ∈ (0...(♯‘𝑊)) ∧ (♯‘𝑊) ∈ (0...(♯‘𝑊))) → ((𝑊 prefix 1) ++ (𝑊 substr ⟨1, (♯‘𝑊)⟩)) = (𝑊 prefix (♯‘𝑊)))
12	1, 4, 10, 11	syl3anc 1369	. 2 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑊 ≠ ∅) → ((𝑊 prefix 1) ++ (𝑊 substr ⟨1, (♯‘𝑊)⟩)) = (𝑊 prefix (♯‘𝑊)))
13		pfx1 14686	. . 3 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑊 ≠ ∅) → (𝑊 prefix 1) = ⟨“(𝑊‘0)”⟩)
14	13	oveq1d 7435	. 2 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑊 ≠ ∅) → ((𝑊 prefix 1) ++ (𝑊 substr ⟨1, (♯‘𝑊)⟩)) = (⟨“(𝑊‘0)”⟩ ++ (𝑊 substr ⟨1, (♯‘𝑊)⟩)))
15		pfxid 14667	. . 3 ⊢ (𝑊 ∈ Word 𝐴 → (𝑊 prefix (♯‘𝑊)) = 𝑊)
16	15	adantr 480	. 2 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑊 ≠ ∅) → (𝑊 prefix (♯‘𝑊)) = 𝑊)
17	12, 14, 16	3eqtr3rd 2777	1 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑊 ≠ ∅) → 𝑊 = (⟨“(𝑊‘0)”⟩ ++ (𝑊 substr ⟨1, (♯‘𝑊)⟩)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∅c0 4323 ⟨cop 4635 ‘cfv 6548 (class class class)co 7420 Fincfn 8964 0cc0 11139 1c1 11140 ℕ₀cn0 12503 ℤ_≥cuz 12853 ...cfz 13517 ♯chash 14322 Word cword 14497 ++ cconcat 14553 ⟨“cs1 14578 substr csubstr 14623 prefix cpfx 14653

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-hash 14323 df-word 14498 df-concat 14554 df-s1 14579 df-substr 14624 df-pfx 14654

This theorem is referenced by: (None)

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