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Theorem pfx2 14300

Description: A prefix of length two. (Contributed by AV, 15-May-2020.)

**Assertion**
Ref	Expression
pfx2	⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → (𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩)

Proof of Theorem pfx2 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2nn0 11902	. . . 4 ⊢ 2 ∈ ℕ₀
2	1	a1i 11	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → 2 ∈ ℕ₀)
3		lencl 13876	. . . 4 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ₀)
4	3	adantr 484	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → (♯‘𝑊) ∈ ℕ₀)
5		simpr 488	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → 2 ≤ (♯‘𝑊))
6		elfz2nn0 12993	. . 3 ⊢ (2 ∈ (0...(♯‘𝑊)) ↔ (2 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ₀ ∧ 2 ≤ (♯‘𝑊)))
7	2, 4, 5, 6	syl3anbrc 1340	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → 2 ∈ (0...(♯‘𝑊)))
8		pfxlen 14036	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 2)) = 2)
9		s2len 14242	. . . . . . 7 ⊢ (♯‘⟨“(𝑊‘0)(𝑊‘1)”⟩) = 2
10	9	eqcomi 2807	. . . . . 6 ⊢ 2 = (♯‘⟨“(𝑊‘0)(𝑊‘1)”⟩)
11	10	a1i 11	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(♯‘𝑊))) ∧ (♯‘(𝑊 prefix 2)) = 2) → 2 = (♯‘⟨“(𝑊‘0)(𝑊‘1)”⟩))
12		2nn 11698	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
13		lbfzo0 13072	. . . . . . . . . 10 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
14	12, 13	mpbir 234	. . . . . . . . 9 ⊢ 0 ∈ (0..^2)
15		pfxfv 14035	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(♯‘𝑊)) ∧ 0 ∈ (0..^2)) → ((𝑊 prefix 2)‘0) = (𝑊‘0))
16	14, 15	mp3an3 1447	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(♯‘𝑊))) → ((𝑊 prefix 2)‘0) = (𝑊‘0))
17	16	adantr 484	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(♯‘𝑊))) ∧ (♯‘(𝑊 prefix 2)) = 2) → ((𝑊 prefix 2)‘0) = (𝑊‘0))
18		fvex 6658	. . . . . . . 8 ⊢ (𝑊‘0) ∈ V
19		s2fv0 14240	. . . . . . . 8 ⊢ ((𝑊‘0) ∈ V → (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0) = (𝑊‘0))
20	18, 19	ax-mp 5	. . . . . . 7 ⊢ (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0) = (𝑊‘0)
21	17, 20	eqtr4di 2851	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(♯‘𝑊))) ∧ (♯‘(𝑊 prefix 2)) = 2) → ((𝑊 prefix 2)‘0) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0))
22		1nn0 11901	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
23		1lt2 11796	. . . . . . . . . 10 ⊢ 1 < 2
24		elfzo0 13073	. . . . . . . . . 10 ⊢ (1 ∈ (0..^2) ↔ (1 ∈ ℕ₀ ∧ 2 ∈ ℕ ∧ 1 < 2))
25	22, 12, 23, 24	mpbir3an 1338	. . . . . . . . 9 ⊢ 1 ∈ (0..^2)
26		pfxfv 14035	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(♯‘𝑊)) ∧ 1 ∈ (0..^2)) → ((𝑊 prefix 2)‘1) = (𝑊‘1))
27	25, 26	mp3an3 1447	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(♯‘𝑊))) → ((𝑊 prefix 2)‘1) = (𝑊‘1))
28		fvex 6658	. . . . . . . . 9 ⊢ (𝑊‘1) ∈ V
29		s2fv1 14241	. . . . . . . . 9 ⊢ ((𝑊‘1) ∈ V → (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1) = (𝑊‘1))
30	28, 29	ax-mp 5	. . . . . . . 8 ⊢ (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1) = (𝑊‘1)
31	27, 30	eqtr4di 2851	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(♯‘𝑊))) → ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))
32	31	adantr 484	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(♯‘𝑊))) ∧ (♯‘(𝑊 prefix 2)) = 2) → ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))
33		0nn0 11900	. . . . . . 7 ⊢ 0 ∈ ℕ₀
34		fveq2 6645	. . . . . . . . 9 ⊢ (𝑖 = 0 → ((𝑊 prefix 2)‘𝑖) = ((𝑊 prefix 2)‘0))
35		fveq2 6645	. . . . . . . . 9 ⊢ (𝑖 = 0 → (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0))
36	34, 35	eqeq12d 2814	. . . . . . . 8 ⊢ (𝑖 = 0 → (((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ ((𝑊 prefix 2)‘0) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0)))
37		fveq2 6645	. . . . . . . . 9 ⊢ (𝑖 = 1 → ((𝑊 prefix 2)‘𝑖) = ((𝑊 prefix 2)‘1))
38		fveq2 6645	. . . . . . . . 9 ⊢ (𝑖 = 1 → (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))
39	37, 38	eqeq12d 2814	. . . . . . . 8 ⊢ (𝑖 = 1 → (((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1)))
40	36, 39	ralprg 4592	. . . . . . 7 ⊢ ((0 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → (∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ (((𝑊 prefix 2)‘0) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0) ∧ ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))))
41	33, 22, 40	mp2an 691	. . . . . 6 ⊢ (∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ (((𝑊 prefix 2)‘0) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0) ∧ ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1)))
42	21, 32, 41	sylanbrc 586	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(♯‘𝑊))) ∧ (♯‘(𝑊 prefix 2)) = 2) → ∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖))
43		eqeq1 2802	. . . . . . 7 ⊢ ((♯‘(𝑊 prefix 2)) = 2 → ((♯‘(𝑊 prefix 2)) = (♯‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ↔ 2 = (♯‘⟨“(𝑊‘0)(𝑊‘1)”⟩)))
44		oveq2 7143	. . . . . . . . 9 ⊢ ((♯‘(𝑊 prefix 2)) = 2 → (0..^(♯‘(𝑊 prefix 2))) = (0..^2))
45		fzo0to2pr 13117	. . . . . . . . 9 ⊢ (0..^2) = {0, 1}
46	44, 45	eqtrdi 2849	. . . . . . . 8 ⊢ ((♯‘(𝑊 prefix 2)) = 2 → (0..^(♯‘(𝑊 prefix 2))) = {0, 1})
47	46	raleqdv 3364	. . . . . . 7 ⊢ ((♯‘(𝑊 prefix 2)) = 2 → (∀𝑖 ∈ (0..^(♯‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ ∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖)))
48	43, 47	anbi12d 633	. . . . . 6 ⊢ ((♯‘(𝑊 prefix 2)) = 2 → (((♯‘(𝑊 prefix 2)) = (♯‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(♯‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖)) ↔ (2 = (♯‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖))))
49	48	adantl 485	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(♯‘𝑊))) ∧ (♯‘(𝑊 prefix 2)) = 2) → (((♯‘(𝑊 prefix 2)) = (♯‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(♯‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖)) ↔ (2 = (♯‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖))))
50	11, 42, 49	mpbir2and 712	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(♯‘𝑊))) ∧ (♯‘(𝑊 prefix 2)) = 2) → ((♯‘(𝑊 prefix 2)) = (♯‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(♯‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖)))
51	8, 50	mpdan 686	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(♯‘𝑊))) → ((♯‘(𝑊 prefix 2)) = (♯‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(♯‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖)))
52		pfxcl 14030	. . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 prefix 2) ∈ Word 𝑉)
53		s2cli 14233	. . . . 5 ⊢ ⟨“(𝑊‘0)(𝑊‘1)”⟩ ∈ Word V
54		eqwrd 13900	. . . . 5 ⊢ (((𝑊 prefix 2) ∈ Word 𝑉 ∧ ⟨“(𝑊‘0)(𝑊‘1)”⟩ ∈ Word V) → ((𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩ ↔ ((♯‘(𝑊 prefix 2)) = (♯‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(♯‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖))))
55	52, 53, 54	sylancl 589	. . . 4 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩ ↔ ((♯‘(𝑊 prefix 2)) = (♯‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(♯‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖))))
56	55	adantr 484	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(♯‘𝑊))) → ((𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩ ↔ ((♯‘(𝑊 prefix 2)) = (♯‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(♯‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖))))
57	51, 56	mpbird 260	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(♯‘𝑊))) → (𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩)
58	7, 57	syldan 594	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → (𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 {cpr 4527 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 < clt 10664 ≤ cle 10665 ℕcn 11625 2c2 11680 ℕ₀cn0 11885 ...cfz 12885 ..^cfzo 13028 ♯chash 13686 Word cword 13857 prefix cpfx 14023 ⟨“cs2 14194

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-substr 13994 df-pfx 14024 df-s2 14201

This theorem is referenced by: (None)

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