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Theorem ccat2s1fvwd 11335

Description: Extract a symbol of a word from the concatenation of the word with two single symbols. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 13-Jan-2020.) (Proof shortened by AV, 1-May-2020.) (Revised by AV, 28-Jan-2024.)

**Hypotheses**
Ref	Expression
ccat2s1fvwd.w	⊢ (𝜑 → 𝑊 ∈ Word 𝑉)
ccat2s1fvwd.i	⊢ (𝜑 → 𝐼 ∈ ℕ₀)
ccat2s1fvwd.1	⊢ (𝜑 → 𝐼 < (♯‘𝑊))
ccat2s1fvwd.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
ccat2s1fvwd.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
ccat2s1fvwd	⊢ (𝜑 → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = (𝑊‘𝐼))

**Proof of Theorem ccat2s1fvwd**
Step	Hyp	Ref	Expression
1		ccat2s1fvwd.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ Word 𝑉)
2		wrdv 11240	. . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → 𝑊 ∈ Word V)
3	1, 2	syl 14	. . . 4 ⊢ (𝜑 → 𝑊 ∈ Word V)
4		ccat2s1fvwd.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
5	4	elexd 2827	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ V)
6	5	s1cld 11310	. . . 4 ⊢ (𝜑 → ⟨“𝑋”⟩ ∈ Word V)
7		ccat2s1fvwd.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8	7	elexd 2827	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ V)
9	8	s1cld 11310	. . . 4 ⊢ (𝜑 → ⟨“𝑌”⟩ ∈ Word V)
10		ccatass 11296	. . . 4 ⊢ ((𝑊 ∈ Word V ∧ ⟨“𝑋”⟩ ∈ Word V ∧ ⟨“𝑌”⟩ ∈ Word V) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) = (𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)))
11	3, 6, 9, 10	syl3anc 1274	. . 3 ⊢ (𝜑 → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) = (𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)))
12	11	fveq1d 5672	. 2 ⊢ (𝜑 → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = ((𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩))‘𝐼))
13		ccatws1cl 11320	. . . 4 ⊢ ((⟨“𝑋”⟩ ∈ Word V ∧ 𝑌 ∈ V) → (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩) ∈ Word V)
14	6, 8, 13	syl2anc 411	. . 3 ⊢ (𝜑 → (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩) ∈ Word V)
15		ccat2s1fvwd.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ ℕ₀)
16		ccat2s1fvwd.1	. . . 4 ⊢ (𝜑 → 𝐼 < (♯‘𝑊))
17		simp2 1025	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → 𝐼 ∈ ℕ₀)
18		lencl 11228	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ₀)
19	18	3ad2ant1 1045	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → (♯‘𝑊) ∈ ℕ₀)
20		nn0ge0 9521	. . . . . . . . 9 ⊢ (𝐼 ∈ ℕ₀ → 0 ≤ 𝐼)
21	20	adantl 277	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀) → 0 ≤ 𝐼)
22		0red 8275	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀) → 0 ∈ ℝ)
23		nn0re 9505	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℕ₀ → 𝐼 ∈ ℝ)
24	23	adantl 277	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀) → 𝐼 ∈ ℝ)
25	18	nn0red 9554	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℝ)
26	25	adantr 276	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀) → (♯‘𝑊) ∈ ℝ)
27		lelttr 8362	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 𝐼 ∈ ℝ ∧ (♯‘𝑊) ∈ ℝ) → ((0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝑊)) → 0 < (♯‘𝑊)))
28	22, 24, 26, 27	syl3anc 1274	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀) → ((0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝑊)) → 0 < (♯‘𝑊)))
29	21, 28	mpand 429	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀) → (𝐼 < (♯‘𝑊) → 0 < (♯‘𝑊)))
30	29	3impia 1227	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → 0 < (♯‘𝑊))
31		elnnnn0b 9540	. . . . . 6 ⊢ ((♯‘𝑊) ∈ ℕ ↔ ((♯‘𝑊) ∈ ℕ₀ ∧ 0 < (♯‘𝑊)))
32	19, 30, 31	sylanbrc 417	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → (♯‘𝑊) ∈ ℕ)
33		simp3 1026	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → 𝐼 < (♯‘𝑊))
34		elfzo0 10520	. . . . 5 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) ↔ (𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐼 < (♯‘𝑊)))
35	17, 32, 33, 34	syl3anbrc 1208	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → 𝐼 ∈ (0..^(♯‘𝑊)))
36	1, 15, 16, 35	syl3anc 1274	. . 3 ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊)))
37		ccatval1 11285	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩) ∈ Word V ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ((𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩))‘𝐼) = (𝑊‘𝐼))
38	1, 14, 36, 37	syl3anc 1274	. 2 ⊢ (𝜑 → ((𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩))‘𝐼) = (𝑊‘𝐼))
39	12, 38	eqtrd 2265	1 ⊢ (𝜑 → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = (𝑊‘𝐼))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 1005 = wceq 1398 ∈ wcel 2203 Vcvv 2813 class class class wbr 4109 ‘cfv 5352 (class class class)co 6050 ℝcr 8126 0cc0 8127 < clt 8308 ≤ cle 8309 ℕcn 9237 ℕ₀cn0 9496 ..^cfzo 10476 ♯chash 11138 Word cword 11224 ++ cconcat 11278 ⟨“cs1 11303

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-addcom 8227 ax-addass 8229 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-0id 8235 ax-rnegex 8236 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-frec 6622 df-1o 6647 df-er 6767 df-en 6976 df-dom 6977 df-fin 6978 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-inn 9238 df-n0 9497 df-z 9578 df-uz 9854 df-fz 10343 df-fzo 10477 df-ihash 11139 df-word 11225 df-concat 11279 df-s1 11304

This theorem is referenced by: ccat2s1fstg 11336 clwwlknonex2lem2 16433

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