ccat2s1fvwd - Intuitionistic Logic Explorer

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Theorem ccat2s1fvwd 11211

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 11116	. . . . 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 2814	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ V)
6	5	s1cld 11186	. . . 4 ⊢ (𝜑 → ⟨“𝑋”⟩ ∈ Word V)
7		ccat2s1fvwd.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8	7	elexd 2814	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ V)
9	8	s1cld 11186	. . . 4 ⊢ (𝜑 → ⟨“𝑌”⟩ ∈ Word V)
10		ccatass 11172	. . . 4 ⊢ ((𝑊 ∈ Word V ∧ ⟨“𝑋”⟩ ∈ Word V ∧ ⟨“𝑌”⟩ ∈ Word V) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) = (𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)))
11	3, 6, 9, 10	syl3anc 1271	. . 3 ⊢ (𝜑 → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) = (𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)))
12	11	fveq1d 5635	. 2 ⊢ (𝜑 → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = ((𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩))‘𝐼))
13		ccatws1cl 11196	. . . 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 1022	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → 𝐼 ∈ ℕ₀)
18		lencl 11104	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ₀)
19	18	3ad2ant1 1042	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → (♯‘𝑊) ∈ ℕ₀)
20		nn0ge0 9415	. . . . . . . . 9 ⊢ (𝐼 ∈ ℕ₀ → 0 ≤ 𝐼)
21	20	adantl 277	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀) → 0 ≤ 𝐼)
22		0red 8168	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀) → 0 ∈ ℝ)
23		nn0re 9399	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℕ₀ → 𝐼 ∈ ℝ)
24	23	adantl 277	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀) → 𝐼 ∈ ℝ)
25	18	nn0red 9444	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℝ)
26	25	adantr 276	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀) → (♯‘𝑊) ∈ ℝ)
27		lelttr 8256	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 𝐼 ∈ ℝ ∧ (♯‘𝑊) ∈ ℝ) → ((0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝑊)) → 0 < (♯‘𝑊)))
28	22, 24, 26, 27	syl3anc 1271	. . . . . . . 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 1224	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → 0 < (♯‘𝑊))
31		elnnnn0b 9434	. . . . . 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 1023	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → 𝐼 < (♯‘𝑊))
34		elfzo0 10409	. . . . 5 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) ↔ (𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐼 < (♯‘𝑊)))
35	17, 32, 33, 34	syl3anbrc 1205	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → 𝐼 ∈ (0..^(♯‘𝑊)))
36	1, 15, 16, 35	syl3anc 1271	. . 3 ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊)))
37		ccatval1 11161	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩) ∈ Word V ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ((𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩))‘𝐼) = (𝑊‘𝐼))
38	1, 14, 36, 37	syl3anc 1271	. 2 ⊢ (𝜑 → ((𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩))‘𝐼) = (𝑊‘𝐼))
39	12, 38	eqtrd 2262	1 ⊢ (𝜑 → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = (𝑊‘𝐼))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 Vcvv 2800 class class class wbr 4084 ‘cfv 5322 (class class class)co 6011 ℝcr 8019 0cc0 8020 < clt 8202 ≤ cle 8203 ℕcn 9131 ℕ₀cn0 9390 ..^cfzo 10365 ♯chash 11025 Word cword 11100 ++ cconcat 11154 ⟨“cs1 11179

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4200 ax-sep 4203 ax-nul 4211 ax-pow 4260 ax-pr 4295 ax-un 4526 ax-setind 4631 ax-iinf 4682 ax-cnex 8111 ax-resscn 8112 ax-1cn 8113 ax-1re 8114 ax-icn 8115 ax-addcl 8116 ax-addrcl 8117 ax-mulcl 8118 ax-addcom 8120 ax-addass 8122 ax-distr 8124 ax-i2m1 8125 ax-0lt1 8126 ax-0id 8128 ax-rnegex 8129 ax-cnre 8131 ax-pre-ltirr 8132 ax-pre-ltwlin 8133 ax-pre-lttrn 8134 ax-pre-apti 8135 ax-pre-ltadd 8136

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3890 df-int 3925 df-iun 3968 df-br 4085 df-opab 4147 df-mpt 4148 df-tr 4184 df-id 4386 df-iord 4459 df-on 4461 df-ilim 4462 df-suc 4464 df-iom 4685 df-xp 4727 df-rel 4728 df-cnv 4729 df-co 4730 df-dm 4731 df-rn 4732 df-res 4733 df-ima 4734 df-iota 5282 df-fun 5324 df-fn 5325 df-f 5326 df-f1 5327 df-fo 5328 df-f1o 5329 df-fv 5330 df-riota 5964 df-ov 6014 df-oprab 6015 df-mpo 6016 df-1st 6296 df-2nd 6297 df-recs 6464 df-frec 6550 df-1o 6575 df-er 6695 df-en 6903 df-dom 6904 df-fin 6905 df-pnf 8204 df-mnf 8205 df-xr 8206 df-ltxr 8207 df-le 8208 df-sub 8340 df-neg 8341 df-inn 9132 df-n0 9391 df-z 9468 df-uz 9744 df-fz 10232 df-fzo 10366 df-ihash 11026 df-word 11101 df-concat 11155 df-s1 11180

This theorem is referenced by: ccat2s1fstg 11212 clwwlknonex2lem2 16223

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