ccat2s1fvwd - Intuitionistic Logic Explorer

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Theorem ccat2s1fvwd 11415

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 11320	. . . . 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 2835	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ V)
6	5	s1cld 11390	. . . 4 ⊢ (𝜑 → ⟨“𝑋”⟩ ∈ Word V)
7		ccat2s1fvwd.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8	7	elexd 2835	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ V)
9	8	s1cld 11390	. . . 4 ⊢ (𝜑 → ⟨“𝑌”⟩ ∈ Word V)
10		ccatass 11376	. . . 4 ⊢ ((𝑊 ∈ Word V ∧ ⟨“𝑋”⟩ ∈ Word V ∧ ⟨“𝑌”⟩ ∈ Word V) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) = (𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)))
11	3, 6, 9, 10	syl3anc 1278	. . 3 ⊢ (𝜑 → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) = (𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)))
12	11	fveq1d 5697	. 2 ⊢ (𝜑 → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = ((𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩))‘𝐼))
13		ccatws1cl 11400	. . . 4 ⊢ ((⟨“𝑋”⟩ ∈ Word V ∧ 𝑌 ∈ V) → (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩) ∈ Word V)
14	6, 8, 13	syl2anc 415	. . 3 ⊢ (𝜑 → (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩) ∈ Word V)
15		ccat2s1fvwd.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ ℕ₀)
16		ccat2s1fvwd.1	. . . 4 ⊢ (𝜑 → 𝐼 < (♯‘𝑊))
17		simp2 1029	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → 𝐼 ∈ ℕ₀)
18		lencl 11308	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ₀)
19	18	3ad2ant1 1049	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → (♯‘𝑊) ∈ ℕ₀)
20		nn0ge0 9588	. . . . . . . . 9 ⊢ (𝐼 ∈ ℕ₀ → 0 ≤ 𝐼)
21	20	adantl 277	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀) → 0 ≤ 𝐼)
22		0red 8327	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀) → 0 ∈ ℝ)
23		nn0re 9572	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℕ₀ → 𝐼 ∈ ℝ)
24	23	adantl 277	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀) → 𝐼 ∈ ℝ)
25	18	nn0red 9621	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℝ)
26	25	adantr 276	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀) → (♯‘𝑊) ∈ ℝ)
27		lelttr 8414	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 𝐼 ∈ ℝ ∧ (♯‘𝑊) ∈ ℝ) → ((0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝑊)) → 0 < (♯‘𝑊)))
28	22, 24, 26, 27	syl3anc 1278	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀) → ((0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝑊)) → 0 < (♯‘𝑊)))
29	21, 28	mpand 433	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀) → (𝐼 < (♯‘𝑊) → 0 < (♯‘𝑊)))
30	29	3impia 1231	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → 0 < (♯‘𝑊))
31		elnnnn0b 9607	. . . . . 6 ⊢ ((♯‘𝑊) ∈ ℕ ↔ ((♯‘𝑊) ∈ ℕ₀ ∧ 0 < (♯‘𝑊)))
32	19, 30, 31	sylanbrc 421	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → (♯‘𝑊) ∈ ℕ)
33		simp3 1030	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → 𝐼 < (♯‘𝑊))
34		elfzo0 10593	. . . . 5 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) ↔ (𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐼 < (♯‘𝑊)))
35	17, 32, 33, 34	syl3anbrc 1212	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ₀ ∧ 𝐼 < (♯‘𝑊)) → 𝐼 ∈ (0..^(♯‘𝑊)))
36	1, 15, 16, 35	syl3anc 1278	. . 3 ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊)))
37		ccatval1 11365	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩) ∈ Word V ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ((𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩))‘𝐼) = (𝑊‘𝐼))
38	1, 14, 36, 37	syl3anc 1278	. 2 ⊢ (𝜑 → ((𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩))‘𝐼) = (𝑊‘𝐼))
39	12, 38	eqtrd 2271	1 ⊢ (𝜑 → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = (𝑊‘𝐼))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 Vcvv 2821 class class class wbr 4130 ‘cfv 5377 (class class class)co 6085 ℝcr 8178 0cc0 8179 < clt 8360 ≤ cle 8361 ℕcn 9304 ℕ₀cn0 9563 ..^cfzo 10549 ♯chash 11214 Word cword 11304 ++ cconcat 11358 ⟨“cs1 11383

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-addcom 8279 ax-addass 8281 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-0id 8287 ax-rnegex 8288 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-1o 6687 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-inn 9305 df-n0 9564 df-z 9645 df-uz 9922 df-fz 10412 df-fzo 10550 df-ihash 11215 df-word 11305 df-concat 11359 df-s1 11384

This theorem is used by: ccat2s1fstg 11416 clwwlknonex2lem2 16679

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