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Theorem ccatpfx 40272
 Description: Joining a prefix with an adjacent subword makes a longer prefix. (Contributed by AV, 7-May-2020.)
Assertion
Ref Expression
ccatpfx ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 prefix 𝑍))

Proof of Theorem ccatpfx
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pfxcl 40249 . . . . . 6 (𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝑌) ∈ Word 𝐴)
213ad2ant1 1075 . . . . 5 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑆 prefix 𝑌) ∈ Word 𝐴)
3 swrdcl 13271 . . . . . 6 (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
433ad2ant1 1075 . . . . 5 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
5 ccatcl 13212 . . . . 5 (((𝑆 prefix 𝑌) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
62, 4, 5syl2anc 691 . . . 4 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
7 wrdf 13165 . . . 4 (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴 → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)):(0..^(#‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))))⟶𝐴)
8 ffn 5958 . . . 4 (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)):(0..^(#‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))))⟶𝐴 → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(#‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))))
96, 7, 83syl 18 . . 3 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(#‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))))
10 ccatlen 13213 . . . . . . 7 (((𝑆 prefix 𝑌) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴) → (#‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = ((#‘(𝑆 prefix 𝑌)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩))))
112, 4, 10syl2anc 691 . . . . . 6 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (#‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = ((#‘(𝑆 prefix 𝑌)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩))))
12 simp1 1054 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → 𝑆 ∈ Word 𝐴)
13 fzass4 12250 . . . . . . . . . . . 12 ((𝑌 ∈ (0...(#‘𝑆)) ∧ 𝑍 ∈ (𝑌...(#‘𝑆))) ↔ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))))
1413biimpri 217 . . . . . . . . . . 11 ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑌 ∈ (0...(#‘𝑆)) ∧ 𝑍 ∈ (𝑌...(#‘𝑆))))
1514simpld 474 . . . . . . . . . 10 ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → 𝑌 ∈ (0...(#‘𝑆)))
16153adant1 1072 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → 𝑌 ∈ (0...(#‘𝑆)))
17 pfxlen 40254 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...(#‘𝑆))) → (#‘(𝑆 prefix 𝑌)) = 𝑌)
1812, 16, 17syl2anc 691 . . . . . . . 8 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (#‘(𝑆 prefix 𝑌)) = 𝑌)
19 swrdlen 13275 . . . . . . . 8 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (#‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍𝑌))
2018, 19oveq12d 6567 . . . . . . 7 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((#‘(𝑆 prefix 𝑌)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑌 + (𝑍𝑌)))
21 elfzelz 12213 . . . . . . . . . . 11 (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℤ)
2221ad2antrl 760 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ ℤ)
2322zcnd 11359 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ ℂ)
24233impb 1252 . . . . . . . 8 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → 𝑌 ∈ ℂ)
25 elfzelz 12213 . . . . . . . . . . 11 (𝑍 ∈ (0...(#‘𝑆)) → 𝑍 ∈ ℤ)
2625ad2antll 761 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑍 ∈ ℤ)
2726zcnd 11359 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑍 ∈ ℂ)
28273impb 1252 . . . . . . . 8 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → 𝑍 ∈ ℂ)
2924, 28pncan3d 10274 . . . . . . 7 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑌 + (𝑍𝑌)) = 𝑍)
3020, 29eqtrd 2644 . . . . . 6 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((#‘(𝑆 prefix 𝑌)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = 𝑍)
3111, 30eqtrd 2644 . . . . 5 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (#‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = 𝑍)
3231oveq2d 6565 . . . 4 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (0..^(#‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))) = (0..^𝑍))
3332fneq2d 5896 . . 3 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(#‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))) ↔ ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^𝑍)))
349, 33mpbid 221 . 2 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^𝑍))
35 pfxfn 40253 . . 3 ((𝑆 ∈ Word 𝐴𝑍 ∈ (0...(#‘𝑆))) → (𝑆 prefix 𝑍) Fn (0..^𝑍))
36353adant2 1073 . 2 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑆 prefix 𝑍) Fn (0..^𝑍))
37 simpr 476 . . . . 5 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → 𝑥 ∈ (0..^𝑍))
38213ad2ant2 1076 . . . . . 6 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → 𝑌 ∈ ℤ)
3938adantr 480 . . . . 5 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → 𝑌 ∈ ℤ)
40 fzospliti 12369 . . . . 5 ((𝑥 ∈ (0..^𝑍) ∧ 𝑌 ∈ ℤ) → (𝑥 ∈ (0..^𝑌) ∨ 𝑥 ∈ (𝑌..^𝑍)))
4137, 39, 40syl2anc 691 . . . 4 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → (𝑥 ∈ (0..^𝑌) ∨ 𝑥 ∈ (𝑌..^𝑍)))
422adantr 480 . . . . . . 7 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → (𝑆 prefix 𝑌) ∈ Word 𝐴)
434adantr 480 . . . . . . 7 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
4418oveq2d 6565 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (0..^(#‘(𝑆 prefix 𝑌))) = (0..^𝑌))
4544eleq2d 2673 . . . . . . . 8 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑥 ∈ (0..^(#‘(𝑆 prefix 𝑌))) ↔ 𝑥 ∈ (0..^𝑌)))
4645biimpar 501 . . . . . . 7 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → 𝑥 ∈ (0..^(#‘(𝑆 prefix 𝑌))))
47 ccatval1 13214 . . . . . . 7 (((𝑆 prefix 𝑌) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴𝑥 ∈ (0..^(#‘(𝑆 prefix 𝑌)))) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 prefix 𝑌)‘𝑥))
4842, 43, 46, 47syl3anc 1318 . . . . . 6 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 prefix 𝑌)‘𝑥))
4912adantr 480 . . . . . . 7 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → 𝑆 ∈ Word 𝐴)
5016adantr 480 . . . . . . 7 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → 𝑌 ∈ (0...(#‘𝑆)))
51 simpr 476 . . . . . . 7 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → 𝑥 ∈ (0..^𝑌))
52 pfxfv 40262 . . . . . . 7 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...(#‘𝑆)) ∧ 𝑥 ∈ (0..^𝑌)) → ((𝑆 prefix 𝑌)‘𝑥) = (𝑆𝑥))
5349, 50, 51, 52syl3anc 1318 . . . . . 6 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → ((𝑆 prefix 𝑌)‘𝑥) = (𝑆𝑥))
5448, 53eqtrd 2644 . . . . 5 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆𝑥))
552adantr 480 . . . . . . 7 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑆 prefix 𝑌) ∈ Word 𝐴)
564adantr 480 . . . . . . 7 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
5718, 30oveq12d 6567 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((#‘(𝑆 prefix 𝑌))..^((#‘(𝑆 prefix 𝑌)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩)))) = (𝑌..^𝑍))
5857eleq2d 2673 . . . . . . . 8 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑥 ∈ ((#‘(𝑆 prefix 𝑌))..^((#‘(𝑆 prefix 𝑌)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩)))) ↔ 𝑥 ∈ (𝑌..^𝑍)))
5958biimpar 501 . . . . . . 7 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → 𝑥 ∈ ((#‘(𝑆 prefix 𝑌))..^((#‘(𝑆 prefix 𝑌)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩)))))
60 ccatval2 13215 . . . . . . 7 (((𝑆 prefix 𝑌) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴𝑥 ∈ ((#‘(𝑆 prefix 𝑌))..^((#‘(𝑆 prefix 𝑌)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩))))) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (#‘(𝑆 prefix 𝑌)))))
6155, 56, 59, 60syl3anc 1318 . . . . . 6 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (#‘(𝑆 prefix 𝑌)))))
6218oveq2d 6565 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑥 − (#‘(𝑆 prefix 𝑌))) = (𝑥𝑌))
6362adantr 480 . . . . . . . 8 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − (#‘(𝑆 prefix 𝑌))) = (𝑥𝑌))
6438anim1i 590 . . . . . . . . . . 11 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑌 ∈ ℤ ∧ 𝑥 ∈ (𝑌..^𝑍)))
6564ancomd 466 . . . . . . . . . 10 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 ∈ (𝑌..^𝑍) ∧ 𝑌 ∈ ℤ))
66 fzosubel 12394 . . . . . . . . . 10 ((𝑥 ∈ (𝑌..^𝑍) ∧ 𝑌 ∈ ℤ) → (𝑥𝑌) ∈ ((𝑌𝑌)..^(𝑍𝑌)))
6765, 66syl 17 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥𝑌) ∈ ((𝑌𝑌)..^(𝑍𝑌)))
6821zcnd 11359 . . . . . . . . . . . . . . 15 (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℂ)
6968subidd 10259 . . . . . . . . . . . . . 14 (𝑌 ∈ (0...𝑍) → (𝑌𝑌) = 0)
7069eqcomd 2616 . . . . . . . . . . . . 13 (𝑌 ∈ (0...𝑍) → 0 = (𝑌𝑌))
71703ad2ant2 1076 . . . . . . . . . . . 12 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → 0 = (𝑌𝑌))
7271oveq1d 6564 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (0..^(𝑍𝑌)) = ((𝑌𝑌)..^(𝑍𝑌)))
7372eleq2d 2673 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((𝑥𝑌) ∈ (0..^(𝑍𝑌)) ↔ (𝑥𝑌) ∈ ((𝑌𝑌)..^(𝑍𝑌))))
7473adantr 480 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥𝑌) ∈ (0..^(𝑍𝑌)) ↔ (𝑥𝑌) ∈ ((𝑌𝑌)..^(𝑍𝑌))))
7567, 74mpbird 246 . . . . . . . 8 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥𝑌) ∈ (0..^(𝑍𝑌)))
7663, 75eqeltrd 2688 . . . . . . 7 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − (#‘(𝑆 prefix 𝑌))) ∈ (0..^(𝑍𝑌)))
77 swrdfv 13276 . . . . . . 7 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ (𝑥 − (#‘(𝑆 prefix 𝑌))) ∈ (0..^(𝑍𝑌))) → ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (#‘(𝑆 prefix 𝑌)))) = (𝑆‘((𝑥 − (#‘(𝑆 prefix 𝑌))) + 𝑌)))
7876, 77syldan 486 . . . . . 6 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (#‘(𝑆 prefix 𝑌)))) = (𝑆‘((𝑥 − (#‘(𝑆 prefix 𝑌))) + 𝑌)))
7963oveq1d 6564 . . . . . . . 8 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − (#‘(𝑆 prefix 𝑌))) + 𝑌) = ((𝑥𝑌) + 𝑌))
80 elfzoelz 12339 . . . . . . . . . . 11 (𝑥 ∈ (𝑌..^𝑍) → 𝑥 ∈ ℤ)
8180zcnd 11359 . . . . . . . . . 10 (𝑥 ∈ (𝑌..^𝑍) → 𝑥 ∈ ℂ)
8281adantl 481 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → 𝑥 ∈ ℂ)
8324adantr 480 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → 𝑌 ∈ ℂ)
8482, 83npcand 10275 . . . . . . . 8 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥𝑌) + 𝑌) = 𝑥)
8579, 84eqtrd 2644 . . . . . . 7 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − (#‘(𝑆 prefix 𝑌))) + 𝑌) = 𝑥)
8685fveq2d 6107 . . . . . 6 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑆‘((𝑥 − (#‘(𝑆 prefix 𝑌))) + 𝑌)) = (𝑆𝑥))
8761, 78, 863eqtrd 2648 . . . . 5 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆𝑥))
8854, 87jaodan 822 . . . 4 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ (𝑥 ∈ (0..^𝑌) ∨ 𝑥 ∈ (𝑌..^𝑍))) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆𝑥))
8941, 88syldan 486 . . 3 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆𝑥))
9012adantr 480 . . . 4 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → 𝑆 ∈ Word 𝐴)
91 simpl3 1059 . . . 4 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → 𝑍 ∈ (0...(#‘𝑆)))
92 pfxfv 40262 . . . 4 ((𝑆 ∈ Word 𝐴𝑍 ∈ (0...(#‘𝑆)) ∧ 𝑥 ∈ (0..^𝑍)) → ((𝑆 prefix 𝑍)‘𝑥) = (𝑆𝑥))
9390, 91, 37, 92syl3anc 1318 . . 3 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → ((𝑆 prefix 𝑍)‘𝑥) = (𝑆𝑥))
9489, 93eqtr4d 2647 . 2 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 prefix 𝑍)‘𝑥))
9534, 36, 94eqfnfvd 6222 1 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 prefix 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815   + caddc 9818   − cmin 10145  ℤcz 11254  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   ++ cconcat 13148   substr csubstr 13150   prefix cpfx 40244 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-pfx 40245 This theorem is referenced by:  pfxcctswrd  40280
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