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Theorem ccatswrd 13308
 Description: Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.)
Assertion
Ref Expression
ccatswrd ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 substr ⟨𝑋, 𝑍⟩))

Proof of Theorem ccatswrd
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 swrdcl 13271 . . . . . 6 (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
21adantr 480 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
3 swrdcl 13271 . . . . . 6 (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
43adantr 480 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
5 ccatcl 13212 . . . . 5 (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
62, 4, 5syl2anc 691 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
7 wrdf 13165 . . . 4 (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴 → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)):(0..^(#‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))))⟶𝐴)
8 ffn 5958 . . . 4 (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)):(0..^(#‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))))⟶𝐴 → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(#‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))))
96, 7, 83syl 18 . . 3 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(#‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))))
10 ccatlen 13213 . . . . . . 7 (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴) → (#‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = ((#‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩))))
112, 4, 10syl2anc 691 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (#‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = ((#‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩))))
12 simpl 472 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑆 ∈ Word 𝐴)
13 simpr1 1060 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑋 ∈ (0...𝑌))
14 simpr2 1061 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ (0...𝑍))
15 simpr3 1062 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑍 ∈ (0...(#‘𝑆)))
16 fzass4 12250 . . . . . . . . . . . 12 ((𝑌 ∈ (0...(#‘𝑆)) ∧ 𝑍 ∈ (𝑌...(#‘𝑆))) ↔ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))))
1716biimpri 217 . . . . . . . . . . 11 ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑌 ∈ (0...(#‘𝑆)) ∧ 𝑍 ∈ (𝑌...(#‘𝑆))))
1817simpld 474 . . . . . . . . . 10 ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → 𝑌 ∈ (0...(#‘𝑆)))
1914, 15, 18syl2anc 691 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ (0...(#‘𝑆)))
20 swrdlen 13275 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆))) → (#‘(𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑌𝑋))
2112, 13, 19, 20syl3anc 1318 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (#‘(𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑌𝑋))
22 swrdlen 13275 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (#‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍𝑌))
2312, 14, 15, 22syl3anc 1318 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (#‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍𝑌))
2421, 23oveq12d 6567 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((#‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = ((𝑌𝑋) + (𝑍𝑌)))
25 elfzelz 12213 . . . . . . . . . 10 (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℤ)
2614, 25syl 17 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ ℤ)
2726zcnd 11359 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ ℂ)
28 elfzelz 12213 . . . . . . . . . 10 (𝑋 ∈ (0...𝑌) → 𝑋 ∈ ℤ)
2913, 28syl 17 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑋 ∈ ℤ)
3029zcnd 11359 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑋 ∈ ℂ)
31 elfzelz 12213 . . . . . . . . . 10 (𝑍 ∈ (0...(#‘𝑆)) → 𝑍 ∈ ℤ)
3215, 31syl 17 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑍 ∈ ℤ)
3332zcnd 11359 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑍 ∈ ℂ)
3427, 30, 33npncan3d 10307 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((𝑌𝑋) + (𝑍𝑌)) = (𝑍𝑋))
3524, 34eqtrd 2644 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((#‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑍𝑋))
3611, 35eqtrd 2644 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (#‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑍𝑋))
3736oveq2d 6565 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (0..^(#‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))) = (0..^(𝑍𝑋)))
3837fneq2d 5896 . . 3 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(#‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))) ↔ ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(𝑍𝑋))))
399, 38mpbid 221 . 2 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(𝑍𝑋)))
40 swrdcl 13271 . . . . 5 (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴)
4140adantr 480 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴)
42 wrdf 13165 . . . 4 ((𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴 → (𝑆 substr ⟨𝑋, 𝑍⟩):(0..^(#‘(𝑆 substr ⟨𝑋, 𝑍⟩)))⟶𝐴)
43 ffn 5958 . . . 4 ((𝑆 substr ⟨𝑋, 𝑍⟩):(0..^(#‘(𝑆 substr ⟨𝑋, 𝑍⟩)))⟶𝐴 → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(#‘(𝑆 substr ⟨𝑋, 𝑍⟩))))
4441, 42, 433syl 18 . . 3 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(#‘(𝑆 substr ⟨𝑋, 𝑍⟩))))
45 fzass4 12250 . . . . . . . . 9 ((𝑋 ∈ (0...𝑍) ∧ 𝑌 ∈ (𝑋...𝑍)) ↔ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)))
4645biimpri 217 . . . . . . . 8 ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)) → (𝑋 ∈ (0...𝑍) ∧ 𝑌 ∈ (𝑋...𝑍)))
4746simpld 474 . . . . . . 7 ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)) → 𝑋 ∈ (0...𝑍))
4813, 14, 47syl2anc 691 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑋 ∈ (0...𝑍))
49 swrdlen 13275 . . . . . 6 ((𝑆 ∈ Word 𝐴𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (#‘(𝑆 substr ⟨𝑋, 𝑍⟩)) = (𝑍𝑋))
5012, 48, 15, 49syl3anc 1318 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (#‘(𝑆 substr ⟨𝑋, 𝑍⟩)) = (𝑍𝑋))
5150oveq2d 6565 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (0..^(#‘(𝑆 substr ⟨𝑋, 𝑍⟩))) = (0..^(𝑍𝑋)))
5251fneq2d 5896 . . 3 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(#‘(𝑆 substr ⟨𝑋, 𝑍⟩))) ↔ (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(𝑍𝑋))))
5344, 52mpbid 221 . 2 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(𝑍𝑋)))
54 simpr 476 . . . . 5 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → 𝑥 ∈ (0..^(𝑍𝑋)))
5526, 29zsubcld 11363 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑌𝑋) ∈ ℤ)
5655adantr 480 . . . . 5 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → (𝑌𝑋) ∈ ℤ)
57 fzospliti 12369 . . . . 5 ((𝑥 ∈ (0..^(𝑍𝑋)) ∧ (𝑌𝑋) ∈ ℤ) → (𝑥 ∈ (0..^(𝑌𝑋)) ∨ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))))
5854, 56, 57syl2anc 691 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → (𝑥 ∈ (0..^(𝑌𝑋)) ∨ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))))
592adantr 480 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
604adantr 480 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
6121oveq2d 6565 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (0..^(#‘(𝑆 substr ⟨𝑋, 𝑌⟩))) = (0..^(𝑌𝑋)))
6261eleq2d 2673 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑥 ∈ (0..^(#‘(𝑆 substr ⟨𝑋, 𝑌⟩))) ↔ 𝑥 ∈ (0..^(𝑌𝑋))))
6362biimpar 501 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → 𝑥 ∈ (0..^(#‘(𝑆 substr ⟨𝑋, 𝑌⟩))))
64 ccatval1 13214 . . . . . . 7 (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴𝑥 ∈ (0..^(#‘(𝑆 substr ⟨𝑋, 𝑌⟩)))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥))
6559, 60, 63, 64syl3anc 1318 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥))
66 simpll 786 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → 𝑆 ∈ Word 𝐴)
67 simplr1 1096 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → 𝑋 ∈ (0...𝑌))
6819adantr 480 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → 𝑌 ∈ (0...(#‘𝑆)))
69 simpr 476 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → 𝑥 ∈ (0..^(𝑌𝑋)))
70 swrdfv 13276 . . . . . . 7 (((𝑆 ∈ Word 𝐴𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
7166, 67, 68, 69, 70syl31anc 1321 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
7265, 71eqtrd 2644 . . . . 5 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
732adantr 480 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
744adantr 480 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
7521, 35oveq12d 6567 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((#‘(𝑆 substr ⟨𝑋, 𝑌⟩))..^((#‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩)))) = ((𝑌𝑋)..^(𝑍𝑋)))
7675eleq2d 2673 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑥 ∈ ((#‘(𝑆 substr ⟨𝑋, 𝑌⟩))..^((#‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩)))) ↔ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))))
7776biimpar 501 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → 𝑥 ∈ ((#‘(𝑆 substr ⟨𝑋, 𝑌⟩))..^((#‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩)))))
78 ccatval2 13215 . . . . . . 7 (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴𝑥 ∈ ((#‘(𝑆 substr ⟨𝑋, 𝑌⟩))..^((#‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩))))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (#‘(𝑆 substr ⟨𝑋, 𝑌⟩)))))
7973, 74, 77, 78syl3anc 1318 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (#‘(𝑆 substr ⟨𝑋, 𝑌⟩)))))
80 simpll 786 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → 𝑆 ∈ Word 𝐴)
81 simplr2 1097 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → 𝑌 ∈ (0...𝑍))
82 simplr3 1098 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → 𝑍 ∈ (0...(#‘𝑆)))
8321oveq2d 6565 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑥 − (#‘(𝑆 substr ⟨𝑋, 𝑌⟩))) = (𝑥 − (𝑌𝑋)))
8483adantr 480 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑥 − (#‘(𝑆 substr ⟨𝑋, 𝑌⟩))) = (𝑥 − (𝑌𝑋)))
8534oveq2d 6565 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((𝑌𝑋)..^((𝑌𝑋) + (𝑍𝑌))) = ((𝑌𝑋)..^(𝑍𝑋)))
8685eleq2d 2673 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑥 ∈ ((𝑌𝑋)..^((𝑌𝑋) + (𝑍𝑌))) ↔ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))))
8786biimpar 501 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → 𝑥 ∈ ((𝑌𝑋)..^((𝑌𝑋) + (𝑍𝑌))))
8832, 26zsubcld 11363 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑍𝑌) ∈ ℤ)
8988adantr 480 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑍𝑌) ∈ ℤ)
90 fzosubel3 12396 . . . . . . . . 9 ((𝑥 ∈ ((𝑌𝑋)..^((𝑌𝑋) + (𝑍𝑌))) ∧ (𝑍𝑌) ∈ ℤ) → (𝑥 − (𝑌𝑋)) ∈ (0..^(𝑍𝑌)))
9187, 89, 90syl2anc 691 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑥 − (𝑌𝑋)) ∈ (0..^(𝑍𝑌)))
9284, 91eqeltrd 2688 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑥 − (#‘(𝑆 substr ⟨𝑋, 𝑌⟩))) ∈ (0..^(𝑍𝑌)))
93 swrdfv 13276 . . . . . . 7 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ (𝑥 − (#‘(𝑆 substr ⟨𝑋, 𝑌⟩))) ∈ (0..^(𝑍𝑌))) → ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (#‘(𝑆 substr ⟨𝑋, 𝑌⟩)))) = (𝑆‘((𝑥 − (#‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌)))
9480, 81, 82, 92, 93syl31anc 1321 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (#‘(𝑆 substr ⟨𝑋, 𝑌⟩)))) = (𝑆‘((𝑥 − (#‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌)))
9583oveq1d 6564 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((𝑥 − (#‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌) = ((𝑥 − (𝑌𝑋)) + 𝑌))
9695adantr 480 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → ((𝑥 − (#‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌) = ((𝑥 − (𝑌𝑋)) + 𝑌))
97 elfzoelz 12339 . . . . . . . . . . 11 (𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋)) → 𝑥 ∈ ℤ)
9897zcnd 11359 . . . . . . . . . 10 (𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋)) → 𝑥 ∈ ℂ)
9998adantl 481 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → 𝑥 ∈ ℂ)
10027, 30subcld 10271 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑌𝑋) ∈ ℂ)
101100adantr 480 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑌𝑋) ∈ ℂ)
10227adantr 480 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → 𝑌 ∈ ℂ)
10399, 101, 102subadd23d 10293 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → ((𝑥 − (𝑌𝑋)) + 𝑌) = (𝑥 + (𝑌 − (𝑌𝑋))))
10427, 30nncand 10276 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑌 − (𝑌𝑋)) = 𝑋)
105104oveq2d 6565 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑥 + (𝑌 − (𝑌𝑋))) = (𝑥 + 𝑋))
106105adantr 480 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑥 + (𝑌 − (𝑌𝑋))) = (𝑥 + 𝑋))
10796, 103, 1063eqtrd 2648 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → ((𝑥 − (#‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌) = (𝑥 + 𝑋))
108107fveq2d 6107 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑆‘((𝑥 − (#‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌)) = (𝑆‘(𝑥 + 𝑋)))
10979, 94, 1083eqtrd 2648 . . . . 5 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
11072, 109jaodan 822 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ (𝑥 ∈ (0..^(𝑌𝑋)) ∨ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋)))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
11158, 110syldan 486 . . 3 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
112 simpll 786 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → 𝑆 ∈ Word 𝐴)
11348adantr 480 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → 𝑋 ∈ (0...𝑍))
114 simplr3 1098 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → 𝑍 ∈ (0...(#‘𝑆)))
115 swrdfv 13276 . . . 4 (((𝑆 ∈ Word 𝐴𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → ((𝑆 substr ⟨𝑋, 𝑍⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
116112, 113, 114, 54, 115syl31anc 1321 . . 3 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → ((𝑆 substr ⟨𝑋, 𝑍⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
117111, 116eqtr4d 2647 . 2 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑋, 𝑍⟩)‘𝑥))
11839, 53, 117eqfnfvd 6222 1 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 substr ⟨𝑋, 𝑍⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ 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 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 This theorem is referenced by:  wrdcctswrd  13317  swrdccatwrd  13320  wrdeqs1cat  13326  splid  13355  splval2  13359  swrds2  13533  efgredleme  17979  efgredlemc  17981  efgcpbllemb  17991  frgpuplem  18008  wrdsplex  29944
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