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Theorem swrdccatin1 14086
 Description: The subword of a concatenation of two words within the first of the concatenated words. (Contributed by Alexander van der Vekens, 28-Mar-2018.)
Assertion
Ref Expression
swrdccatin1 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))

Proof of Theorem swrdccatin1
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 oveq2 7163 . . . . . 6 ((♯‘𝐴) = 0 → (0...(♯‘𝐴)) = (0...0))
21eleq2d 2898 . . . . 5 ((♯‘𝐴) = 0 → (𝑁 ∈ (0...(♯‘𝐴)) ↔ 𝑁 ∈ (0...0)))
3 elfz1eq 12917 . . . . . 6 (𝑁 ∈ (0...0) → 𝑁 = 0)
4 elfz1eq 12917 . . . . . . . 8 (𝑀 ∈ (0...0) → 𝑀 = 0)
5 swrd00 14005 . . . . . . . . . 10 ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = ∅
6 swrd00 14005 . . . . . . . . . 10 (𝐴 substr ⟨0, 0⟩) = ∅
75, 6eqtr4i 2847 . . . . . . . . 9 ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = (𝐴 substr ⟨0, 0⟩)
8 opeq1 4802 . . . . . . . . . 10 (𝑀 = 0 → ⟨𝑀, 0⟩ = ⟨0, 0⟩)
98oveq2d 7171 . . . . . . . . 9 (𝑀 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = ((𝐴 ++ 𝐵) substr ⟨0, 0⟩))
108oveq2d 7171 . . . . . . . . 9 (𝑀 = 0 → (𝐴 substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨0, 0⟩))
117, 9, 103eqtr4a 2882 . . . . . . . 8 (𝑀 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩))
124, 11syl 17 . . . . . . 7 (𝑀 ∈ (0...0) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩))
13 oveq2 7163 . . . . . . . . 9 (𝑁 = 0 → (0...𝑁) = (0...0))
1413eleq2d 2898 . . . . . . . 8 (𝑁 = 0 → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...0)))
15 opeq2 4803 . . . . . . . . . 10 (𝑁 = 0 → ⟨𝑀, 𝑁⟩ = ⟨𝑀, 0⟩)
1615oveq2d 7171 . . . . . . . . 9 (𝑁 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩))
1715oveq2d 7171 . . . . . . . . 9 (𝑁 = 0 → (𝐴 substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 0⟩))
1816, 17eqeq12d 2837 . . . . . . . 8 (𝑁 = 0 → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩) ↔ ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩)))
1914, 18imbi12d 347 . . . . . . 7 (𝑁 = 0 → ((𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)) ↔ (𝑀 ∈ (0...0) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩))))
2012, 19mpbiri 260 . . . . . 6 (𝑁 = 0 → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
213, 20syl 17 . . . . 5 (𝑁 ∈ (0...0) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
222, 21syl6bi 255 . . . 4 ((♯‘𝐴) = 0 → (𝑁 ∈ (0...(♯‘𝐴)) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
2322impcomd 414 . . 3 ((♯‘𝐴) = 0 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
2423adantl 484 . 2 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) = 0) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
25 ccatcl 13925 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
2625ad2antrr 724 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
27 simprl 769 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → 𝑀 ∈ (0...𝑁))
28 elfzelfzccat 13933 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(♯‘𝐴)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
2928imp 409 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
3029ad2ant2rl 747 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
31 swrdvalfn 14012 . . . . 5 (((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
3226, 27, 30, 31syl3anc 1367 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
33 3anass 1091 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) ↔ (𝐴 ∈ Word 𝑉 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
3433simplbi2 503 . . . . . . 7 (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → (𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
3534ad2antrr 724 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → (𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
3635imp 409 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))))
37 swrdvalfn 14012 . . . . 5 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → (𝐴 substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
3836, 37syl 17 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝐴 substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
39 simp-4l 781 . . . . . 6 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝐴 ∈ Word 𝑉)
40 simp-4r 782 . . . . . 6 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝐵 ∈ Word 𝑉)
41 elfznn0 12999 . . . . . . . . . 10 (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ0)
42 nn0addcl 11931 . . . . . . . . . . 11 ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℕ0)
4342expcom 416 . . . . . . . . . 10 (𝑀 ∈ ℕ0 → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
4441, 43syl 17 . . . . . . . . 9 (𝑀 ∈ (0...𝑁) → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
4544ad2antrl 726 . . . . . . . 8 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
46 elfzonn0 13081 . . . . . . . 8 (𝑘 ∈ (0..^(𝑁𝑀)) → 𝑘 ∈ ℕ0)
4745, 46impel 508 . . . . . . 7 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (𝑘 + 𝑀) ∈ ℕ0)
48 lencl 13882 . . . . . . . . . . 11 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
49 elnnne0 11910 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ ℕ ↔ ((♯‘𝐴) ∈ ℕ0 ∧ (♯‘𝐴) ≠ 0))
5049simplbi2 503 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
5148, 50syl 17 . . . . . . . . . 10 (𝐴 ∈ Word 𝑉 → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
5251adantr 483 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
5352imp 409 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) → (♯‘𝐴) ∈ ℕ)
5453ad2antrr 724 . . . . . . 7 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (♯‘𝐴) ∈ ℕ)
55 elfzo0 13077 . . . . . . . . 9 (𝑘 ∈ (0..^(𝑁𝑀)) ↔ (𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)))
56 elfz2nn0 12997 . . . . . . . . . . . 12 (𝑁 ∈ (0...(♯‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0𝑁 ≤ (♯‘𝐴)))
57 nn0re 11905 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
5857ad2antrl 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → 𝑘 ∈ ℝ)
59 nn0re 11905 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
6059ad2antll 727 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → 𝑀 ∈ ℝ)
61 nn0re 11905 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
6261ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → 𝑁 ∈ ℝ)
6358, 60, 62ltaddsubd 11239 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁𝑘 < (𝑁𝑀)))
64 nn0readdcl 11960 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℝ)
6564adantl 484 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → (𝑘 + 𝑀) ∈ ℝ)
66 nn0re 11905 . . . . . . . . . . . . . . . . . . . . . . 23 ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℝ)
6766ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → (♯‘𝐴) ∈ ℝ)
68 ltletr 10731 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑘 + 𝑀) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (♯‘𝐴) ∈ ℝ) → (((𝑘 + 𝑀) < 𝑁𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴)))
6965, 62, 67, 68syl3anc 1367 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → (((𝑘 + 𝑀) < 𝑁𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴)))
7069expd 418 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁 → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴))))
7163, 70sylbird 262 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → (𝑘 < (𝑁𝑀) → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴))))
7271ex 415 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 < (𝑁𝑀) → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴)))))
7372com24 95 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → (𝑁 ≤ (♯‘𝐴) → (𝑘 < (𝑁𝑀) → ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 + 𝑀) < (♯‘𝐴)))))
74733impia 1113 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0𝑁 ≤ (♯‘𝐴)) → (𝑘 < (𝑁𝑀) → ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 + 𝑀) < (♯‘𝐴))))
7574com13 88 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 < (𝑁𝑀) → ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴))))
7675impancom 454 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℕ0𝑘 < (𝑁𝑀)) → (𝑀 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴))))
77763adant2 1127 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑀 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴))))
7877com13 88 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0𝑁 ≤ (♯‘𝐴)) → (𝑀 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴))))
7956, 78sylbi 219 . . . . . . . . . . 11 (𝑁 ∈ (0...(♯‘𝐴)) → (𝑀 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴))))
8041, 79mpan9 509 . . . . . . . . . 10 ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴)))
8180adantl 484 . . . . . . . . 9 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴)))
8255, 81syl5bi 244 . . . . . . . 8 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝑘 ∈ (0..^(𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴)))
8382imp 409 . . . . . . 7 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (𝑘 + 𝑀) < (♯‘𝐴))
84 elfzo0 13077 . . . . . . 7 ((𝑘 + 𝑀) ∈ (0..^(♯‘𝐴)) ↔ ((𝑘 + 𝑀) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ ∧ (𝑘 + 𝑀) < (♯‘𝐴)))
8547, 54, 83, 84syl3anbrc 1339 . . . . . 6 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (𝑘 + 𝑀) ∈ (0..^(♯‘𝐴)))
86 ccatval1 13929 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐴‘(𝑘 + 𝑀)))
8739, 40, 85, 86syl3anc 1367 . . . . 5 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐴‘(𝑘 + 𝑀)))
8825ad5ant12 754 . . . . . 6 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
89 simplrl 775 . . . . . 6 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝑀 ∈ (0...𝑁))
9030adantr 483 . . . . . 6 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
91 simpr 487 . . . . . 6 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝑘 ∈ (0..^(𝑁𝑀)))
92 swrdfv 14009 . . . . . 6 ((((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)))
9388, 89, 90, 91, 92syl31anc 1369 . . . . 5 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)))
94 swrdfv 14009 . . . . . 6 (((𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → ((𝐴 substr ⟨𝑀, 𝑁⟩)‘𝑘) = (𝐴‘(𝑘 + 𝑀)))
9536, 94sylan 582 . . . . 5 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → ((𝐴 substr ⟨𝑀, 𝑁⟩)‘𝑘) = (𝐴‘(𝑘 + 𝑀)))
9687, 93, 953eqtr4d 2866 . . . 4 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 substr ⟨𝑀, 𝑁⟩)‘𝑘))
9732, 38, 96eqfnfvd 6804 . . 3 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))
9897ex 415 . 2 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
9924, 98pm2.61dane 3104 1 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∅c0 4290  ⟨cop 4572   class class class wbr 5065   Fn wfn 6349  ‘cfv 6354  (class class class)co 7155  ℝcr 10535  0cc0 10536   + caddc 10539   < clt 10674   ≤ cle 10675   − cmin 10869  ℕcn 11637  ℕ0cn0 11896  ...cfz 12891  ..^cfzo 13032  ♯chash 13689  Word cword 13860   ++ cconcat 13921   substr csubstr 14001 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-n0 11897  df-z 11981  df-uz 12243  df-fz 12892  df-fzo 13033  df-hash 13690  df-word 13861  df-concat 13922  df-substr 14002 This theorem is referenced by:  pfxccat3  14095  pfxccatpfx1  14097  swrdccatin1d  14104
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