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Theorem swrdccatin1 13828
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 6918 . . . . . . 7 ((♯‘𝐴) = 0 → (0...(♯‘𝐴)) = (0...0))
21eleq2d 2892 . . . . . 6 ((♯‘𝐴) = 0 → (𝑁 ∈ (0...(♯‘𝐴)) ↔ 𝑁 ∈ (0...0)))
3 elfz1eq 12652 . . . . . . 7 (𝑁 ∈ (0...0) → 𝑁 = 0)
4 elfz1eq 12652 . . . . . . . . 9 (𝑀 ∈ (0...0) → 𝑀 = 0)
5 swrd00 13711 . . . . . . . . . . 11 ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = ∅
6 swrd00 13711 . . . . . . . . . . 11 (𝐴 substr ⟨0, 0⟩) = ∅
75, 6eqtr4i 2852 . . . . . . . . . 10 ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = (𝐴 substr ⟨0, 0⟩)
8 opeq1 4625 . . . . . . . . . . 11 (𝑀 = 0 → ⟨𝑀, 0⟩ = ⟨0, 0⟩)
98oveq2d 6926 . . . . . . . . . 10 (𝑀 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = ((𝐴 ++ 𝐵) substr ⟨0, 0⟩))
108oveq2d 6926 . . . . . . . . . 10 (𝑀 = 0 → (𝐴 substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨0, 0⟩))
117, 9, 103eqtr4a 2887 . . . . . . . . 9 (𝑀 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩))
124, 11syl 17 . . . . . . . 8 (𝑀 ∈ (0...0) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩))
13 oveq2 6918 . . . . . . . . . 10 (𝑁 = 0 → (0...𝑁) = (0...0))
1413eleq2d 2892 . . . . . . . . 9 (𝑁 = 0 → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...0)))
15 opeq2 4626 . . . . . . . . . . 11 (𝑁 = 0 → ⟨𝑀, 𝑁⟩ = ⟨𝑀, 0⟩)
1615oveq2d 6926 . . . . . . . . . 10 (𝑁 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩))
1715oveq2d 6926 . . . . . . . . . 10 (𝑁 = 0 → (𝐴 substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 0⟩))
1816, 17eqeq12d 2840 . . . . . . . . 9 (𝑁 = 0 → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩) ↔ ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩)))
1914, 18imbi12d 336 . . . . . . . 8 (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 250 . . . . . . 7 (𝑁 = 0 → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
213, 20syl 17 . . . . . 6 (𝑁 ∈ (0...0) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
222, 21syl6bi 245 . . . . 5 ((♯‘𝐴) = 0 → (𝑁 ∈ (0...(♯‘𝐴)) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
2322com23 86 . . . 4 ((♯‘𝐴) = 0 → (𝑀 ∈ (0...𝑁) → (𝑁 ∈ (0...(♯‘𝐴)) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
2423impd 400 . . 3 ((♯‘𝐴) = 0 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
2524a1d 25 . 2 ((♯‘𝐴) = 0 → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
26 ccatcl 13641 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
2726adantl 475 . . . . . . 7 ((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
2827adantr 474 . . . . . 6 (((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
29 simprl 787 . . . . . 6 (((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → 𝑀 ∈ (0...𝑁))
30 elfzelfzccat 13647 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(♯‘𝐴)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
3130adantl 475 . . . . . . . . 9 ((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (𝑁 ∈ (0...(♯‘𝐴)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
3231com12 32 . . . . . . . 8 (𝑁 ∈ (0...(♯‘𝐴)) → ((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
3332adantl 475 . . . . . . 7 ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
3433impcom 398 . . . . . 6 (((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
35 swrdvalfn 13720 . . . . . 6 (((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
3628, 29, 34, 35syl3anc 1494 . . . . 5 (((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
37 3anass 1120 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) ↔ (𝐴 ∈ Word 𝑉 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
3837simplbi2 496 . . . . . . . 8 (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → (𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
3938ad2antrl 719 . . . . . . 7 ((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → (𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
4039imp 397 . . . . . 6 (((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))))
41 swrdvalfn 13720 . . . . . 6 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → (𝐴 substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
4240, 41syl 17 . . . . 5 (((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝐴 substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
43 simprl 787 . . . . . . . 8 ((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝐴 ∈ Word 𝑉)
4443ad2antrr 717 . . . . . . 7 ((((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝐴 ∈ Word 𝑉)
45 simprr 789 . . . . . . . 8 ((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝐵 ∈ Word 𝑉)
4645ad2antrr 717 . . . . . . 7 ((((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝐵 ∈ Word 𝑉)
47 elfzo0 12811 . . . . . . . . . 10 (𝑘 ∈ (0..^(𝑁𝑀)) ↔ (𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)))
48 elfz2nn0 12732 . . . . . . . . . . . . . 14 (𝑀 ∈ (0...𝑁) ↔ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁))
49 nn0addcl 11662 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℕ0)
5049expcom 404 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ0 → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
51503ad2ant1 1167 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
5248, 51sylbi 209 . . . . . . . . . . . . 13 (𝑀 ∈ (0...𝑁) → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
5352ad2antrl 719 . . . . . . . . . . . 12 (((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
5453com12 32 . . . . . . . . . . 11 (𝑘 ∈ ℕ0 → (((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝑘 + 𝑀) ∈ ℕ0))
55543ad2ant1 1167 . . . . . . . . . 10 ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝑘 + 𝑀) ∈ ℕ0))
5647, 55sylbi 209 . . . . . . . . 9 (𝑘 ∈ (0..^(𝑁𝑀)) → (((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝑘 + 𝑀) ∈ ℕ0))
5756impcom 398 . . . . . . . 8 ((((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (𝑘 + 𝑀) ∈ ℕ0)
58 lencl 13600 . . . . . . . . . . . 12 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
59 df-ne 3000 . . . . . . . . . . . . 13 ((♯‘𝐴) ≠ 0 ↔ ¬ (♯‘𝐴) = 0)
60 elnnne0 11641 . . . . . . . . . . . . . 14 ((♯‘𝐴) ∈ ℕ ↔ ((♯‘𝐴) ∈ ℕ0 ∧ (♯‘𝐴) ≠ 0))
6160simplbi2 496 . . . . . . . . . . . . 13 ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
6259, 61syl5bir 235 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ ℕ0 → (¬ (♯‘𝐴) = 0 → (♯‘𝐴) ∈ ℕ))
6358, 62syl 17 . . . . . . . . . . 11 (𝐴 ∈ Word 𝑉 → (¬ (♯‘𝐴) = 0 → (♯‘𝐴) ∈ ℕ))
6463adantr 474 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (¬ (♯‘𝐴) = 0 → (♯‘𝐴) ∈ ℕ))
6564impcom 398 . . . . . . . . 9 ((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (♯‘𝐴) ∈ ℕ)
6665ad2antrr 717 . . . . . . . 8 ((((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (♯‘𝐴) ∈ ℕ)
67 elfz2nn0 12732 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (0...(♯‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0𝑁 ≤ (♯‘𝐴)))
68 nn0re 11635 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
6968ad2antrl 719 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → 𝑘 ∈ ℝ)
70 nn0re 11635 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
7170adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → 𝑀 ∈ ℝ)
7271adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → 𝑀 ∈ ℝ)
73 nn0re 11635 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
7473ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → 𝑁 ∈ ℝ)
7569, 72, 74ltaddsubd 10959 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁𝑘 < (𝑁𝑀)))
76 nn0re 11635 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 + 𝑀) ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℝ)
7749, 76syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℝ)
7877adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → (𝑘 + 𝑀) ∈ ℝ)
79 nn0re 11635 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℝ)
8079adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → (♯‘𝐴) ∈ ℝ)
8180adantr 474 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → (♯‘𝐴) ∈ ℝ)
82 ltletr 10455 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 + 𝑀) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (♯‘𝐴) ∈ ℝ) → (((𝑘 + 𝑀) < 𝑁𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴)))
8378, 74, 81, 82syl3anc 1494 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → (((𝑘 + 𝑀) < 𝑁𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴)))
8483expd 406 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁 → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴))))
8575, 84sylbird 252 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → (𝑘 < (𝑁𝑀) → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴))))
8685ex 403 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 < (𝑁𝑀) → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴)))))
8786com24 95 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → (𝑁 ≤ (♯‘𝐴) → (𝑘 < (𝑁𝑀) → ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 + 𝑀) < (♯‘𝐴)))))
88873impia 1149 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0𝑁 ≤ (♯‘𝐴)) → (𝑘 < (𝑁𝑀) → ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 + 𝑀) < (♯‘𝐴))))
8988com13 88 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 < (𝑁𝑀) → ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴))))
9089impancom 445 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ ℕ0𝑘 < (𝑁𝑀)) → (𝑀 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴))))
91903adant2 1165 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑀 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴))))
9291com13 88 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0𝑁 ≤ (♯‘𝐴)) → (𝑀 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴))))
9367, 92sylbi 209 . . . . . . . . . . . . . . 15 (𝑁 ∈ (0...(♯‘𝐴)) → (𝑀 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴))))
9493com12 32 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ0 → (𝑁 ∈ (0...(♯‘𝐴)) → ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴))))
95943ad2ant1 1167 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝑁 ∈ (0...(♯‘𝐴)) → ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴))))
9648, 95sylbi 209 . . . . . . . . . . . 12 (𝑀 ∈ (0...𝑁) → (𝑁 ∈ (0...(♯‘𝐴)) → ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴))))
9796a1i 11 . . . . . . . . . . 11 ((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (𝑀 ∈ (0...𝑁) → (𝑁 ∈ (0...(♯‘𝐴)) → ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴)))))
9897imp32 411 . . . . . . . . . 10 (((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴)))
9947, 98syl5bi 234 . . . . . . . . 9 (((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝑘 ∈ (0..^(𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴)))
10099imp 397 . . . . . . . 8 ((((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (𝑘 + 𝑀) < (♯‘𝐴))
101 elfzo0 12811 . . . . . . . 8 ((𝑘 + 𝑀) ∈ (0..^(♯‘𝐴)) ↔ ((𝑘 + 𝑀) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ ∧ (𝑘 + 𝑀) < (♯‘𝐴)))
10257, 66, 100, 101syl3anbrc 1447 . . . . . . 7 ((((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (𝑘 + 𝑀) ∈ (0..^(♯‘𝐴)))
103 ccatval1 13644 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐴‘(𝑘 + 𝑀)))
10444, 46, 102, 103syl3anc 1494 . . . . . 6 ((((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐴‘(𝑘 + 𝑀)))
10527ad2antrr 717 . . . . . . . 8 ((((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
10629adantr 474 . . . . . . . 8 ((((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝑀 ∈ (0...𝑁))
10734adantr 474 . . . . . . . 8 ((((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
108105, 106, 1073jca 1162 . . . . . . 7 ((((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → ((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
109 swrdfv 13717 . . . . . . 7 ((((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)))
110108, 109sylancom 582 . . . . . 6 ((((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)))
111 swrdfv 13717 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → ((𝐴 substr ⟨𝑀, 𝑁⟩)‘𝑘) = (𝐴‘(𝑘 + 𝑀)))
11240, 111sylan 575 . . . . . 6 ((((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → ((𝐴 substr ⟨𝑀, 𝑁⟩)‘𝑘) = (𝐴‘(𝑘 + 𝑀)))
113104, 110, 1123eqtr4d 2871 . . . . 5 ((((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 substr ⟨𝑀, 𝑁⟩)‘𝑘))
11436, 42, 113eqfnfvd 6568 . . . 4 (((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))
115114ex 403 . . 3 ((¬ (♯‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
116115ex 403 . 2 (¬ (♯‘𝐴) = 0 → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
11725, 116pm2.61i 177 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:  ¬ wn 3  wi 4  wa 386  w3a 1111   = wceq 1656  wcel 2164  wne 2999  c0 4146  cop 4405   class class class wbr 4875   Fn wfn 6122  cfv 6127  (class class class)co 6910  cr 10258  0cc0 10259   + caddc 10262   < clt 10398  cle 10399  cmin 10592  cn 11357  0cn0 11625  ...cfz 12626  ..^cfzo 12767  chash 13417  Word cword 13581   ++ cconcat 13637   substr csubstr 13707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-oadd 7835  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-card 9085  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-n0 11626  df-z 11712  df-uz 11976  df-fz 12627  df-fzo 12768  df-hash 13418  df-word 13582  df-concat 13638  df-substr 13708
This theorem is referenced by:  pfxccat3  13840  swrdccat3OLD  13841  pfxccatpfx1  13844  swrdccatin1d  13855
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