Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  swrdccatin1 Structured version   Visualization version   GIF version

Theorem swrdccatin1 13334
 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 6557 . . . . . . 7 ((#‘𝐴) = 0 → (0...(#‘𝐴)) = (0...0))
21eleq2d 2673 . . . . . 6 ((#‘𝐴) = 0 → (𝑁 ∈ (0...(#‘𝐴)) ↔ 𝑁 ∈ (0...0)))
3 elfz1eq 12223 . . . . . . 7 (𝑁 ∈ (0...0) → 𝑁 = 0)
4 elfz1eq 12223 . . . . . . . . 9 (𝑀 ∈ (0...0) → 𝑀 = 0)
5 swrd00 13270 . . . . . . . . . . 11 ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = ∅
6 swrd00 13270 . . . . . . . . . . 11 (𝐴 substr ⟨0, 0⟩) = ∅
75, 6eqtr4i 2635 . . . . . . . . . 10 ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = (𝐴 substr ⟨0, 0⟩)
8 opeq1 4340 . . . . . . . . . . 11 (𝑀 = 0 → ⟨𝑀, 0⟩ = ⟨0, 0⟩)
98oveq2d 6565 . . . . . . . . . 10 (𝑀 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = ((𝐴 ++ 𝐵) substr ⟨0, 0⟩))
108oveq2d 6565 . . . . . . . . . 10 (𝑀 = 0 → (𝐴 substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨0, 0⟩))
117, 9, 103eqtr4a 2670 . . . . . . . . 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 6557 . . . . . . . . . 10 (𝑁 = 0 → (0...𝑁) = (0...0))
1413eleq2d 2673 . . . . . . . . 9 (𝑁 = 0 → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...0)))
15 opeq2 4341 . . . . . . . . . . 11 (𝑁 = 0 → ⟨𝑀, 𝑁⟩ = ⟨𝑀, 0⟩)
1615oveq2d 6565 . . . . . . . . . 10 (𝑁 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩))
1715oveq2d 6565 . . . . . . . . . 10 (𝑁 = 0 → (𝐴 substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 0⟩))
1816, 17eqeq12d 2625 . . . . . . . . 9 (𝑁 = 0 → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩) ↔ ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩)))
1914, 18imbi12d 333 . . . . . . . 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 247 . . . . . . 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 242 . . . . 5 ((#‘𝐴) = 0 → (𝑁 ∈ (0...(#‘𝐴)) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
2322com23 84 . . . 4 ((#‘𝐴) = 0 → (𝑀 ∈ (0...𝑁) → (𝑁 ∈ (0...(#‘𝐴)) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
2423impd 446 . . 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 13212 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
2726adantl 481 . . . . . . 7 ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
2827adantr 480 . . . . . 6 (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
29 simprl 790 . . . . . 6 (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → 𝑀 ∈ (0...𝑁))
30 elfzelfzccat 13217 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(#‘𝐴)) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
3130adantl 481 . . . . . . . . 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 481 . . . . . . 7 ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
3433impcom 445 . . . . . 6 (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵))))
35 swrdvalfn 13278 . . . . . 6 (((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
3628, 29, 34, 35syl3anc 1318 . . . . 5 (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
37 3anass 1035 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) ↔ (𝐴 ∈ Word 𝑉 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))))
3837simplbi2 653 . . . . . . . 8 (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → (𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))))
3938ad2antrl 760 . . . . . . 7 ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → (𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))))
4039imp 444 . . . . . 6 (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → (𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))))
41 swrdvalfn 13278 . . . . . 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 790 . . . . . . . 8 ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝐴 ∈ Word 𝑉)
4443ad2antrr 758 . . . . . . 7 ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝐴 ∈ Word 𝑉)
45 simprr 792 . . . . . . . 8 ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝐵 ∈ Word 𝑉)
4645ad2antrr 758 . . . . . . 7 ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝐵 ∈ Word 𝑉)
47 elfzo0 12376 . . . . . . . . . 10 (𝑘 ∈ (0..^(𝑁𝑀)) ↔ (𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)))
48 elfz2nn0 12300 . . . . . . . . . . . . . 14 (𝑀 ∈ (0...𝑁) ↔ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁))
49 nn0addcl 11205 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℕ0)
5049expcom 450 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ0 → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
51503ad2ant1 1075 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
5248, 51sylbi 206 . . . . . . . . . . . . 13 (𝑀 ∈ (0...𝑁) → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
5352ad2antrl 760 . . . . . . . . . . . 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 1075 . . . . . . . . . 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 206 . . . . . . . . 9 (𝑘 ∈ (0..^(𝑁𝑀)) → (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → (𝑘 + 𝑀) ∈ ℕ0))
5756impcom 445 . . . . . . . 8 ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (𝑘 + 𝑀) ∈ ℕ0)
58 lencl 13179 . . . . . . . . . . . 12 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
59 df-ne 2782 . . . . . . . . . . . . 13 ((#‘𝐴) ≠ 0 ↔ ¬ (#‘𝐴) = 0)
60 elnnne0 11183 . . . . . . . . . . . . . 14 ((#‘𝐴) ∈ ℕ ↔ ((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐴) ≠ 0))
6160simplbi2 653 . . . . . . . . . . . . 13 ((#‘𝐴) ∈ ℕ0 → ((#‘𝐴) ≠ 0 → (#‘𝐴) ∈ ℕ))
6259, 61syl5bir 232 . . . . . . . . . . . 12 ((#‘𝐴) ∈ ℕ0 → (¬ (#‘𝐴) = 0 → (#‘𝐴) ∈ ℕ))
6358, 62syl 17 . . . . . . . . . . 11 (𝐴 ∈ Word 𝑉 → (¬ (#‘𝐴) = 0 → (#‘𝐴) ∈ ℕ))
6463adantr 480 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (¬ (#‘𝐴) = 0 → (#‘𝐴) ∈ ℕ))
6564impcom 445 . . . . . . . . 9 ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (#‘𝐴) ∈ ℕ)
6665ad2antrr 758 . . . . . . . 8 ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (#‘𝐴) ∈ ℕ)
67 elfz2nn0 12300 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (0...(#‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0𝑁 ≤ (#‘𝐴)))
68 nn0re 11178 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
6968ad2antrl 760 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → 𝑘 ∈ ℝ)
70 nn0re 11178 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
7170adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → 𝑀 ∈ ℝ)
7271adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → 𝑀 ∈ ℝ)
73 nn0re 11178 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
7473ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → 𝑁 ∈ ℝ)
7569, 72, 74ltaddsubd 10506 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁𝑘 < (𝑁𝑀)))
76 nn0re 11178 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 + 𝑀) ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℝ)
7749, 76syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℝ)
7877adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → (𝑘 + 𝑀) ∈ ℝ)
79 nn0re 11178 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ ℝ)
8079adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → (#‘𝐴) ∈ ℝ)
8180adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → (#‘𝐴) ∈ ℝ)
82 ltletr 10008 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 + 𝑀) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ) → (((𝑘 + 𝑀) < 𝑁𝑁 ≤ (#‘𝐴)) → (𝑘 + 𝑀) < (#‘𝐴)))
8378, 74, 81, 82syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → (((𝑘 + 𝑀) < 𝑁𝑁 ≤ (#‘𝐴)) → (𝑘 + 𝑀) < (#‘𝐴)))
8483expd 451 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁 → (𝑁 ≤ (#‘𝐴) → (𝑘 + 𝑀) < (#‘𝐴))))
8575, 84sylbird 249 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → (𝑘 < (𝑁𝑀) → (𝑁 ≤ (#‘𝐴) → (𝑘 + 𝑀) < (#‘𝐴))))
8685ex 449 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 < (𝑁𝑀) → (𝑁 ≤ (#‘𝐴) → (𝑘 + 𝑀) < (#‘𝐴)))))
8786com24 93 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → (𝑁 ≤ (#‘𝐴) → (𝑘 < (𝑁𝑀) → ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 + 𝑀) < (#‘𝐴)))))
88873impia 1253 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0𝑁 ≤ (#‘𝐴)) → (𝑘 < (𝑁𝑀) → ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 + 𝑀) < (#‘𝐴))))
8988com13 86 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 < (𝑁𝑀) → ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0𝑁 ≤ (#‘𝐴)) → (𝑘 + 𝑀) < (#‘𝐴))))
9089impancom 455 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ ℕ0𝑘 < (𝑁𝑀)) → (𝑀 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0𝑁 ≤ (#‘𝐴)) → (𝑘 + 𝑀) < (#‘𝐴))))
91903adant2 1073 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑀 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0𝑁 ≤ (#‘𝐴)) → (𝑘 + 𝑀) < (#‘𝐴))))
9291com13 86 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0𝑁 ≤ (#‘𝐴)) → (𝑀 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑘 + 𝑀) < (#‘𝐴))))
9367, 92sylbi 206 . . . . . . . . . . . . . . 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 1075 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝑁 ∈ (0...(#‘𝐴)) → ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑘 + 𝑀) < (#‘𝐴))))
9648, 95sylbi 206 . . . . . . . . . . . 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 448 . . . . . . . . . 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 231 . . . . . . . . 9 (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → (𝑘 ∈ (0..^(𝑁𝑀)) → (𝑘 + 𝑀) < (#‘𝐴)))
10099imp 444 . . . . . . . 8 ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (𝑘 + 𝑀) < (#‘𝐴))
101 elfzo0 12376 . . . . . . . 8 ((𝑘 + 𝑀) ∈ (0..^(#‘𝐴)) ↔ ((𝑘 + 𝑀) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ ∧ (𝑘 + 𝑀) < (#‘𝐴)))
10257, 66, 100, 101syl3anbrc 1239 . . . . . . 7 ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (𝑘 + 𝑀) ∈ (0..^(#‘𝐴)))
103 ccatval1 13214 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ (0..^(#‘𝐴))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐴‘(𝑘 + 𝑀)))
10444, 46, 102, 103syl3anc 1318 . . . . . 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 758 . . . . . . . 8 ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
10629adantr 480 . . . . . . . 8 ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝑀 ∈ (0...𝑁))
10734adantr 480 . . . . . . . 8 ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵))))
108105, 106, 1073jca 1235 . . . . . . 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 13276 . . . . . . 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 698 . . . . . 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 13276 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → ((𝐴 substr ⟨𝑀, 𝑁⟩)‘𝑘) = (𝐴‘(𝑘 + 𝑀)))
11240, 111sylan 487 . . . . . 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 2654 . . . . 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 6222 . . . 4 (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))
115114ex 449 . . 3 ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
116115ex 449 . 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 175 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  ⟨cop 4131   class class class wbr 4583   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ...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:  swrdccat3  13343  swrdccatin1d  13350  pfxccat3  40289  pfxccatpfx1  40290
 Copyright terms: Public domain W3C validator