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Theorem swrdccatin1 14702
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 7421 . . . . . 6 ((♯‘𝐴) = 0 → (0...(♯‘𝐴)) = (0...0))
21eleq2d 2811 . . . . 5 ((♯‘𝐴) = 0 → (𝑁 ∈ (0...(♯‘𝐴)) ↔ 𝑁 ∈ (0...0)))
3 elfz1eq 13539 . . . . . 6 (𝑁 ∈ (0...0) → 𝑁 = 0)
4 elfz1eq 13539 . . . . . . . 8 (𝑀 ∈ (0...0) → 𝑀 = 0)
5 swrd00 14621 . . . . . . . . . 10 ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = ∅
6 swrd00 14621 . . . . . . . . . 10 (𝐴 substr ⟨0, 0⟩) = ∅
75, 6eqtr4i 2756 . . . . . . . . 9 ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = (𝐴 substr ⟨0, 0⟩)
8 opeq1 4870 . . . . . . . . . 10 (𝑀 = 0 → ⟨𝑀, 0⟩ = ⟨0, 0⟩)
98oveq2d 7429 . . . . . . . . 9 (𝑀 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = ((𝐴 ++ 𝐵) substr ⟨0, 0⟩))
108oveq2d 7429 . . . . . . . . 9 (𝑀 = 0 → (𝐴 substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨0, 0⟩))
117, 9, 103eqtr4a 2791 . . . . . . . 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 7421 . . . . . . . . 9 (𝑁 = 0 → (0...𝑁) = (0...0))
1413eleq2d 2811 . . . . . . . 8 (𝑁 = 0 → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...0)))
15 opeq2 4871 . . . . . . . . . 10 (𝑁 = 0 → ⟨𝑀, 𝑁⟩ = ⟨𝑀, 0⟩)
1615oveq2d 7429 . . . . . . . . 9 (𝑁 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩))
1715oveq2d 7429 . . . . . . . . 9 (𝑁 = 0 → (𝐴 substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 0⟩))
1816, 17eqeq12d 2741 . . . . . . . 8 (𝑁 = 0 → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩) ↔ ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩)))
1914, 18imbi12d 343 . . . . . . 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 257 . . . . . 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, 21biimtrdi 252 . . . 4 ((♯‘𝐴) = 0 → (𝑁 ∈ (0...(♯‘𝐴)) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
2322impcomd 410 . . 3 ((♯‘𝐴) = 0 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
2423adantl 480 . 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 14551 . . . . . 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 14557 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(♯‘𝐴)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
2928imp 405 . . . . . 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 14628 . . . . 5 (((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
3226, 27, 30, 31syl3anc 1368 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
33 3anass 1092 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) ↔ (𝐴 ∈ Word 𝑉 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
3433simplbi2 499 . . . . . . 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 405 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))))
37 swrdvalfn 14628 . . . . 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 13621 . . . . . . . . . 10 (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ0)
42 nn0addcl 12532 . . . . . . . . . . 11 ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℕ0)
4342expcom 412 . . . . . . . . . 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 13704 . . . . . . . 8 (𝑘 ∈ (0..^(𝑁𝑀)) → 𝑘 ∈ ℕ0)
4745, 46impel 504 . . . . . . 7 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (𝑘 + 𝑀) ∈ ℕ0)
48 lencl 14510 . . . . . . . . . . 11 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
49 elnnne0 12511 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ ℕ ↔ ((♯‘𝐴) ∈ ℕ0 ∧ (♯‘𝐴) ≠ 0))
5049simplbi2 499 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
5148, 50syl 17 . . . . . . . . . 10 (𝐴 ∈ Word 𝑉 → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
5251adantr 479 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
5352imp 405 . . . . . . . 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 13700 . . . . . . . . 9 (𝑘 ∈ (0..^(𝑁𝑀)) ↔ (𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)))
56 elfz2nn0 13619 . . . . . . . . . . . 12 (𝑁 ∈ (0...(♯‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0𝑁 ≤ (♯‘𝐴)))
57 nn0re 12506 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
5857ad2antrl 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → 𝑘 ∈ ℝ)
59 nn0re 12506 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
6059ad2antll 727 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → 𝑀 ∈ ℝ)
61 nn0re 12506 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
6261ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → 𝑁 ∈ ℝ)
6358, 60, 62ltaddsubd 11839 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁𝑘 < (𝑁𝑀)))
64 nn0readdcl 12563 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℝ)
6564adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → (𝑘 + 𝑀) ∈ ℝ)
66 nn0re 12506 . . . . . . . . . . . . . . . . . . . . . . 23 ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℝ)
6766ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → (♯‘𝐴) ∈ ℝ)
68 ltletr 11331 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑘 + 𝑀) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (♯‘𝐴) ∈ ℝ) → (((𝑘 + 𝑀) < 𝑁𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴)))
6965, 62, 67, 68syl3anc 1368 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → (((𝑘 + 𝑀) < 𝑁𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴)))
7069expd 414 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁 → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴))))
7163, 70sylbird 259 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0𝑀 ∈ ℕ0)) → (𝑘 < (𝑁𝑀) → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴))))
7271ex 411 . . . . . . . . . . . . . . . . . 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 1114 . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℕ0𝑘 < (𝑁𝑀)) → (𝑀 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴))))
77763adant2 1128 . . . . . . . . . . . . 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 216 . . . . . . . . . . 11 (𝑁 ∈ (0...(♯‘𝐴)) → (𝑀 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴))))
8041, 79mpan9 505 . . . . . . . . . 10 ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴)))
8180adantl 480 . . . . . . . . 9 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → ((𝑘 ∈ ℕ0 ∧ (𝑁𝑀) ∈ ℕ ∧ 𝑘 < (𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴)))
8255, 81biimtrid 241 . . . . . . . 8 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝑘 ∈ (0..^(𝑁𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴)))
8382imp 405 . . . . . . 7 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (𝑘 + 𝑀) < (♯‘𝐴))
84 elfzo0 13700 . . . . . . 7 ((𝑘 + 𝑀) ∈ (0..^(♯‘𝐴)) ↔ ((𝑘 + 𝑀) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ ∧ (𝑘 + 𝑀) < (♯‘𝐴)))
8547, 54, 83, 84syl3anbrc 1340 . . . . . 6 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (𝑘 + 𝑀) ∈ (0..^(♯‘𝐴)))
86 ccatval1 14554 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐴‘(𝑘 + 𝑀)))
8739, 40, 85, 86syl3anc 1368 . . . . 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 479 . . . . . 6 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
91 simpr 483 . . . . . 6 (((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝑘 ∈ (0..^(𝑁𝑀)))
92 swrdfv 14625 . . . . . 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 1370 . . . . 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 14625 . . . . . 6 (((𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → ((𝐴 substr ⟨𝑀, 𝑁⟩)‘𝑘) = (𝐴‘(𝑘 + 𝑀)))
9536, 94sylan 578 . . . . 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 2775 . . . 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 7036 . . 3 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))
9897ex 411 . 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 3019 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 394  w3a 1084   = wceq 1533  wcel 2098  wne 2930  c0 4319  cop 4631   class class class wbr 5144   Fn wfn 6538  cfv 6543  (class class class)co 7413  cr 11132  0cc0 11133   + caddc 11136   < clt 11273  cle 11274  cmin 11469  cn 12237  0cn0 12497  ...cfz 13511  ..^cfzo 13654  chash 14316  Word cword 14491   ++ cconcat 14547   substr csubstr 14617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-n0 12498  df-z 12584  df-uz 12848  df-fz 13512  df-fzo 13655  df-hash 14317  df-word 14492  df-concat 14548  df-substr 14618
This theorem is referenced by:  pfxccat3  14711  pfxccatpfx1  14713  swrdccatin1d  14720
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