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Theorem swrdccat3blem 13749
Description: Lemma for swrdccat3b 13750. (Contributed by AV, 30-May-2018.)
Hypothesis
Ref Expression
swrdccatin2.l 𝐿 = (♯‘𝐴)
Assertion
Ref Expression
swrdccat3blem ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (♯‘𝐵)))) ∧ (𝐿 + (♯‘𝐵)) ≤ 𝐿) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩))

Proof of Theorem swrdccat3blem
StepHypRef Expression
1 lencl 13505 . . . . . . . 8 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ0)
2 nn0le0eq0 11568 . . . . . . . . 9 ((♯‘𝐵) ∈ ℕ0 → ((♯‘𝐵) ≤ 0 ↔ (♯‘𝐵) = 0))
32biimpd 220 . . . . . . . 8 ((♯‘𝐵) ∈ ℕ0 → ((♯‘𝐵) ≤ 0 → (♯‘𝐵) = 0))
41, 3syl 17 . . . . . . 7 (𝐵 ∈ Word 𝑉 → ((♯‘𝐵) ≤ 0 → (♯‘𝐵) = 0))
54adantl 473 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((♯‘𝐵) ≤ 0 → (♯‘𝐵) = 0))
6 hasheq0 13356 . . . . . . . . . . 11 (𝐵 ∈ Word 𝑉 → ((♯‘𝐵) = 0 ↔ 𝐵 = ∅))
76biimpd 220 . . . . . . . . . 10 (𝐵 ∈ Word 𝑉 → ((♯‘𝐵) = 0 → 𝐵 = ∅))
87adantl 473 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((♯‘𝐵) = 0 → 𝐵 = ∅))
98imp 395 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐵) = 0) → 𝐵 = ∅)
10 lencl 13505 . . . . . . . . . . . . . . . 16 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
11 swrdccatin2.l . . . . . . . . . . . . . . . . . . 19 𝐿 = (♯‘𝐴)
1211eqcomi 2774 . . . . . . . . . . . . . . . . . 18 (♯‘𝐴) = 𝐿
1312eleq1i 2835 . . . . . . . . . . . . . . . . 17 ((♯‘𝐴) ∈ ℕ0𝐿 ∈ ℕ0)
14 nn0re 11548 . . . . . . . . . . . . . . . . . 18 (𝐿 ∈ ℕ0𝐿 ∈ ℝ)
15 elfz2nn0 12638 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ (0...(𝐿 + 0)) ↔ (𝑀 ∈ ℕ0 ∧ (𝐿 + 0) ∈ ℕ0𝑀 ≤ (𝐿 + 0)))
16 recn 10279 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐿 ∈ ℝ → 𝐿 ∈ ℂ)
1716addid1d 10490 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ ℝ → (𝐿 + 0) = 𝐿)
1817breq2d 4821 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐿 ∈ ℝ → (𝑀 ≤ (𝐿 + 0) ↔ 𝑀𝐿))
19 nn0re 11548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
2019anim1i 608 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑀 ∈ ℕ0𝐿 ∈ ℝ) → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
2120ancoms 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
22 letri3 10377 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 = 𝐿 ↔ (𝑀𝐿𝐿𝑀)))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝑀 = 𝐿 ↔ (𝑀𝐿𝐿𝑀)))
2423biimprd 239 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → ((𝑀𝐿𝐿𝑀) → 𝑀 = 𝐿))
2524exp4b 421 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ ℝ → (𝑀 ∈ ℕ0 → (𝑀𝐿 → (𝐿𝑀𝑀 = 𝐿))))
2625com23 86 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐿 ∈ ℝ → (𝑀𝐿 → (𝑀 ∈ ℕ0 → (𝐿𝑀𝑀 = 𝐿))))
2718, 26sylbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐿 ∈ ℝ → (𝑀 ≤ (𝐿 + 0) → (𝑀 ∈ ℕ0 → (𝐿𝑀𝑀 = 𝐿))))
2827com3l 89 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ≤ (𝐿 + 0) → (𝑀 ∈ ℕ0 → (𝐿 ∈ ℝ → (𝐿𝑀𝑀 = 𝐿))))
2928impcom 396 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 ∈ ℕ0𝑀 ≤ (𝐿 + 0)) → (𝐿 ∈ ℝ → (𝐿𝑀𝑀 = 𝐿)))
30293adant2 1161 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ ℕ0 ∧ (𝐿 + 0) ∈ ℕ0𝑀 ≤ (𝐿 + 0)) → (𝐿 ∈ ℝ → (𝐿𝑀𝑀 = 𝐿)))
3130com12 32 . . . . . . . . . . . . . . . . . . 19 (𝐿 ∈ ℝ → ((𝑀 ∈ ℕ0 ∧ (𝐿 + 0) ∈ ℕ0𝑀 ≤ (𝐿 + 0)) → (𝐿𝑀𝑀 = 𝐿)))
3215, 31syl5bi 233 . . . . . . . . . . . . . . . . . 18 (𝐿 ∈ ℝ → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿𝑀𝑀 = 𝐿)))
3314, 32syl 17 . . . . . . . . . . . . . . . . 17 (𝐿 ∈ ℕ0 → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿𝑀𝑀 = 𝐿)))
3413, 33sylbi 208 . . . . . . . . . . . . . . . 16 ((♯‘𝐴) ∈ ℕ0 → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿𝑀𝑀 = 𝐿)))
3510, 34syl 17 . . . . . . . . . . . . . . 15 (𝐴 ∈ Word 𝑉 → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿𝑀𝑀 = 𝐿)))
3635imp 395 . . . . . . . . . . . . . 14 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) → (𝐿𝑀𝑀 = 𝐿))
37 elfznn0 12640 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (0...(𝐿 + 0)) → 𝑀 ∈ ℕ0)
38 swrd00 13619 . . . . . . . . . . . . . . . . . . . . . 22 (∅ substr ⟨0, 0⟩) = ∅
39 swrd00 13619 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 substr ⟨𝐿, 𝐿⟩) = ∅
4038, 39eqtr4i 2790 . . . . . . . . . . . . . . . . . . . . 21 (∅ substr ⟨0, 0⟩) = (𝐴 substr ⟨𝐿, 𝐿⟩)
41 nn0cn 11549 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐿 ∈ ℕ0𝐿 ∈ ℂ)
4241subidd 10634 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐿 ∈ ℕ0 → (𝐿𝐿) = 0)
4342opeq1d 4565 . . . . . . . . . . . . . . . . . . . . . 22 (𝐿 ∈ ℕ0 → ⟨(𝐿𝐿), 0⟩ = ⟨0, 0⟩)
4443oveq2d 6858 . . . . . . . . . . . . . . . . . . . . 21 (𝐿 ∈ ℕ0 → (∅ substr ⟨(𝐿𝐿), 0⟩) = (∅ substr ⟨0, 0⟩))
4541addid1d 10490 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐿 ∈ ℕ0 → (𝐿 + 0) = 𝐿)
4645opeq2d 4566 . . . . . . . . . . . . . . . . . . . . . 22 (𝐿 ∈ ℕ0 → ⟨𝐿, (𝐿 + 0)⟩ = ⟨𝐿, 𝐿⟩)
4746oveq2d 6858 . . . . . . . . . . . . . . . . . . . . 21 (𝐿 ∈ ℕ0 → (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩) = (𝐴 substr ⟨𝐿, 𝐿⟩))
4840, 44, 473eqtr4a 2825 . . . . . . . . . . . . . . . . . . . 20 (𝐿 ∈ ℕ0 → (∅ substr ⟨(𝐿𝐿), 0⟩) = (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩))
4948a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑀 = 𝐿 → (𝐿 ∈ ℕ0 → (∅ substr ⟨(𝐿𝐿), 0⟩) = (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩)))
50 eleq1 2832 . . . . . . . . . . . . . . . . . . 19 (𝑀 = 𝐿 → (𝑀 ∈ ℕ0𝐿 ∈ ℕ0))
51 oveq1 6849 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 = 𝐿 → (𝑀𝐿) = (𝐿𝐿))
5251opeq1d 4565 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 = 𝐿 → ⟨(𝑀𝐿), 0⟩ = ⟨(𝐿𝐿), 0⟩)
5352oveq2d 6858 . . . . . . . . . . . . . . . . . . . 20 (𝑀 = 𝐿 → (∅ substr ⟨(𝑀𝐿), 0⟩) = (∅ substr ⟨(𝐿𝐿), 0⟩))
54 opeq1 4559 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 = 𝐿 → ⟨𝑀, (𝐿 + 0)⟩ = ⟨𝐿, (𝐿 + 0)⟩)
5554oveq2d 6858 . . . . . . . . . . . . . . . . . . . 20 (𝑀 = 𝐿 → (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩) = (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩))
5653, 55eqeq12d 2780 . . . . . . . . . . . . . . . . . . 19 (𝑀 = 𝐿 → ((∅ substr ⟨(𝑀𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩) ↔ (∅ substr ⟨(𝐿𝐿), 0⟩) = (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩)))
5749, 50, 563imtr4d 285 . . . . . . . . . . . . . . . . . 18 (𝑀 = 𝐿 → (𝑀 ∈ ℕ0 → (∅ substr ⟨(𝑀𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
5857com12 32 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℕ0 → (𝑀 = 𝐿 → (∅ substr ⟨(𝑀𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
5958a1d 25 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℕ0 → (𝐴 ∈ Word 𝑉 → (𝑀 = 𝐿 → (∅ substr ⟨(𝑀𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))))
6037, 59syl 17 . . . . . . . . . . . . . . 15 (𝑀 ∈ (0...(𝐿 + 0)) → (𝐴 ∈ Word 𝑉 → (𝑀 = 𝐿 → (∅ substr ⟨(𝑀𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))))
6160impcom 396 . . . . . . . . . . . . . 14 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) → (𝑀 = 𝐿 → (∅ substr ⟨(𝑀𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
6236, 61syld 47 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) → (𝐿𝑀 → (∅ substr ⟨(𝑀𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
6362imp 395 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) ∧ 𝐿𝑀) → (∅ substr ⟨(𝑀𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
64 swrdcl 13620 . . . . . . . . . . . . . . . 16 (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉)
65 ccatrid 13558 . . . . . . . . . . . . . . . 16 ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅) = (𝐴 substr ⟨𝑀, 𝐿⟩))
6664, 65syl 17 . . . . . . . . . . . . . . 15 (𝐴 ∈ Word 𝑉 → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅) = (𝐴 substr ⟨𝑀, 𝐿⟩))
6713, 41sylbi 208 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝐴) ∈ ℕ0𝐿 ∈ ℂ)
6810, 67syl 17 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ Word 𝑉𝐿 ∈ ℂ)
69 addid1 10470 . . . . . . . . . . . . . . . . . . 19 (𝐿 ∈ ℂ → (𝐿 + 0) = 𝐿)
7069eqcomd 2771 . . . . . . . . . . . . . . . . . 18 (𝐿 ∈ ℂ → 𝐿 = (𝐿 + 0))
7168, 70syl 17 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ Word 𝑉𝐿 = (𝐿 + 0))
7271opeq2d 4566 . . . . . . . . . . . . . . . 16 (𝐴 ∈ Word 𝑉 → ⟨𝑀, 𝐿⟩ = ⟨𝑀, (𝐿 + 0)⟩)
7372oveq2d 6858 . . . . . . . . . . . . . . 15 (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
7466, 73eqtrd 2799 . . . . . . . . . . . . . 14 (𝐴 ∈ Word 𝑉 → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
7574adantr 472 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
7675adantr 472 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) ∧ ¬ 𝐿𝑀) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
7763, 76ifeqda 4278 . . . . . . . . . . 11 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) → if(𝐿𝑀, (∅ substr ⟨(𝑀𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
7877ex 401 . . . . . . . . . 10 (𝐴 ∈ Word 𝑉 → (𝑀 ∈ (0...(𝐿 + 0)) → if(𝐿𝑀, (∅ substr ⟨(𝑀𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
7978ad3antrrr 721 . . . . . . . . 9 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐵) = 0) ∧ 𝐵 = ∅) → (𝑀 ∈ (0...(𝐿 + 0)) → if(𝐿𝑀, (∅ substr ⟨(𝑀𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
80 oveq2 6850 . . . . . . . . . . . . . 14 ((♯‘𝐵) = 0 → (𝐿 + (♯‘𝐵)) = (𝐿 + 0))
8180oveq2d 6858 . . . . . . . . . . . . 13 ((♯‘𝐵) = 0 → (0...(𝐿 + (♯‘𝐵))) = (0...(𝐿 + 0)))
8281eleq2d 2830 . . . . . . . . . . . 12 ((♯‘𝐵) = 0 → (𝑀 ∈ (0...(𝐿 + (♯‘𝐵))) ↔ 𝑀 ∈ (0...(𝐿 + 0))))
8382adantr 472 . . . . . . . . . . 11 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → (𝑀 ∈ (0...(𝐿 + (♯‘𝐵))) ↔ 𝑀 ∈ (0...(𝐿 + 0))))
84 simpr 477 . . . . . . . . . . . . . 14 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → 𝐵 = ∅)
85 opeq2 4560 . . . . . . . . . . . . . . 15 ((♯‘𝐵) = 0 → ⟨(𝑀𝐿), (♯‘𝐵)⟩ = ⟨(𝑀𝐿), 0⟩)
8685adantr 472 . . . . . . . . . . . . . 14 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → ⟨(𝑀𝐿), (♯‘𝐵)⟩ = ⟨(𝑀𝐿), 0⟩)
8784, 86oveq12d 6860 . . . . . . . . . . . . 13 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩) = (∅ substr ⟨(𝑀𝐿), 0⟩))
88 oveq2 6850 . . . . . . . . . . . . . 14 (𝐵 = ∅ → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅))
8988adantl 473 . . . . . . . . . . . . 13 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅))
9087, 89ifeq12d 4263 . . . . . . . . . . . 12 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = if(𝐿𝑀, (∅ substr ⟨(𝑀𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)))
9180opeq2d 4566 . . . . . . . . . . . . . 14 ((♯‘𝐵) = 0 → ⟨𝑀, (𝐿 + (♯‘𝐵))⟩ = ⟨𝑀, (𝐿 + 0)⟩)
9291oveq2d 6858 . . . . . . . . . . . . 13 ((♯‘𝐵) = 0 → (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
9392adantr 472 . . . . . . . . . . . 12 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
9490, 93eqeq12d 2780 . . . . . . . . . . 11 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → (if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩) ↔ if(𝐿𝑀, (∅ substr ⟨(𝑀𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
9583, 94imbi12d 335 . . . . . . . . . 10 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → ((𝑀 ∈ (0...(𝐿 + (♯‘𝐵))) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩)) ↔ (𝑀 ∈ (0...(𝐿 + 0)) → if(𝐿𝑀, (∅ substr ⟨(𝑀𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))))
9695adantll 705 . . . . . . . . 9 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐵) = 0) ∧ 𝐵 = ∅) → ((𝑀 ∈ (0...(𝐿 + (♯‘𝐵))) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩)) ↔ (𝑀 ∈ (0...(𝐿 + 0)) → if(𝐿𝑀, (∅ substr ⟨(𝑀𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))))
9779, 96mpbird 248 . . . . . . . 8 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐵) = 0) ∧ 𝐵 = ∅) → (𝑀 ∈ (0...(𝐿 + (♯‘𝐵))) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩)))
989, 97mpdan 678 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐵) = 0) → (𝑀 ∈ (0...(𝐿 + (♯‘𝐵))) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩)))
9998ex 401 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((♯‘𝐵) = 0 → (𝑀 ∈ (0...(𝐿 + (♯‘𝐵))) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩))))
1005, 99syld 47 . . . . 5 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((♯‘𝐵) ≤ 0 → (𝑀 ∈ (0...(𝐿 + (♯‘𝐵))) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩))))
101100com23 86 . . . 4 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑀 ∈ (0...(𝐿 + (♯‘𝐵))) → ((♯‘𝐵) ≤ 0 → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩))))
102101imp 395 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((♯‘𝐵) ≤ 0 → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩)))
103102adantr 472 . 2 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (♯‘𝐵)))) ∧ (𝐿 + (♯‘𝐵)) ≤ 𝐿) → ((♯‘𝐵) ≤ 0 → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩)))
10411eleq1i 2835 . . . . . . . 8 (𝐿 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0)
105104, 14sylbir 226 . . . . . . 7 ((♯‘𝐴) ∈ ℕ0𝐿 ∈ ℝ)
10610, 105syl 17 . . . . . 6 (𝐴 ∈ Word 𝑉𝐿 ∈ ℝ)
1071nn0red 11599 . . . . . 6 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℝ)
108 leaddle0 10797 . . . . . 6 ((𝐿 ∈ ℝ ∧ (♯‘𝐵) ∈ ℝ) → ((𝐿 + (♯‘𝐵)) ≤ 𝐿 ↔ (♯‘𝐵) ≤ 0))
109106, 107, 108syl2an 589 . . . . 5 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝐿 + (♯‘𝐵)) ≤ 𝐿 ↔ (♯‘𝐵) ≤ 0))
110 pm2.24 122 . . . . 5 ((♯‘𝐵) ≤ 0 → (¬ (♯‘𝐵) ≤ 0 → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩)))
111109, 110syl6bi 244 . . . 4 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝐿 + (♯‘𝐵)) ≤ 𝐿 → (¬ (♯‘𝐵) ≤ 0 → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩))))
112111adantr 472 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((𝐿 + (♯‘𝐵)) ≤ 𝐿 → (¬ (♯‘𝐵) ≤ 0 → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩))))
113112imp 395 . 2 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (♯‘𝐵)))) ∧ (𝐿 + (♯‘𝐵)) ≤ 𝐿) → (¬ (♯‘𝐵) ≤ 0 → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩)))
114103, 113pm2.61d 171 1 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (♯‘𝐵)))) ∧ (𝐿 + (♯‘𝐵)) ≤ 𝐿) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  c0 4079  ifcif 4243  cop 4340   class class class wbr 4809  cfv 6068  (class class class)co 6842  cc 10187  cr 10188  0cc0 10189   + caddc 10192  cle 10329  cmin 10520  0cn0 11538  ...cfz 12533  chash 13321  Word cword 13486   ++ cconcat 13541   substr csubstr 13616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13487  df-concat 13542  df-substr 13617
This theorem is referenced by:  swrdccat3b  13750  swrdccat3bOLD  13751
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