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Theorem swrdccat3blem 14627
Description: Lemma for swrdccat3b 14628. (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 14421 . . . . . . . 8 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ0)
2 nn0le0eq0 12441 . . . . . . . . 9 ((♯‘𝐵) ∈ ℕ0 → ((♯‘𝐵) ≤ 0 ↔ (♯‘𝐵) = 0))
32biimpd 228 . . . . . . . 8 ((♯‘𝐵) ∈ ℕ0 → ((♯‘𝐵) ≤ 0 → (♯‘𝐵) = 0))
41, 3syl 17 . . . . . . 7 (𝐵 ∈ Word 𝑉 → ((♯‘𝐵) ≤ 0 → (♯‘𝐵) = 0))
54adantl 482 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((♯‘𝐵) ≤ 0 → (♯‘𝐵) = 0))
6 hasheq0 14263 . . . . . . . . . . 11 (𝐵 ∈ Word 𝑉 → ((♯‘𝐵) = 0 ↔ 𝐵 = ∅))
76biimpd 228 . . . . . . . . . 10 (𝐵 ∈ Word 𝑉 → ((♯‘𝐵) = 0 → 𝐵 = ∅))
87adantl 482 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((♯‘𝐵) = 0 → 𝐵 = ∅))
98imp 407 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (♯‘𝐵) = 0) → 𝐵 = ∅)
10 lencl 14421 . . . . . . . . . . . . . . . 16 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
11 swrdccatin2.l . . . . . . . . . . . . . . . . . . 19 𝐿 = (♯‘𝐴)
1211eqcomi 2745 . . . . . . . . . . . . . . . . . 18 (♯‘𝐴) = 𝐿
1312eleq1i 2828 . . . . . . . . . . . . . . . . 17 ((♯‘𝐴) ∈ ℕ0𝐿 ∈ ℕ0)
14 nn0re 12422 . . . . . . . . . . . . . . . . . 18 (𝐿 ∈ ℕ0𝐿 ∈ ℝ)
15 elfz2nn0 13532 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ (0...(𝐿 + 0)) ↔ (𝑀 ∈ ℕ0 ∧ (𝐿 + 0) ∈ ℕ0𝑀 ≤ (𝐿 + 0)))
16 recn 11141 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐿 ∈ ℝ → 𝐿 ∈ ℂ)
1716addid1d 11355 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ ℝ → (𝐿 + 0) = 𝐿)
1817breq2d 5117 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐿 ∈ ℝ → (𝑀 ≤ (𝐿 + 0) ↔ 𝑀𝐿))
19 nn0re 12422 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
2019anim1i 615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑀 ∈ ℕ0𝐿 ∈ ℝ) → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
2120ancoms 459 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
22 letri3 11240 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 = 𝐿 ↔ (𝑀𝐿𝐿𝑀)))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝑀 = 𝐿 ↔ (𝑀𝐿𝐿𝑀)))
2423biimprd 247 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → ((𝑀𝐿𝐿𝑀) → 𝑀 = 𝐿))
2524exp4b 431 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ ℝ → (𝑀 ∈ ℕ0 → (𝑀𝐿 → (𝐿𝑀𝑀 = 𝐿))))
2625com23 86 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐿 ∈ ℝ → (𝑀𝐿 → (𝑀 ∈ ℕ0 → (𝐿𝑀𝑀 = 𝐿))))
2718, 26sylbid 239 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐿 ∈ ℝ → (𝑀 ≤ (𝐿 + 0) → (𝑀 ∈ ℕ0 → (𝐿𝑀𝑀 = 𝐿))))
2827com3l 89 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ≤ (𝐿 + 0) → (𝑀 ∈ ℕ0 → (𝐿 ∈ ℝ → (𝐿𝑀𝑀 = 𝐿))))
2928impcom 408 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 ∈ ℕ0𝑀 ≤ (𝐿 + 0)) → (𝐿 ∈ ℝ → (𝐿𝑀𝑀 = 𝐿)))
30293adant2 1131 . . . . . . . . . . . . . . . . . . . 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, 31biimtrid 241 . . . . . . . . . . . . . . . . . 18 (𝐿 ∈ ℝ → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿𝑀𝑀 = 𝐿)))
3314, 32syl 17 . . . . . . . . . . . . . . . . 17 (𝐿 ∈ ℕ0 → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿𝑀𝑀 = 𝐿)))
3413, 33sylbi 216 . . . . . . . . . . . . . . . 16 ((♯‘𝐴) ∈ ℕ0 → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿𝑀𝑀 = 𝐿)))
3510, 34syl 17 . . . . . . . . . . . . . . 15 (𝐴 ∈ Word 𝑉 → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿𝑀𝑀 = 𝐿)))
3635imp 407 . . . . . . . . . . . . . 14 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) → (𝐿𝑀𝑀 = 𝐿))
37 elfznn0 13534 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (0...(𝐿 + 0)) → 𝑀 ∈ ℕ0)
38 swrd00 14532 . . . . . . . . . . . . . . . . . . . . . 22 (∅ substr ⟨0, 0⟩) = ∅
39 swrd00 14532 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 substr ⟨𝐿, 𝐿⟩) = ∅
4038, 39eqtr4i 2767 . . . . . . . . . . . . . . . . . . . . 21 (∅ substr ⟨0, 0⟩) = (𝐴 substr ⟨𝐿, 𝐿⟩)
41 nn0cn 12423 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐿 ∈ ℕ0𝐿 ∈ ℂ)
4241subidd 11500 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐿 ∈ ℕ0 → (𝐿𝐿) = 0)
4342opeq1d 4836 . . . . . . . . . . . . . . . . . . . . . 22 (𝐿 ∈ ℕ0 → ⟨(𝐿𝐿), 0⟩ = ⟨0, 0⟩)
4443oveq2d 7373 . . . . . . . . . . . . . . . . . . . . 21 (𝐿 ∈ ℕ0 → (∅ substr ⟨(𝐿𝐿), 0⟩) = (∅ substr ⟨0, 0⟩))
4541addid1d 11355 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐿 ∈ ℕ0 → (𝐿 + 0) = 𝐿)
4645opeq2d 4837 . . . . . . . . . . . . . . . . . . . . . 22 (𝐿 ∈ ℕ0 → ⟨𝐿, (𝐿 + 0)⟩ = ⟨𝐿, 𝐿⟩)
4746oveq2d 7373 . . . . . . . . . . . . . . . . . . . . 21 (𝐿 ∈ ℕ0 → (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩) = (𝐴 substr ⟨𝐿, 𝐿⟩))
4840, 44, 473eqtr4a 2802 . . . . . . . . . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . . . . . . 19 (𝑀 = 𝐿 → (𝑀 ∈ ℕ0𝐿 ∈ ℕ0))
51 oveq1 7364 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 = 𝐿 → (𝑀𝐿) = (𝐿𝐿))
5251opeq1d 4836 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 = 𝐿 → ⟨(𝑀𝐿), 0⟩ = ⟨(𝐿𝐿), 0⟩)
5352oveq2d 7373 . . . . . . . . . . . . . . . . . . . 20 (𝑀 = 𝐿 → (∅ substr ⟨(𝑀𝐿), 0⟩) = (∅ substr ⟨(𝐿𝐿), 0⟩))
54 opeq1 4830 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 = 𝐿 → ⟨𝑀, (𝐿 + 0)⟩ = ⟨𝐿, (𝐿 + 0)⟩)
5554oveq2d 7373 . . . . . . . . . . . . . . . . . . . 20 (𝑀 = 𝐿 → (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩) = (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩))
5653, 55eqeq12d 2752 . . . . . . . . . . . . . . . . . . 19 (𝑀 = 𝐿 → ((∅ substr ⟨(𝑀𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩) ↔ (∅ substr ⟨(𝐿𝐿), 0⟩) = (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩)))
5749, 50, 563imtr4d 293 . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . 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 407 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) ∧ 𝐿𝑀) → (∅ substr ⟨(𝑀𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
64 swrdcl 14533 . . . . . . . . . . . . . . . 16 (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉)
65 ccatrid 14475 . . . . . . . . . . . . . . . 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 216 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝐴) ∈ ℕ0𝐿 ∈ ℂ)
6810, 67syl 17 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ Word 𝑉𝐿 ∈ ℂ)
69 addid1 11335 . . . . . . . . . . . . . . . . . . 19 (𝐿 ∈ ℂ → (𝐿 + 0) = 𝐿)
7069eqcomd 2742 . . . . . . . . . . . . . . . . . 18 (𝐿 ∈ ℂ → 𝐿 = (𝐿 + 0))
7168, 70syl 17 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ Word 𝑉𝐿 = (𝐿 + 0))
7271opeq2d 4837 . . . . . . . . . . . . . . . 16 (𝐴 ∈ Word 𝑉 → ⟨𝑀, 𝐿⟩ = ⟨𝑀, (𝐿 + 0)⟩)
7372oveq2d 7373 . . . . . . . . . . . . . . 15 (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
7466, 73eqtrd 2776 . . . . . . . . . . . . . 14 (𝐴 ∈ Word 𝑉 → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
7574adantr 481 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
7675adantr 481 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) ∧ ¬ 𝐿𝑀) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
7763, 76ifeqda 4522 . . . . . . . . . . 11 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) → if(𝐿𝑀, (∅ substr ⟨(𝑀𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
7877ex 413 . . . . . . . . . 10 (𝐴 ∈ Word 𝑉 → (𝑀 ∈ (0...(𝐿 + 0)) → if(𝐿𝑀, (∅ substr ⟨(𝑀𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
7978ad3antrrr 728 . . . . . . . . 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 7365 . . . . . . . . . . . . . 14 ((♯‘𝐵) = 0 → (𝐿 + (♯‘𝐵)) = (𝐿 + 0))
8180oveq2d 7373 . . . . . . . . . . . . 13 ((♯‘𝐵) = 0 → (0...(𝐿 + (♯‘𝐵))) = (0...(𝐿 + 0)))
8281eleq2d 2823 . . . . . . . . . . . 12 ((♯‘𝐵) = 0 → (𝑀 ∈ (0...(𝐿 + (♯‘𝐵))) ↔ 𝑀 ∈ (0...(𝐿 + 0))))
8382adantr 481 . . . . . . . . . . 11 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → (𝑀 ∈ (0...(𝐿 + (♯‘𝐵))) ↔ 𝑀 ∈ (0...(𝐿 + 0))))
84 simpr 485 . . . . . . . . . . . . . 14 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → 𝐵 = ∅)
85 opeq2 4831 . . . . . . . . . . . . . . 15 ((♯‘𝐵) = 0 → ⟨(𝑀𝐿), (♯‘𝐵)⟩ = ⟨(𝑀𝐿), 0⟩)
8685adantr 481 . . . . . . . . . . . . . 14 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → ⟨(𝑀𝐿), (♯‘𝐵)⟩ = ⟨(𝑀𝐿), 0⟩)
8784, 86oveq12d 7375 . . . . . . . . . . . . 13 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩) = (∅ substr ⟨(𝑀𝐿), 0⟩))
88 oveq2 7365 . . . . . . . . . . . . . 14 (𝐵 = ∅ → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅))
8988adantl 482 . . . . . . . . . . . . 13 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅))
9087, 89ifeq12d 4507 . . . . . . . . . . . 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 4837 . . . . . . . . . . . . . 14 ((♯‘𝐵) = 0 → ⟨𝑀, (𝐿 + (♯‘𝐵))⟩ = ⟨𝑀, (𝐿 + 0)⟩)
9291oveq2d 7373 . . . . . . . . . . . . 13 ((♯‘𝐵) = 0 → (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
9392adantr 481 . . . . . . . . . . . 12 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
9490, 93eqeq12d 2752 . . . . . . . . . . 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 344 . . . . . . . . . 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 712 . . . . . . . . 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 256 . . . . . . . 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 685 . . . . . . 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 413 . . . . . 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 407 . . 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 481 . 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 2828 . . . . . . . 8 (𝐿 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0)
105104, 14sylbir 234 . . . . . . 7 ((♯‘𝐴) ∈ ℕ0𝐿 ∈ ℝ)
10610, 105syl 17 . . . . . 6 (𝐴 ∈ Word 𝑉𝐿 ∈ ℝ)
1071nn0red 12474 . . . . . 6 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℝ)
108 leaddle0 11670 . . . . . 6 ((𝐿 ∈ ℝ ∧ (♯‘𝐵) ∈ ℝ) → ((𝐿 + (♯‘𝐵)) ≤ 𝐿 ↔ (♯‘𝐵) ≤ 0))
109106, 107, 108syl2an 596 . . . . 5 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝐿 + (♯‘𝐵)) ≤ 𝐿 ↔ (♯‘𝐵) ≤ 0))
110 pm2.24 124 . . . . 5 ((♯‘𝐵) ≤ 0 → (¬ (♯‘𝐵) ≤ 0 → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩)))
111109, 110syl6bi 252 . . . 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 481 . . 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 407 . 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 179 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  c0 4282  ifcif 4486  cop 4592   class class class wbr 5105  cfv 6496  (class class class)co 7357  cc 11049  cr 11050  0cc0 11051   + caddc 11054  cle 11190  cmin 11385  0cn0 12413  ...cfz 13424  chash 14230  Word cword 14402   ++ cconcat 14458   substr csubstr 14528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-concat 14459  df-substr 14529
This theorem is referenced by:  swrdccat3b  14628
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