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Theorem swrdccat3blem 14091
Description: Lemma for swrdccat3b 14092. (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 13873 . . . . . . . 8 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ0)
2 nn0le0eq0 11914 . . . . . . . . 9 ((♯‘𝐵) ∈ ℕ0 → ((♯‘𝐵) ≤ 0 ↔ (♯‘𝐵) = 0))
32biimpd 230 . . . . . . . 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 13714 . . . . . . . . . . 11 (𝐵 ∈ Word 𝑉 → ((♯‘𝐵) = 0 ↔ 𝐵 = ∅))
76biimpd 230 . . . . . . . . . 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 13873 . . . . . . . . . . . . . . . 16 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
11 swrdccatin2.l . . . . . . . . . . . . . . . . . . 19 𝐿 = (♯‘𝐴)
1211eqcomi 2835 . . . . . . . . . . . . . . . . . 18 (♯‘𝐴) = 𝐿
1312eleq1i 2908 . . . . . . . . . . . . . . . . 17 ((♯‘𝐴) ∈ ℕ0𝐿 ∈ ℕ0)
14 nn0re 11895 . . . . . . . . . . . . . . . . . 18 (𝐿 ∈ ℕ0𝐿 ∈ ℝ)
15 elfz2nn0 12988 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ (0...(𝐿 + 0)) ↔ (𝑀 ∈ ℕ0 ∧ (𝐿 + 0) ∈ ℕ0𝑀 ≤ (𝐿 + 0)))
16 recn 10616 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐿 ∈ ℝ → 𝐿 ∈ ℂ)
1716addid1d 10829 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ ℝ → (𝐿 + 0) = 𝐿)
1817breq2d 5075 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐿 ∈ ℝ → (𝑀 ≤ (𝐿 + 0) ↔ 𝑀𝐿))
19 nn0re 11895 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
2019anim1i 614 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑀 ∈ ℕ0𝐿 ∈ ℝ) → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
2120ancoms 459 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
22 letri3 10715 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 = 𝐿 ↔ (𝑀𝐿𝐿𝑀)))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝑀 = 𝐿 ↔ (𝑀𝐿𝐿𝑀)))
2423biimprd 249 . . . . . . . . . . . . . . . . . . . . . . . . . 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 241 . . . . . . . . . . . . . . . . . . . . . . 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 1125 . . . . . . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . . . . . 18 (𝐿 ∈ ℝ → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿𝑀𝑀 = 𝐿)))
3314, 32syl 17 . . . . . . . . . . . . . . . . 17 (𝐿 ∈ ℕ0 → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿𝑀𝑀 = 𝐿)))
3413, 33sylbi 218 . . . . . . . . . . . . . . . 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 12990 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (0...(𝐿 + 0)) → 𝑀 ∈ ℕ0)
38 swrd00 13996 . . . . . . . . . . . . . . . . . . . . . 22 (∅ substr ⟨0, 0⟩) = ∅
39 swrd00 13996 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 substr ⟨𝐿, 𝐿⟩) = ∅
4038, 39eqtr4i 2852 . . . . . . . . . . . . . . . . . . . . 21 (∅ substr ⟨0, 0⟩) = (𝐴 substr ⟨𝐿, 𝐿⟩)
41 nn0cn 11896 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐿 ∈ ℕ0𝐿 ∈ ℂ)
4241subidd 10974 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐿 ∈ ℕ0 → (𝐿𝐿) = 0)
4342opeq1d 4808 . . . . . . . . . . . . . . . . . . . . . 22 (𝐿 ∈ ℕ0 → ⟨(𝐿𝐿), 0⟩ = ⟨0, 0⟩)
4443oveq2d 7164 . . . . . . . . . . . . . . . . . . . . 21 (𝐿 ∈ ℕ0 → (∅ substr ⟨(𝐿𝐿), 0⟩) = (∅ substr ⟨0, 0⟩))
4541addid1d 10829 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐿 ∈ ℕ0 → (𝐿 + 0) = 𝐿)
4645opeq2d 4809 . . . . . . . . . . . . . . . . . . . . . 22 (𝐿 ∈ ℕ0 → ⟨𝐿, (𝐿 + 0)⟩ = ⟨𝐿, 𝐿⟩)
4746oveq2d 7164 . . . . . . . . . . . . . . . . . . . . 21 (𝐿 ∈ ℕ0 → (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩) = (𝐴 substr ⟨𝐿, 𝐿⟩))
4840, 44, 473eqtr4a 2887 . . . . . . . . . . . . . . . . . . . 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 2905 . . . . . . . . . . . . . . . . . . 19 (𝑀 = 𝐿 → (𝑀 ∈ ℕ0𝐿 ∈ ℕ0))
51 oveq1 7155 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 = 𝐿 → (𝑀𝐿) = (𝐿𝐿))
5251opeq1d 4808 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 = 𝐿 → ⟨(𝑀𝐿), 0⟩ = ⟨(𝐿𝐿), 0⟩)
5352oveq2d 7164 . . . . . . . . . . . . . . . . . . . 20 (𝑀 = 𝐿 → (∅ substr ⟨(𝑀𝐿), 0⟩) = (∅ substr ⟨(𝐿𝐿), 0⟩))
54 opeq1 4802 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 = 𝐿 → ⟨𝑀, (𝐿 + 0)⟩ = ⟨𝐿, (𝐿 + 0)⟩)
5554oveq2d 7164 . . . . . . . . . . . . . . . . . . . 20 (𝑀 = 𝐿 → (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩) = (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩))
5653, 55eqeq12d 2842 . . . . . . . . . . . . . . . . . . 19 (𝑀 = 𝐿 → ((∅ substr ⟨(𝑀𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩) ↔ (∅ substr ⟨(𝐿𝐿), 0⟩) = (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩)))
5749, 50, 563imtr4d 295 . . . . . . . . . . . . . . . . . 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 13997 . . . . . . . . . . . . . . . 16 (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉)
65 ccatrid 13931 . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝐴) ∈ ℕ0𝐿 ∈ ℂ)
6810, 67syl 17 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ Word 𝑉𝐿 ∈ ℂ)
69 addid1 10809 . . . . . . . . . . . . . . . . . . 19 (𝐿 ∈ ℂ → (𝐿 + 0) = 𝐿)
7069eqcomd 2832 . . . . . . . . . . . . . . . . . 18 (𝐿 ∈ ℂ → 𝐿 = (𝐿 + 0))
7168, 70syl 17 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ Word 𝑉𝐿 = (𝐿 + 0))
7271opeq2d 4809 . . . . . . . . . . . . . . . 16 (𝐴 ∈ Word 𝑉 → ⟨𝑀, 𝐿⟩ = ⟨𝑀, (𝐿 + 0)⟩)
7372oveq2d 7164 . . . . . . . . . . . . . . 15 (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
7466, 73eqtrd 2861 . . . . . . . . . . . . . 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 4505 . . . . . . . . . . 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 726 . . . . . . . . 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 7156 . . . . . . . . . . . . . 14 ((♯‘𝐵) = 0 → (𝐿 + (♯‘𝐵)) = (𝐿 + 0))
8180oveq2d 7164 . . . . . . . . . . . . 13 ((♯‘𝐵) = 0 → (0...(𝐿 + (♯‘𝐵))) = (0...(𝐿 + 0)))
8281eleq2d 2903 . . . . . . . . . . . 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 4803 . . . . . . . . . . . . . . 15 ((♯‘𝐵) = 0 → ⟨(𝑀𝐿), (♯‘𝐵)⟩ = ⟨(𝑀𝐿), 0⟩)
8685adantr 481 . . . . . . . . . . . . . 14 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → ⟨(𝑀𝐿), (♯‘𝐵)⟩ = ⟨(𝑀𝐿), 0⟩)
8784, 86oveq12d 7166 . . . . . . . . . . . . 13 (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩) = (∅ substr ⟨(𝑀𝐿), 0⟩))
88 oveq2 7156 . . . . . . . . . . . . . 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 4490 . . . . . . . . . . . 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 4809 . . . . . . . . . . . . . 14 ((♯‘𝐵) = 0 → ⟨𝑀, (𝐿 + (♯‘𝐵))⟩ = ⟨𝑀, (𝐿 + 0)⟩)
9291oveq2d 7164 . . . . . . . . . . . . 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 2842 . . . . . . . . . . 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 346 . . . . . . . . . 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 710 . . . . . . . . 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 258 . . . . . . . 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 683 . . . . . . 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 2908 . . . . . . . 8 (𝐿 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0)
105104, 14sylbir 236 . . . . . . 7 ((♯‘𝐴) ∈ ℕ0𝐿 ∈ ℝ)
10610, 105syl 17 . . . . . 6 (𝐴 ∈ Word 𝑉𝐿 ∈ ℝ)
1071nn0red 11945 . . . . . 6 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℝ)
108 leaddle0 11144 . . . . . 6 ((𝐿 ∈ ℝ ∧ (♯‘𝐵) ∈ ℝ) → ((𝐿 + (♯‘𝐵)) ≤ 𝐿 ↔ (♯‘𝐵) ≤ 0))
109106, 107, 108syl2an 595 . . . . 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 254 . . . 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 180 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  c0 4295  ifcif 4470  cop 4570   class class class wbr 5063  cfv 6352  (class class class)co 7148  cc 10524  cr 10525  0cc0 10526   + caddc 10529  cle 10665  cmin 10859  0cn0 11886  ...cfz 12882  chash 13680  Word cword 13851   ++ cconcat 13912   substr csubstr 13992
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-n0 11887  df-z 11971  df-uz 12233  df-fz 12883  df-fzo 13024  df-hash 13681  df-word 13852  df-concat 13913  df-substr 13993
This theorem is referenced by:  swrdccat3b  14092
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