MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  swrdccat3blem Structured version   Visualization version   GIF version

Theorem swrdccat3blem 13541
Description: Lemma for swrdccat3b 13542. (Contributed by AV, 30-May-2018.)
Hypothesis
Ref Expression
swrdccatin12.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 13356 . . . . . . . 8 (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℕ0)
2 nn0le0eq0 11359 . . . . . . . . 9 ((#‘𝐵) ∈ ℕ0 → ((#‘𝐵) ≤ 0 ↔ (#‘𝐵) = 0))
32biimpd 219 . . . . . . . 8 ((#‘𝐵) ∈ ℕ0 → ((#‘𝐵) ≤ 0 → (#‘𝐵) = 0))
41, 3syl 17 . . . . . . 7 (𝐵 ∈ Word 𝑉 → ((#‘𝐵) ≤ 0 → (#‘𝐵) = 0))
54adantl 481 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((#‘𝐵) ≤ 0 → (#‘𝐵) = 0))
6 hasheq0 13192 . . . . . . . . . . 11 (𝐵 ∈ Word 𝑉 → ((#‘𝐵) = 0 ↔ 𝐵 = ∅))
76biimpd 219 . . . . . . . . . 10 (𝐵 ∈ Word 𝑉 → ((#‘𝐵) = 0 → 𝐵 = ∅))
87adantl 481 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((#‘𝐵) = 0 → 𝐵 = ∅))
98imp 444 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (#‘𝐵) = 0) → 𝐵 = ∅)
10 lencl 13356 . . . . . . . . . . . . . . . 16 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
11 swrdccatin12.l . . . . . . . . . . . . . . . . . . 19 𝐿 = (#‘𝐴)
1211eqcomi 2660 . . . . . . . . . . . . . . . . . 18 (#‘𝐴) = 𝐿
1312eleq1i 2721 . . . . . . . . . . . . . . . . 17 ((#‘𝐴) ∈ ℕ0𝐿 ∈ ℕ0)
14 nn0re 11339 . . . . . . . . . . . . . . . . . 18 (𝐿 ∈ ℕ0𝐿 ∈ ℝ)
15 elfz2nn0 12469 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ (0...(𝐿 + 0)) ↔ (𝑀 ∈ ℕ0 ∧ (𝐿 + 0) ∈ ℕ0𝑀 ≤ (𝐿 + 0)))
16 recn 10064 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐿 ∈ ℝ → 𝐿 ∈ ℂ)
1716addid1d 10274 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ ℝ → (𝐿 + 0) = 𝐿)
1817breq2d 4697 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐿 ∈ ℝ → (𝑀 ≤ (𝐿 + 0) ↔ 𝑀𝐿))
19 nn0re 11339 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
2019anim1i 591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑀 ∈ ℕ0𝐿 ∈ ℝ) → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
2120ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
22 letri3 10161 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 = 𝐿 ↔ (𝑀𝐿𝐿𝑀)))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝑀 = 𝐿 ↔ (𝑀𝐿𝐿𝑀)))
2423biimprd 238 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → ((𝑀𝐿𝐿𝑀) → 𝑀 = 𝐿))
2524exp4b 631 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ ℝ → (𝑀 ∈ ℕ0 → (𝑀𝐿 → (𝐿𝑀𝑀 = 𝐿))))
2625com23 86 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐿 ∈ ℝ → (𝑀𝐿 → (𝑀 ∈ ℕ0 → (𝐿𝑀𝑀 = 𝐿))))
2718, 26sylbid 230 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐿 ∈ ℝ → (𝑀 ≤ (𝐿 + 0) → (𝑀 ∈ ℕ0 → (𝐿𝑀𝑀 = 𝐿))))
2827com3l 89 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ≤ (𝐿 + 0) → (𝑀 ∈ ℕ0 → (𝐿 ∈ ℝ → (𝐿𝑀𝑀 = 𝐿))))
2928impcom 445 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 ∈ ℕ0𝑀 ≤ (𝐿 + 0)) → (𝐿 ∈ ℝ → (𝐿𝑀𝑀 = 𝐿)))
30293adant2 1100 . . . . . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . . . . . 18 (𝐿 ∈ ℝ → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿𝑀𝑀 = 𝐿)))
3314, 32syl 17 . . . . . . . . . . . . . . . . 17 (𝐿 ∈ ℕ0 → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿𝑀𝑀 = 𝐿)))
3413, 33sylbi 207 . . . . . . . . . . . . . . . 16 ((#‘𝐴) ∈ ℕ0 → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿𝑀𝑀 = 𝐿)))
3510, 34syl 17 . . . . . . . . . . . . . . 15 (𝐴 ∈ Word 𝑉 → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿𝑀𝑀 = 𝐿)))
3635imp 444 . . . . . . . . . . . . . 14 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) → (𝐿𝑀𝑀 = 𝐿))
37 elfznn0 12471 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (0...(𝐿 + 0)) → 𝑀 ∈ ℕ0)
38 swrd00 13463 . . . . . . . . . . . . . . . . . . . . . 22 (∅ substr ⟨0, 0⟩) = ∅
39 swrd00 13463 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 substr ⟨𝐿, 𝐿⟩) = ∅
4038, 39eqtr4i 2676 . . . . . . . . . . . . . . . . . . . . 21 (∅ substr ⟨0, 0⟩) = (𝐴 substr ⟨𝐿, 𝐿⟩)
41 nn0cn 11340 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐿 ∈ ℕ0𝐿 ∈ ℂ)
4241subidd 10418 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐿 ∈ ℕ0 → (𝐿𝐿) = 0)
4342opeq1d 4439 . . . . . . . . . . . . . . . . . . . . . 22 (𝐿 ∈ ℕ0 → ⟨(𝐿𝐿), 0⟩ = ⟨0, 0⟩)
4443oveq2d 6706 . . . . . . . . . . . . . . . . . . . . 21 (𝐿 ∈ ℕ0 → (∅ substr ⟨(𝐿𝐿), 0⟩) = (∅ substr ⟨0, 0⟩))
4541addid1d 10274 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐿 ∈ ℕ0 → (𝐿 + 0) = 𝐿)
4645opeq2d 4440 . . . . . . . . . . . . . . . . . . . . . 22 (𝐿 ∈ ℕ0 → ⟨𝐿, (𝐿 + 0)⟩ = ⟨𝐿, 𝐿⟩)
4746oveq2d 6706 . . . . . . . . . . . . . . . . . . . . 21 (𝐿 ∈ ℕ0 → (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩) = (𝐴 substr ⟨𝐿, 𝐿⟩))
4840, 44, 473eqtr4a 2711 . . . . . . . . . . . . . . . . . . . 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 2718 . . . . . . . . . . . . . . . . . . 19 (𝑀 = 𝐿 → (𝑀 ∈ ℕ0𝐿 ∈ ℕ0))
51 oveq1 6697 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 = 𝐿 → (𝑀𝐿) = (𝐿𝐿))
5251opeq1d 4439 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 = 𝐿 → ⟨(𝑀𝐿), 0⟩ = ⟨(𝐿𝐿), 0⟩)
5352oveq2d 6706 . . . . . . . . . . . . . . . . . . . 20 (𝑀 = 𝐿 → (∅ substr ⟨(𝑀𝐿), 0⟩) = (∅ substr ⟨(𝐿𝐿), 0⟩))
54 opeq1 4433 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 = 𝐿 → ⟨𝑀, (𝐿 + 0)⟩ = ⟨𝐿, (𝐿 + 0)⟩)
5554oveq2d 6706 . . . . . . . . . . . . . . . . . . . 20 (𝑀 = 𝐿 → (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩) = (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩))
5653, 55eqeq12d 2666 . . . . . . . . . . . . . . . . . . 19 (𝑀 = 𝐿 → ((∅ substr ⟨(𝑀𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩) ↔ (∅ substr ⟨(𝐿𝐿), 0⟩) = (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩)))
5749, 50, 563imtr4d 283 . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . 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 444 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) ∧ 𝐿𝑀) → (∅ substr ⟨(𝑀𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
64 swrdcl 13464 . . . . . . . . . . . . . . . 16 (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉)
65 ccatrid 13405 . . . . . . . . . . . . . . . 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 207 . . . . . . . . . . . . . . . . . . 19 ((#‘𝐴) ∈ ℕ0𝐿 ∈ ℂ)
6810, 67syl 17 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ Word 𝑉𝐿 ∈ ℂ)
69 addid1 10254 . . . . . . . . . . . . . . . . . . 19 (𝐿 ∈ ℂ → (𝐿 + 0) = 𝐿)
7069eqcomd 2657 . . . . . . . . . . . . . . . . . 18 (𝐿 ∈ ℂ → 𝐿 = (𝐿 + 0))
7168, 70syl 17 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ Word 𝑉𝐿 = (𝐿 + 0))
7271opeq2d 4440 . . . . . . . . . . . . . . . 16 (𝐴 ∈ Word 𝑉 → ⟨𝑀, 𝐿⟩ = ⟨𝑀, (𝐿 + 0)⟩)
7372oveq2d 6706 . . . . . . . . . . . . . . 15 (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
7466, 73eqtrd 2685 . . . . . . . . . . . . . 14 (𝐴 ∈ Word 𝑉 → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
7574adantr 480 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
7675adantr 480 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) ∧ ¬ 𝐿𝑀) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
7763, 76ifeqda 4154 . . . . . . . . . . 11 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...(𝐿 + 0))) → if(𝐿𝑀, (∅ substr ⟨(𝑀𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
7877ex 449 . . . . . . . . . 10 (𝐴 ∈ Word 𝑉 → (𝑀 ∈ (0...(𝐿 + 0)) → if(𝐿𝑀, (∅ substr ⟨(𝑀𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
7978ad3antrrr 766 . . . . . . . . 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 6698 . . . . . . . . . . . . . 14 ((#‘𝐵) = 0 → (𝐿 + (#‘𝐵)) = (𝐿 + 0))
8180oveq2d 6706 . . . . . . . . . . . . 13 ((#‘𝐵) = 0 → (0...(𝐿 + (#‘𝐵))) = (0...(𝐿 + 0)))
8281eleq2d 2716 . . . . . . . . . . . 12 ((#‘𝐵) = 0 → (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) ↔ 𝑀 ∈ (0...(𝐿 + 0))))
8382adantr 480 . . . . . . . . . . 11 (((#‘𝐵) = 0 ∧ 𝐵 = ∅) → (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) ↔ 𝑀 ∈ (0...(𝐿 + 0))))
84 simpr 476 . . . . . . . . . . . . . 14 (((#‘𝐵) = 0 ∧ 𝐵 = ∅) → 𝐵 = ∅)
85 opeq2 4434 . . . . . . . . . . . . . . 15 ((#‘𝐵) = 0 → ⟨(𝑀𝐿), (#‘𝐵)⟩ = ⟨(𝑀𝐿), 0⟩)
8685adantr 480 . . . . . . . . . . . . . 14 (((#‘𝐵) = 0 ∧ 𝐵 = ∅) → ⟨(𝑀𝐿), (#‘𝐵)⟩ = ⟨(𝑀𝐿), 0⟩)
8784, 86oveq12d 6708 . . . . . . . . . . . . 13 (((#‘𝐵) = 0 ∧ 𝐵 = ∅) → (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩) = (∅ substr ⟨(𝑀𝐿), 0⟩))
88 oveq2 6698 . . . . . . . . . . . . . 14 (𝐵 = ∅ → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅))
8988adantl 481 . . . . . . . . . . . . 13 (((#‘𝐵) = 0 ∧ 𝐵 = ∅) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅))
9087, 89ifeq12d 4139 . . . . . . . . . . . 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 4440 . . . . . . . . . . . . . 14 ((#‘𝐵) = 0 → ⟨𝑀, (𝐿 + (#‘𝐵))⟩ = ⟨𝑀, (𝐿 + 0)⟩)
9291oveq2d 6706 . . . . . . . . . . . . 13 ((#‘𝐵) = 0 → (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
9392adantr 480 . . . . . . . . . . . 12 (((#‘𝐵) = 0 ∧ 𝐵 = ∅) → (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
9490, 93eqeq12d 2666 . . . . . . . . . . 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 333 . . . . . . . . . 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 750 . . . . . . . . 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 247 . . . . . . . 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 703 . . . . . . 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 449 . . . . . 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 444 . . 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 480 . 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 2721 . . . . . . . 8 (𝐿 ∈ ℕ0 ↔ (#‘𝐴) ∈ ℕ0)
105104, 14sylbir 225 . . . . . . 7 ((#‘𝐴) ∈ ℕ0𝐿 ∈ ℝ)
10610, 105syl 17 . . . . . 6 (𝐴 ∈ Word 𝑉𝐿 ∈ ℝ)
1071nn0red 11390 . . . . . 6 (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℝ)
108 leaddle0 10581 . . . . . 6 ((𝐿 ∈ ℝ ∧ (#‘𝐵) ∈ ℝ) → ((𝐿 + (#‘𝐵)) ≤ 𝐿 ↔ (#‘𝐵) ≤ 0))
109106, 107, 108syl2an 493 . . . . 5 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝐿 + (#‘𝐵)) ≤ 𝐿 ↔ (#‘𝐵) ≤ 0))
110 pm2.24 121 . . . . 5 ((#‘𝐵) ≤ 0 → (¬ (#‘𝐵) ≤ 0 → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩)))
111109, 110syl6bi 243 . . . 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 480 . . 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 444 . 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 170 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  c0 3948  ifcif 4119  cop 4216   class class class wbr 4685  cfv 5926  (class class class)co 6690  cc 9972  cr 9973  0cc0 9974   + caddc 9977  cle 10113  cmin 10304  0cn0 11330  ...cfz 12364  #chash 13157  Word cword 13323   ++ cconcat 13325   substr csubstr 13327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-substr 13335
This theorem is referenced by:  swrdccat3b  13542
  Copyright terms: Public domain W3C validator