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Theorem ccatsymb 14617
Description: The symbol at a given position in a concatenated word. (Contributed by AV, 26-May-2018.) (Proof shortened by AV, 24-Nov-2018.)
Assertion
Ref Expression
ccatsymb ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵)‘𝐼) = if(𝐼 < (♯‘𝐴), (𝐴𝐼), (𝐵‘(𝐼 − (♯‘𝐴)))))

Proof of Theorem ccatsymb
StepHypRef Expression
1 simprll 779 . . . . . . . 8 ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
2 simpr 484 . . . . . . . . . 10 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → 𝐼 < (♯‘𝐴))
32anim2i 617 . . . . . . . . 9 ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (0 ≤ 𝐼𝐼 < (♯‘𝐴)))
4 simpr 484 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℤ)
5 0zd 12623 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 0 ∈ ℤ)
6 lencl 14568 . . . . . . . . . . . . 13 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
76nn0zd 12637 . . . . . . . . . . . 12 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℤ)
87ad2antrr 726 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (♯‘𝐴) ∈ ℤ)
9 elfzo 13698 . . . . . . . . . . 11 ((𝐼 ∈ ℤ ∧ 0 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → (𝐼 ∈ (0..^(♯‘𝐴)) ↔ (0 ≤ 𝐼𝐼 < (♯‘𝐴))))
104, 5, 8, 9syl3anc 1370 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ (0..^(♯‘𝐴)) ↔ (0 ≤ 𝐼𝐼 < (♯‘𝐴))))
1110ad2antrl 728 . . . . . . . . 9 ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐼 ∈ (0..^(♯‘𝐴)) ↔ (0 ≤ 𝐼𝐼 < (♯‘𝐴))))
123, 11mpbird 257 . . . . . . . 8 ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → 𝐼 ∈ (0..^(♯‘𝐴)))
13 df-3an 1088 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ (0..^(♯‘𝐴))) ↔ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ (0..^(♯‘𝐴))))
141, 12, 13sylanbrc 583 . . . . . . 7 ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ (0..^(♯‘𝐴))))
15 ccatval1 14612 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐴𝐼))
1615eqcomd 2741 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ (0..^(♯‘𝐴))) → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
1714, 16syl 17 . . . . . 6 ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
1817ex 412 . . . . 5 (0 ≤ 𝐼 → ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
19 zre 12615 . . . . . . . . . 10 (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
20 0red 11262 . . . . . . . . . 10 (𝐼 ∈ ℤ → 0 ∈ ℝ)
2119, 20ltnled 11406 . . . . . . . . 9 (𝐼 ∈ ℤ → (𝐼 < 0 ↔ ¬ 0 ≤ 𝐼))
2221adantl 481 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 < 0 ↔ ¬ 0 ≤ 𝐼))
23 simpl 482 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝐴 ∈ Word 𝑉)
2423anim1i 615 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐴 ∈ Word 𝑉𝐼 ∈ ℤ))
2524adantr 480 . . . . . . . . . . 11 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴 ∈ Word 𝑉𝐼 ∈ ℤ))
26 animorrl 982 . . . . . . . . . . 11 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐼 < 0 ∨ (♯‘𝐴) ≤ 𝐼))
27 wrdsymb0 14584 . . . . . . . . . . 11 ((𝐴 ∈ Word 𝑉𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (♯‘𝐴) ≤ 𝐼) → (𝐴𝐼) = ∅))
2825, 26, 27sylc 65 . . . . . . . . . 10 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴𝐼) = ∅)
29 ccatcl 14609 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
3029anim1i 615 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵) ∈ Word 𝑉𝐼 ∈ ℤ))
3130adantr 480 . . . . . . . . . . 11 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → ((𝐴 ++ 𝐵) ∈ Word 𝑉𝐼 ∈ ℤ))
32 animorrl 982 . . . . . . . . . . 11 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐼 < 0 ∨ (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼))
33 wrdsymb0 14584 . . . . . . . . . . 11 (((𝐴 ++ 𝐵) ∈ Word 𝑉𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = ∅))
3431, 32, 33sylc 65 . . . . . . . . . 10 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → ((𝐴 ++ 𝐵)‘𝐼) = ∅)
3528, 34eqtr4d 2778 . . . . . . . . 9 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
3635ex 412 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 < 0 → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
3722, 36sylbird 260 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (¬ 0 ≤ 𝐼 → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
3837com12 32 . . . . . 6 (¬ 0 ≤ 𝐼 → (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
3938adantrd 491 . . . . 5 (¬ 0 ≤ 𝐼 → ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
4018, 39pm2.61i 182 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
41 simprll 779 . . . . . . . 8 ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
42 id 22 . . . . . . . . . 10 (𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → 𝐼 < ((♯‘𝐴) + (♯‘𝐵)))
436nn0red 12586 . . . . . . . . . . . . 13 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℝ)
44 lenlt 11337 . . . . . . . . . . . . 13 (((♯‘𝐴) ∈ ℝ ∧ 𝐼 ∈ ℝ) → ((♯‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (♯‘𝐴)))
4543, 19, 44syl2an 596 . . . . . . . . . . . 12 ((𝐴 ∈ Word 𝑉𝐼 ∈ ℤ) → ((♯‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (♯‘𝐴)))
4645adantlr 715 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((♯‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (♯‘𝐴)))
4746biimpar 477 . . . . . . . . . 10 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (♯‘𝐴) ≤ 𝐼)
4842, 47anim12ci 614 . . . . . . . . 9 ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → ((♯‘𝐴) ≤ 𝐼𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
49 lencl 14568 . . . . . . . . . . . . . 14 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ0)
5049nn0zd 12637 . . . . . . . . . . . . 13 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℤ)
51 zaddcl 12655 . . . . . . . . . . . . 13 (((♯‘𝐴) ∈ ℤ ∧ (♯‘𝐵) ∈ ℤ) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ)
527, 50, 51syl2an 596 . . . . . . . . . . . 12 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ)
5352adantr 480 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ)
54 elfzo 13698 . . . . . . . . . . 11 ((𝐼 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ) → (𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ↔ ((♯‘𝐴) ≤ 𝐼𝐼 < ((♯‘𝐴) + (♯‘𝐵)))))
554, 8, 53, 54syl3anc 1370 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ↔ ((♯‘𝐴) ≤ 𝐼𝐼 < ((♯‘𝐴) + (♯‘𝐵)))))
5655ad2antrl 728 . . . . . . . . 9 ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → (𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ↔ ((♯‘𝐴) ≤ 𝐼𝐼 < ((♯‘𝐴) + (♯‘𝐵)))))
5748, 56mpbird 257 . . . . . . . 8 ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))))
58 df-3an 1088 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))) ↔ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))))
5941, 57, 58sylanbrc 583 . . . . . . 7 ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))))
60 ccatval2 14613 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (♯‘𝐴))))
6160eqcomd 2741 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
6259, 61syl 17 . . . . . 6 ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
6362ex 412 . . . . 5 (𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
6449nn0red 12586 . . . . . . . . . 10 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℝ)
65 readdcl 11236 . . . . . . . . . 10 (((♯‘𝐴) ∈ ℝ ∧ (♯‘𝐵) ∈ ℝ) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ)
6643, 64, 65syl2an 596 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ)
67 lenlt 11337 . . . . . . . . 9 ((((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ ∧ 𝐼 ∈ ℝ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 ↔ ¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
6866, 19, 67syl2an 596 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 ↔ ¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
69 simplr 769 . . . . . . . . . . . . 13 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 𝐵 ∈ Word 𝑉)
70 simpr 484 . . . . . . . . . . . . . . 15 ((𝐴 ∈ Word 𝑉𝐼 ∈ ℤ) → 𝐼 ∈ ℤ)
717adantr 480 . . . . . . . . . . . . . . 15 ((𝐴 ∈ Word 𝑉𝐼 ∈ ℤ) → (♯‘𝐴) ∈ ℤ)
7270, 71zsubcld 12725 . . . . . . . . . . . . . 14 ((𝐴 ∈ Word 𝑉𝐼 ∈ ℤ) → (𝐼 − (♯‘𝐴)) ∈ ℤ)
7372adantlr 715 . . . . . . . . . . . . 13 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 − (♯‘𝐴)) ∈ ℤ)
7469, 73jca 511 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐵 ∈ Word 𝑉 ∧ (𝐼 − (♯‘𝐴)) ∈ ℤ))
7574adantr 480 . . . . . . . . . . 11 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐵 ∈ Word 𝑉 ∧ (𝐼 − (♯‘𝐴)) ∈ ℤ))
7643ad2antrr 726 . . . . . . . . . . . . . 14 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (♯‘𝐴) ∈ ℝ)
7764ad2antlr 727 . . . . . . . . . . . . . 14 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (♯‘𝐵) ∈ ℝ)
7819adantl 481 . . . . . . . . . . . . . 14 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℝ)
7976, 77, 78leaddsub2d 11863 . . . . . . . . . . . . 13 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 ↔ (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴))))
8079biimpa 476 . . . . . . . . . . . 12 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴)))
8180olcd 874 . . . . . . . . . . 11 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((𝐼 − (♯‘𝐴)) < 0 ∨ (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴))))
82 wrdsymb0 14584 . . . . . . . . . . 11 ((𝐵 ∈ Word 𝑉 ∧ (𝐼 − (♯‘𝐴)) ∈ ℤ) → (((𝐼 − (♯‘𝐴)) < 0 ∨ (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴))) → (𝐵‘(𝐼 − (♯‘𝐴))) = ∅))
8375, 81, 82sylc 65 . . . . . . . . . 10 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐵‘(𝐼 − (♯‘𝐴))) = ∅)
8430adantr 480 . . . . . . . . . . 11 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵) ∈ Word 𝑉𝐼 ∈ ℤ))
85 ccatlen 14610 . . . . . . . . . . . . . 14 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
8685ad2antrr 726 . . . . . . . . . . . . 13 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
87 simpr 484 . . . . . . . . . . . . 13 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼)
8886, 87eqbrtrd 5170 . . . . . . . . . . . 12 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼)
8988olcd 874 . . . . . . . . . . 11 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐼 < 0 ∨ (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼))
9084, 89, 33sylc 65 . . . . . . . . . 10 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = ∅)
9183, 90eqtr4d 2778 . . . . . . . . 9 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
9291ex 412 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
9368, 92sylbird 260 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
9493com12 32 . . . . . 6 𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
9594adantrd 491 . . . . 5 𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
9663, 95pm2.61i 182 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
9740, 96ifeqda 4567 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → if(𝐼 < (♯‘𝐴), (𝐴𝐼), (𝐵‘(𝐼 − (♯‘𝐴)))) = ((𝐴 ++ 𝐵)‘𝐼))
9897eqcomd 2741 . 2 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵)‘𝐼) = if(𝐼 < (♯‘𝐴), (𝐴𝐼), (𝐵‘(𝐼 − (♯‘𝐴)))))
99983impa 1109 1 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵)‘𝐼) = if(𝐼 < (♯‘𝐴), (𝐴𝐼), (𝐵‘(𝐼 − (♯‘𝐴)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  c0 4339  ifcif 4531   class class class wbr 5148  cfv 6563  (class class class)co 7431  cr 11152  0cc0 11153   + caddc 11156   < clt 11293  cle 11294  cmin 11490  cz 12611  ..^cfzo 13691  chash 14366  Word cword 14549   ++ cconcat 14605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-concat 14606
This theorem is referenced by:  swrdccatin2  14764  frlmvscadiccat  42493
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