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Theorem ccatsymb 13219
 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 id 22 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
213adant3 1074 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
32ad2antrl 760 . . . . . . 7 ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
4 simpr 476 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴)) → 𝐼 < (#‘𝐴))
54anim2i 591 . . . . . . . 8 ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → (0 ≤ 𝐼𝐼 < (#‘𝐴)))
6 simp3 1056 . . . . . . . . . . 11 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → 𝐼 ∈ ℤ)
7 0zd 11266 . . . . . . . . . . 11 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → 0 ∈ ℤ)
8 lencl 13179 . . . . . . . . . . . . 13 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
98nn0zd 11356 . . . . . . . . . . . 12 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℤ)
1093ad2ant1 1075 . . . . . . . . . . 11 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (#‘𝐴) ∈ ℤ)
116, 7, 103jca 1235 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (𝐼 ∈ ℤ ∧ 0 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
1211ad2antrl 760 . . . . . . . . 9 ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → (𝐼 ∈ ℤ ∧ 0 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
13 elfzo 12341 . . . . . . . . 9 ((𝐼 ∈ ℤ ∧ 0 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) → (𝐼 ∈ (0..^(#‘𝐴)) ↔ (0 ≤ 𝐼𝐼 < (#‘𝐴))))
1412, 13syl 17 . . . . . . . 8 ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → (𝐼 ∈ (0..^(#‘𝐴)) ↔ (0 ≤ 𝐼𝐼 < (#‘𝐴))))
155, 14mpbird 246 . . . . . . 7 ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → 𝐼 ∈ (0..^(#‘𝐴)))
16 df-3an 1033 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝐴))) ↔ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ (0..^(#‘𝐴))))
173, 15, 16sylanbrc 695 . . . . . 6 ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝐴))))
18 ccatval1 13214 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝐴))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐴𝐼))
1918eqcomd 2616 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝐴))) → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
2017, 19syl 17 . . . . 5 ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
2120ex 449 . . . 4 (0 ≤ 𝐼 → (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴)) → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
22 zre 11258 . . . . . . . . . 10 (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
23 0red 9920 . . . . . . . . . 10 (𝐼 ∈ ℤ → 0 ∈ ℝ)
2422, 23jca 553 . . . . . . . . 9 (𝐼 ∈ ℤ → (𝐼 ∈ ℝ ∧ 0 ∈ ℝ))
25243ad2ant3 1077 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (𝐼 ∈ ℝ ∧ 0 ∈ ℝ))
26 ltnle 9996 . . . . . . . 8 ((𝐼 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐼 < 0 ↔ ¬ 0 ≤ 𝐼))
2725, 26syl 17 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (𝐼 < 0 ↔ ¬ 0 ≤ 𝐼))
28 id 22 . . . . . . . . . . . 12 ((𝐴 ∈ Word 𝑉𝐼 ∈ ℤ) → (𝐴 ∈ Word 𝑉𝐼 ∈ ℤ))
29283adant2 1073 . . . . . . . . . . 11 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (𝐴 ∈ Word 𝑉𝐼 ∈ ℤ))
3029adantr 480 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴 ∈ Word 𝑉𝐼 ∈ ℤ))
31 simpr 476 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < 0) → 𝐼 < 0)
3231orcd 406 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐼 < 0 ∨ (#‘𝐴) ≤ 𝐼))
33 wrdsymb0 13194 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑉𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (#‘𝐴) ≤ 𝐼) → (𝐴𝐼) = ∅))
3430, 32, 33sylc 63 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴𝐼) = ∅)
35 ccatcl 13212 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
36353adant3 1074 . . . . . . . . . . . 12 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
3736, 6jca 553 . . . . . . . . . . 11 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵) ∈ Word 𝑉𝐼 ∈ ℤ))
3837adantr 480 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < 0) → ((𝐴 ++ 𝐵) ∈ Word 𝑉𝐼 ∈ ℤ))
3931orcd 406 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐼 < 0 ∨ (#‘(𝐴 ++ 𝐵)) ≤ 𝐼))
40 wrdsymb0 13194 . . . . . . . . . 10 (((𝐴 ++ 𝐵) ∈ Word 𝑉𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (#‘(𝐴 ++ 𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = ∅))
4138, 39, 40sylc 63 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < 0) → ((𝐴 ++ 𝐵)‘𝐼) = ∅)
4234, 41eqtr4d 2647 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
4342ex 449 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (𝐼 < 0 → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
4427, 43sylbird 249 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (¬ 0 ≤ 𝐼 → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
4544adantr 480 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴)) → (¬ 0 ≤ 𝐼 → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
4645com12 32 . . . 4 (¬ 0 ≤ 𝐼 → (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴)) → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
4721, 46pm2.61i 175 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴)) → (𝐴𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
482ad2antrl 760 . . . . . . . 8 ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
498nn0red 11229 . . . . . . . . . . . . . 14 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℝ)
50 lenlt 9995 . . . . . . . . . . . . . 14 (((#‘𝐴) ∈ ℝ ∧ 𝐼 ∈ ℝ) → ((#‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (#‘𝐴)))
5149, 22, 50syl2an 493 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝐼 ∈ ℤ) → ((#‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (#‘𝐴)))
52513adant2 1073 . . . . . . . . . . . 12 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → ((#‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (#‘𝐴)))
5352biimpar 501 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴)) → (#‘𝐴) ≤ 𝐼)
5453anim2i 591 . . . . . . . . . 10 ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → (𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ (#‘𝐴) ≤ 𝐼))
5554ancomd 466 . . . . . . . . 9 ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → ((#‘𝐴) ≤ 𝐼𝐼 < ((#‘𝐴) + (#‘𝐵))))
56 lencl 13179 . . . . . . . . . . . . . . 15 (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℕ0)
5756nn0zd 11356 . . . . . . . . . . . . . 14 (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℤ)
58 zaddcl 11294 . . . . . . . . . . . . . 14 (((#‘𝐴) ∈ ℤ ∧ (#‘𝐵) ∈ ℤ) → ((#‘𝐴) + (#‘𝐵)) ∈ ℤ)
599, 57, 58syl2an 493 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((#‘𝐴) + (#‘𝐵)) ∈ ℤ)
60593adant3 1074 . . . . . . . . . . . 12 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → ((#‘𝐴) + (#‘𝐵)) ∈ ℤ)
616, 10, 603jca 1235 . . . . . . . . . . 11 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (𝐼 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ ((#‘𝐴) + (#‘𝐵)) ∈ ℤ))
6261ad2antrl 760 . . . . . . . . . 10 ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → (𝐼 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ ((#‘𝐴) + (#‘𝐵)) ∈ ℤ))
63 elfzo 12341 . . . . . . . . . 10 ((𝐼 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ ((#‘𝐴) + (#‘𝐵)) ∈ ℤ) → (𝐼 ∈ ((#‘𝐴)..^((#‘𝐴) + (#‘𝐵))) ↔ ((#‘𝐴) ≤ 𝐼𝐼 < ((#‘𝐴) + (#‘𝐵)))))
6462, 63syl 17 . . . . . . . . 9 ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → (𝐼 ∈ ((#‘𝐴)..^((#‘𝐴) + (#‘𝐵))) ↔ ((#‘𝐴) ≤ 𝐼𝐼 < ((#‘𝐴) + (#‘𝐵)))))
6555, 64mpbird 246 . . . . . . . 8 ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → 𝐼 ∈ ((#‘𝐴)..^((#‘𝐴) + (#‘𝐵))))
66 df-3an 1033 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ((#‘𝐴)..^((#‘𝐴) + (#‘𝐵)))) ↔ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ((#‘𝐴)..^((#‘𝐴) + (#‘𝐵)))))
6748, 65, 66sylanbrc 695 . . . . . . 7 ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ((#‘𝐴)..^((#‘𝐴) + (#‘𝐵)))))
68 ccatval2 13215 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ((#‘𝐴)..^((#‘𝐴) + (#‘𝐵)))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴))))
6967, 68syl 17 . . . . . 6 ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴))))
7069ex 449 . . . . 5 (𝐼 < ((#‘𝐴) + (#‘𝐵)) → (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴)) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴)))))
7156nn0red 11229 . . . . . . . . . . 11 (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℝ)
72 readdcl 9898 . . . . . . . . . . 11 (((#‘𝐴) ∈ ℝ ∧ (#‘𝐵) ∈ ℝ) → ((#‘𝐴) + (#‘𝐵)) ∈ ℝ)
7349, 71, 72syl2an 493 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((#‘𝐴) + (#‘𝐵)) ∈ ℝ)
74733adant3 1074 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → ((#‘𝐴) + (#‘𝐵)) ∈ ℝ)
75223ad2ant3 1077 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → 𝐼 ∈ ℝ)
7674, 75lenltd 10062 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (((#‘𝐴) + (#‘𝐵)) ≤ 𝐼 ↔ ¬ 𝐼 < ((#‘𝐴) + (#‘𝐵))))
7737adantr 480 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵) ∈ Word 𝑉𝐼 ∈ ℤ))
78 ccatlen 13213 . . . . . . . . . . . . . . 15 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
79783adant3 1074 . . . . . . . . . . . . . 14 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
8079adantr 480 . . . . . . . . . . . . 13 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
81 simpr 476 . . . . . . . . . . . . 13 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼)
8280, 81eqbrtrd 4605 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (#‘(𝐴 ++ 𝐵)) ≤ 𝐼)
8382olcd 407 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (𝐼 < 0 ∨ (#‘(𝐴 ++ 𝐵)) ≤ 𝐼))
8477, 83, 40sylc 63 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = ∅)
85 simp2 1055 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → 𝐵 ∈ Word 𝑉)
86 zsubcl 11296 . . . . . . . . . . . . . . . 16 ((𝐼 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) → (𝐼 − (#‘𝐴)) ∈ ℤ)
879, 86sylan2 490 . . . . . . . . . . . . . . 15 ((𝐼 ∈ ℤ ∧ 𝐴 ∈ Word 𝑉) → (𝐼 − (#‘𝐴)) ∈ ℤ)
8887ancoms 468 . . . . . . . . . . . . . 14 ((𝐴 ∈ Word 𝑉𝐼 ∈ ℤ) → (𝐼 − (#‘𝐴)) ∈ ℤ)
89883adant2 1073 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (𝐼 − (#‘𝐴)) ∈ ℤ)
9085, 89jca 553 . . . . . . . . . . . 12 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (𝐵 ∈ Word 𝑉 ∧ (𝐼 − (#‘𝐴)) ∈ ℤ))
9190adantr 480 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (𝐵 ∈ Word 𝑉 ∧ (𝐼 − (#‘𝐴)) ∈ ℤ))
92 leaddsub2 10384 . . . . . . . . . . . . . 14 (((#‘𝐴) ∈ ℝ ∧ (#‘𝐵) ∈ ℝ ∧ 𝐼 ∈ ℝ) → (((#‘𝐴) + (#‘𝐵)) ≤ 𝐼 ↔ (#‘𝐵) ≤ (𝐼 − (#‘𝐴))))
9349, 71, 22, 92syl3an 1360 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (((#‘𝐴) + (#‘𝐵)) ≤ 𝐼 ↔ (#‘𝐵) ≤ (𝐼 − (#‘𝐴))))
9493biimpa 500 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (#‘𝐵) ≤ (𝐼 − (#‘𝐴)))
9594olcd 407 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → ((𝐼 − (#‘𝐴)) < 0 ∨ (#‘𝐵) ≤ (𝐼 − (#‘𝐴))))
96 wrdsymb0 13194 . . . . . . . . . . 11 ((𝐵 ∈ Word 𝑉 ∧ (𝐼 − (#‘𝐴)) ∈ ℤ) → (((𝐼 − (#‘𝐴)) < 0 ∨ (#‘𝐵) ≤ (𝐼 − (#‘𝐴))) → (𝐵‘(𝐼 − (#‘𝐴))) = ∅))
9791, 95, 96sylc 63 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (𝐵‘(𝐼 − (#‘𝐴))) = ∅)
9884, 97eqtr4d 2647 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴))))
9998ex 449 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (((#‘𝐴) + (#‘𝐵)) ≤ 𝐼 → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴)))))
10076, 99sylbird 249 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → (¬ 𝐼 < ((#‘𝐴) + (#‘𝐵)) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴)))))
101100adantr 480 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴)) → (¬ 𝐼 < ((#‘𝐴) + (#‘𝐵)) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴)))))
102101com12 32 . . . . 5 𝐼 < ((#‘𝐴) + (#‘𝐵)) → (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴)) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴)))))
10370, 102pm2.61i 175 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴)) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴))))
104103eqcomd 2616 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴)) → (𝐵‘(𝐼 − (#‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
10547, 104ifeqda 4071 . 2 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → if(𝐼 < (#‘𝐴), (𝐴𝐼), (𝐵‘(𝐼 − (#‘𝐴)))) = ((𝐴 ++ 𝐵)‘𝐼))
106105eqcomd 2616 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 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∅c0 3874  ifcif 4036   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℤcz 11254  ..^cfzo 12334  #chash 12979  Word cword 13146   ++ cconcat 13148 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156 This theorem is referenced by:  swrdccatin2  13338
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