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Definition df-concat 13333
Description: Define the concatenation operator which combines two words. Definition in section 9.1 of [AhoHopUll] p. 318. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 15-Aug-2015.)
Assertion
Ref Expression
df-concat ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((#‘𝑠) + (#‘𝑡))) ↦ if(𝑥 ∈ (0..^(#‘𝑠)), (𝑠𝑥), (𝑡‘(𝑥 − (#‘𝑠))))))
Distinct variable group:   𝑡,𝑠,𝑥

Detailed syntax breakdown of Definition df-concat
StepHypRef Expression
1 cconcat 13325 . 2 class ++
2 vs . . 3 setvar 𝑠
3 vt . . 3 setvar 𝑡
4 cvv 3231 . . 3 class V
5 vx . . . 4 setvar 𝑥
6 cc0 9974 . . . . 5 class 0
72cv 1522 . . . . . . 7 class 𝑠
8 chash 13157 . . . . . . 7 class #
97, 8cfv 5926 . . . . . 6 class (#‘𝑠)
103cv 1522 . . . . . . 7 class 𝑡
1110, 8cfv 5926 . . . . . 6 class (#‘𝑡)
12 caddc 9977 . . . . . 6 class +
139, 11, 12co 6690 . . . . 5 class ((#‘𝑠) + (#‘𝑡))
14 cfzo 12504 . . . . 5 class ..^
156, 13, 14co 6690 . . . 4 class (0..^((#‘𝑠) + (#‘𝑡)))
165cv 1522 . . . . . 6 class 𝑥
176, 9, 14co 6690 . . . . . 6 class (0..^(#‘𝑠))
1816, 17wcel 2030 . . . . 5 wff 𝑥 ∈ (0..^(#‘𝑠))
1916, 7cfv 5926 . . . . 5 class (𝑠𝑥)
20 cmin 10304 . . . . . . 7 class
2116, 9, 20co 6690 . . . . . 6 class (𝑥 − (#‘𝑠))
2221, 10cfv 5926 . . . . 5 class (𝑡‘(𝑥 − (#‘𝑠)))
2318, 19, 22cif 4119 . . . 4 class if(𝑥 ∈ (0..^(#‘𝑠)), (𝑠𝑥), (𝑡‘(𝑥 − (#‘𝑠))))
245, 15, 23cmpt 4762 . . 3 class (𝑥 ∈ (0..^((#‘𝑠) + (#‘𝑡))) ↦ if(𝑥 ∈ (0..^(#‘𝑠)), (𝑠𝑥), (𝑡‘(𝑥 − (#‘𝑠)))))
252, 3, 4, 4, 24cmpt2 6692 . 2 class (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((#‘𝑠) + (#‘𝑡))) ↦ if(𝑥 ∈ (0..^(#‘𝑠)), (𝑠𝑥), (𝑡‘(𝑥 − (#‘𝑠))))))
261, 25wceq 1523 1 wff ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((#‘𝑠) + (#‘𝑡))) ↦ if(𝑥 ∈ (0..^(#‘𝑠)), (𝑠𝑥), (𝑡‘(𝑥 − (#‘𝑠))))))
Colors of variables: wff setvar class
This definition is referenced by:  ccatfn  13390  ccatfval  13391
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