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Definition df-sub 11514
Description: Define subtraction. Theorem subval 11519 shows its value (and describes how this definition works), Theorem subaddi 11616 relates it to addition, and Theorems subcli 11605 and resubcli 11591 prove its closure laws. (Contributed by NM, 26-Nov-1994.)
Assertion
Ref Expression
df-sub − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-sub
StepHypRef Expression
1 cmin 11512 . 2 class
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cc 11169 . . 3 class
53cv 1569 . . . . . 6 class 𝑦
6 vz . . . . . . 7 setvar 𝑧
76cv 1569 . . . . . 6 class 𝑧
8 caddc 11174 . . . . . 6 class +
95, 7, 8co 7408 . . . . 5 class (𝑦 + 𝑧)
102cv 1569 . . . . 5 class 𝑥
119, 10wceq 1570 . . . 4 wff (𝑦 + 𝑧) = 𝑥
1211, 6, 4crio 7364 . . 3 class (𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
132, 3, 4, 4, 12cmpo 7410 . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
141, 13wceq 1570 1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
Colors of variables:    wff setvar class
This definition is used by:  subval  11519  subf  11530  sn-subf  43408
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