Database
SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
Mathbox for Steven Nguyen
Real subtraction
sn-00idlem1
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sn-00idlem2
Metamath Proof Explorer
Ascii
Unicode
Theorem
sn-00idlem1
Description:
Lemma for
sn-00id
.
(Contributed by
SN
, 25-Dec-2023)
Ref
Expression
Assertion
sn-00idlem1
⊢
A
∈
ℝ
→
A
⁢
0
-
ℝ
0
=
A
-
ℝ
A
Proof
Step
Hyp
Ref
Expression
1
1re
⊢
1
∈
ℝ
2
resubdi
⊢
A
∈
ℝ
∧
1
∈
ℝ
∧
1
∈
ℝ
→
A
⁢
1
-
ℝ
1
=
A
⋅
1
-
ℝ
A
⋅
1
3
1
1
2
mp3an23
⊢
A
∈
ℝ
→
A
⁢
1
-
ℝ
1
=
A
⋅
1
-
ℝ
A
⋅
1
4
re1m1e0m0
⊢
1
-
ℝ
1
=
0
-
ℝ
0
5
4
oveq2i
⊢
A
⁢
1
-
ℝ
1
=
A
⁢
0
-
ℝ
0
6
5
a1i
⊢
A
∈
ℝ
→
A
⁢
1
-
ℝ
1
=
A
⁢
0
-
ℝ
0
7
ax-1rid
⊢
A
∈
ℝ
→
A
⋅
1
=
A
8
7
7
oveq12d
⊢
A
∈
ℝ
→
A
⋅
1
-
ℝ
A
⋅
1
=
A
-
ℝ
A
9
3
6
8
3eqtr3d
⊢
A
∈
ℝ
→
A
⁢
0
-
ℝ
0
=
A
-
ℝ
A