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Mirrors > Home > ILE Home > Th. List > addid2 | GIF version |
Description: 0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
addid2 | ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 7019 | . . 3 ⊢ 0 ∈ ℂ | |
2 | addcom 7150 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴 + 0) = (0 + 𝐴)) | |
3 | 1, 2 | mpan2 401 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = (0 + 𝐴)) |
4 | addid1 7151 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) | |
5 | 3, 4 | eqtr3d 2074 | 1 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 (class class class)co 5512 ℂcc 6887 0cc0 6889 + caddc 6892 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 ax-ext 2022 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-mulcl 6982 ax-addcom 6984 ax-i2m1 6989 ax-0id 6992 |
This theorem depends on definitions: df-bi 110 df-cleq 2033 df-clel 2036 |
This theorem is referenced by: readdcan 7153 addid2i 7156 addid2d 7163 cnegexlem1 7186 cnegexlem2 7187 addcan 7191 negneg 7261 fzoaddel2 9049 divfl0 9138 |
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