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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 7111 | . . 3 ⊢ 0 ∈ ℂ | |
2 | addcom 7245 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴 + 0) = (0 + 𝐴)) | |
3 | 1, 2 | mpan2 415 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = (0 + 𝐴)) |
4 | addid1 7246 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) | |
5 | 3, 4 | eqtr3d 2115 | 1 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∈ wcel 1433 (class class class)co 5532 ℂcc 6979 0cc0 6981 + caddc 6984 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-17 1459 ax-ial 1467 ax-ext 2063 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-mulcl 7074 ax-addcom 7076 ax-i2m1 7081 ax-0id 7084 |
This theorem depends on definitions: df-bi 115 df-cleq 2074 df-clel 2077 |
This theorem is referenced by: readdcan 7248 addid2i 7251 addid2d 7258 cnegexlem1 7283 cnegexlem2 7284 addcan 7288 negneg 7358 fzoaddel2 9202 divfl0 9298 modqid 9351 gcdid 10377 |
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