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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 6817 | . . 3 ⊢ 0 ∈ ℂ | |
2 | addcom 6947 | . . 3 ⊢ ((A ∈ ℂ ∧ 0 ∈ ℂ) → (A + 0) = (0 + A)) | |
3 | 1, 2 | mpan2 401 | . 2 ⊢ (A ∈ ℂ → (A + 0) = (0 + A)) |
4 | addid1 6948 | . 2 ⊢ (A ∈ ℂ → (A + 0) = A) | |
5 | 3, 4 | eqtr3d 2071 | 1 ⊢ (A ∈ ℂ → (0 + A) = A) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 (class class class)co 5455 ℂcc 6709 0cc0 6711 + caddc 6714 |
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 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-17 1416 ax-ial 1424 ax-ext 2019 ax-1cn 6776 ax-icn 6778 ax-addcl 6779 ax-mulcl 6781 ax-addcom 6783 ax-i2m1 6788 ax-0id 6791 |
This theorem depends on definitions: df-bi 110 df-cleq 2030 df-clel 2033 |
This theorem is referenced by: readdcan 6950 addid2i 6953 addid2d 6960 cnegexlem1 6983 cnegexlem2 6984 addcan 6988 negneg 7057 fzoaddel2 8819 |
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