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Mirrors > Home > ILE Home > Th. List > addid2i | GIF version |
Description: 0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.) |
Ref | Expression |
---|---|
mul.1 | ⊢ 𝐴 ∈ ℂ |
Ref | Expression |
---|---|
addid2i | ⊢ (0 + 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | addlid 8127 | . 2 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (0 + 𝐴) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 (class class class)co 5897 ℂcc 7840 0cc0 7842 + caddc 7845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-17 1537 ax-ial 1545 ax-ext 2171 ax-1cn 7935 ax-icn 7937 ax-addcl 7938 ax-mulcl 7940 ax-addcom 7942 ax-i2m1 7947 ax-0id 7950 |
This theorem depends on definitions: df-bi 117 df-cleq 2182 df-clel 2185 |
This theorem is referenced by: ine0 8382 inelr 8572 muleqadd 8656 0p1e1 9064 iap0 9173 num0h 9426 nummul1c 9463 decrmac 9472 decmul1 9478 fz0tp 10154 fz0to4untppr 10156 fzo0to3tp 10251 rei 10943 imi 10944 resqrexlemover 11054 ef01bndlem 11799 efhalfpi 14697 sinq34lt0t 14729 ex-fac 14958 |
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