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Mirrors > Home > ILE Home > Th. List > 00id | GIF version |
Description: 0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
00id | ⊢ (0 + 0) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 7765 | . 2 ⊢ 0 ∈ ℂ | |
2 | addid1 7907 | . 2 ⊢ (0 ∈ ℂ → (0 + 0) = 0) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (0 + 0) = 0 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 (class class class)co 5774 ℂcc 7625 0cc0 7627 + caddc 7630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 ax-ext 2121 ax-1cn 7720 ax-icn 7722 ax-addcl 7723 ax-mulcl 7725 ax-i2m1 7732 ax-0id 7735 |
This theorem depends on definitions: df-bi 116 df-cleq 2132 df-clel 2135 |
This theorem is referenced by: negdii 8053 addgt0 8217 addgegt0 8218 addgtge0 8219 addge0 8220 add20 8243 recexaplem2 8420 crap0 8723 iap0 8950 decaddm10 9247 10p10e20 9283 ser0 10294 bcpasc 10519 abs00ap 10841 fsumadd 11182 fsumrelem 11247 arisum 11274 bezoutr1 11728 1kp2ke3k 12946 |
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