Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > add1p1 | GIF version |
Description: Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.) |
Ref | Expression |
---|---|
add1p1 | ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . 3 ⊢ (𝑁 ∈ ℂ → 𝑁 ∈ ℂ) | |
2 | 1cnd 7782 | . . 3 ⊢ (𝑁 ∈ ℂ → 1 ∈ ℂ) | |
3 | 1, 2, 2 | addassd 7788 | . 2 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1))) |
4 | 1p1e2 8837 | . . . 4 ⊢ (1 + 1) = 2 | |
5 | 4 | a1i 9 | . . 3 ⊢ (𝑁 ∈ ℂ → (1 + 1) = 2) |
6 | 5 | oveq2d 5790 | . 2 ⊢ (𝑁 ∈ ℂ → (𝑁 + (1 + 1)) = (𝑁 + 2)) |
7 | 3, 6 | eqtrd 2172 | 1 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 (class class class)co 5774 ℂcc 7618 1c1 7621 + caddc 7623 2c2 8771 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-1cn 7713 ax-addass 7722 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 df-2 8779 |
This theorem is referenced by: nneoor 9153 |
Copyright terms: Public domain | W3C validator |