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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 7407 | . . 3 ⊢ (𝑁 ∈ ℂ → 1 ∈ ℂ) | |
3 | 1, 2, 2 | addassd 7413 | . 2 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1))) |
4 | 1p1e2 8432 | . . . 4 ⊢ (1 + 1) = 2 | |
5 | 4 | a1i 9 | . . 3 ⊢ (𝑁 ∈ ℂ → (1 + 1) = 2) |
6 | 5 | oveq2d 5607 | . 2 ⊢ (𝑁 ∈ ℂ → (𝑁 + (1 + 1)) = (𝑁 + 2)) |
7 | 3, 6 | eqtrd 2115 | 1 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 (class class class)co 5591 ℂcc 7251 1c1 7254 + caddc 7256 2c2 8366 |
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-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-1cn 7341 ax-addass 7350 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-rex 2359 df-v 2614 df-un 2988 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-iota 4934 df-fv 4977 df-ov 5594 df-2 8375 |
This theorem is referenced by: nneoor 8744 |
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