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Mirrors > Home > MPE Home > Th. List > add1p1 | Structured version Visualization version 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 22 | . . 3 ⊢ (𝑁 ∈ ℂ → 𝑁 ∈ ℂ) | |
2 | 1cnd 10625 | . . 3 ⊢ (𝑁 ∈ ℂ → 1 ∈ ℂ) | |
3 | 1, 2, 2 | addassd 10652 | . 2 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1))) |
4 | 1p1e2 11750 | . . . 4 ⊢ (1 + 1) = 2 | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℂ → (1 + 1) = 2) |
6 | 5 | oveq2d 7151 | . 2 ⊢ (𝑁 ∈ ℂ → (𝑁 + (1 + 1)) = (𝑁 + 2)) |
7 | 3, 6 | eqtrd 2833 | 1 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 1c1 10527 + caddc 10529 2c2 11680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 ax-1cn 10584 ax-addass 10591 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-2 11688 |
This theorem is referenced by: nneo 12054 ccatw2s1len 13971 chfacfscmul0 21463 chfacfscmulfsupp 21464 chfacfscmulgsum 21465 chfacfpmmul0 21467 chfacfpmmulfsupp 21468 chfacfpmmulgsum 21469 upgrwlkdvdelem 27525 poimirlem7 35064 fmtnoprmfac2 44084 fmtnofac1 44087 evenltle 44235 |
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