![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 10371 | . . 3 ⊢ (𝑁 ∈ ℂ → 1 ∈ ℂ) | |
3 | 1, 2, 2 | addassd 10399 | . 2 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1))) |
4 | 1p1e2 11507 | . . . 4 ⊢ (1 + 1) = 2 | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℂ → (1 + 1) = 2) |
6 | 5 | oveq2d 6938 | . 2 ⊢ (𝑁 ∈ ℂ → (𝑁 + (1 + 1)) = (𝑁 + 2)) |
7 | 3, 6 | eqtrd 2813 | 1 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 (class class class)co 6922 ℂcc 10270 1c1 10273 + caddc 10275 2c2 11430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-1cn 10330 ax-addass 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-rex 3095 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-iota 6099 df-fv 6143 df-ov 6925 df-2 11438 |
This theorem is referenced by: nneo 11813 ccatw2s1len 13715 chfacfscmul0 21070 chfacfscmulfsupp 21071 chfacfscmulgsum 21072 chfacfpmmul0 21074 chfacfpmmulfsupp 21075 chfacfpmmulgsum 21076 upgrwlkdvdelem 27088 poimirlem7 34026 fmtnoprmfac2 42482 fmtnofac1 42485 evenltle 42633 |
Copyright terms: Public domain | W3C validator |