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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 10638 | . . 3 ⊢ (𝑁 ∈ ℂ → 1 ∈ ℂ) | |
3 | 1, 2, 2 | addassd 10665 | . 2 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1))) |
4 | 1p1e2 11765 | . . . 4 ⊢ (1 + 1) = 2 | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℂ → (1 + 1) = 2) |
6 | 5 | oveq2d 7174 | . 2 ⊢ (𝑁 ∈ ℂ → (𝑁 + (1 + 1)) = (𝑁 + 2)) |
7 | 3, 6 | eqtrd 2858 | 1 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 (class class class)co 7158 ℂcc 10537 1c1 10540 + caddc 10542 2c2 11695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-1cn 10597 ax-addass 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-2 11703 |
This theorem is referenced by: nneo 12069 ccatw2s1len 13982 chfacfscmul0 21468 chfacfscmulfsupp 21469 chfacfscmulgsum 21470 chfacfpmmul0 21472 chfacfpmmulfsupp 21473 chfacfpmmulgsum 21474 upgrwlkdvdelem 27519 poimirlem7 34901 fmtnoprmfac2 43736 fmtnofac1 43739 evenltle 43889 |
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