Theorem add1p1 12463

Description: Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)

Assertion

Ref	Expression
add1p1	⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))

Proof of Theorem add1p1

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝑁 ∈ ℂ → 𝑁 ∈ ℂ)
2		1cnd 11209	. . 3 ⊢ (𝑁 ∈ ℂ → 1 ∈ ℂ)
3	1, 2, 2	addassd 11236	. 2 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
4		1p1e2 12337	. . . 4 ⊢ (1 + 1) = 2
5	4	a1i 11	. . 3 ⊢ (𝑁 ∈ ℂ → (1 + 1) = 2)
6	5	oveq2d 7425	. 2 ⊢ (𝑁 ∈ ℂ → (𝑁 + (1 + 1)) = (𝑁 + 2))
7	3, 6	eqtrd 2773	1 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  (class class class)co 7409  ℂcc 11108  1c1 11111   + caddc 11113  2c2 12267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-1cn 11168  ax-addass 11175

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-2 12275

This theorem is referenced by:  nneo  12646  ccatw2s1len  14575  chfacfscmul0  22360  chfacfscmulfsupp  22361  chfacfscmulgsum  22362  chfacfpmmul0  22364  chfacfpmmulfsupp  22365  chfacfpmmulgsum  22366  upgrwlkdvdelem  28993  poimirlem7  36495  fmtnoprmfac2  46235  fmtnofac1  46238  evenltle  46385