Theorem add1p1 12409

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 11145	. . 3 ⊢ (𝑁 ∈ ℂ → 1 ∈ ℂ)
3	1, 2, 2	addassd 11172	. 2 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
4		1p1e2 12282	. . . 4 ⊢ (1 + 1) = 2
5	4	a1i 11	. . 3 ⊢ (𝑁 ∈ ℂ → (1 + 1) = 2)
6	5	oveq2d 7385	. 2 ⊢ (𝑁 ∈ ℂ → (𝑁 + (1 + 1)) = (𝑁 + 2))
7	3, 6	eqtrd 2764	1 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  (class class class)co 7369  ℂcc 11042  1c1 11045   + caddc 11047  2c2 12217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-1cn 11102  ax-addass 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-iota 6452  df-fv 6507  df-ov 7372  df-2 12225

This theorem is referenced by:  nneo  12594  ccatw2s1len  14566  chfacfscmul0  22778  chfacfscmulfsupp  22779  chfacfscmulgsum  22780  chfacfpmmul0  22782  chfacfpmmulfsupp  22783  chfacfpmmulgsum  22784  upgrwlkdvdelem  29716  poimirlem7  37614  fmtnoprmfac2  47561  fmtnofac1  47564  evenltle  47711  gpg5nbgrvtx03starlem2  48053