Theorem 1p2e3 12263

Description: 1 + 2 = 3. For a shorter proof using addcomli 11305, see 1p2e3ALT 12264. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 12-Dec-2022.)

Assertion

Ref	Expression
1p2e3	⊢ (1 + 2) = 3

Proof of Theorem 1p2e3

Step	Hyp	Ref	Expression
1		df-2 12188	. . 3 ⊢ 2 = (1 + 1)
2	1	oveq2i 7357	. 2 ⊢ (1 + 2) = (1 + (1 + 1))
3		ax-1cn 11064	. . 3 ⊢ 1 ∈ ℂ
4	3, 3, 3	addassi 11122	. 2 ⊢ ((1 + 1) + 1) = (1 + (1 + 1))
5		1p1e2 12245	. . . 4 ⊢ (1 + 1) = 2
6	5	oveq1i 7356	. . 3 ⊢ ((1 + 1) + 1) = (2 + 1)
7		2p1e3 12262	. . 3 ⊢ (2 + 1) = 3
8	6, 7	eqtri 2754	. 2 ⊢ ((1 + 1) + 1) = 3
9	2, 4, 8	3eqtr2i 2760	1 ⊢ (1 + 2) = 3

Syntax hints:   = wceq 1541  (class class class)co 7346  1c1 11007   + caddc 11009  2c2 12180  3c3 12181

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-1cn 11064  ax-addass 11071

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-2 12188  df-3 12189

This theorem is referenced by:  fzo1to4tp  13654  binom3  14131  3lcm2e6woprm  16526  prmgaplem7  16969  2exp16  17002  prmlem1a  17018  23prm  17030  prmlem2  17031  83prm  17034  139prm  17035  163prm  17036  317prm  17037  631prm  17038  1259lem4  17045  1259prm  17047  2503lem2  17049  2503lem3  17050  4001lem2  17053  quart1lem  26792  log2ublem3  26885  log2ub  26886  pntibndlem2  27529  1kp2ke3k  30426  ex-ind-dvds  30441  cos9thpiminplylem2  33796  fib4  34417  2np3bcnp1  42185  1p3e4  42300  ex-decpmul  42347  sn-0ne2  42447  3cubeslem3r  42728  rabren3dioph  42856  modm2nep1  47405  fmtno4nprmfac193  47613  139prmALT  47635  127prm  47638  nnsum4primesodd  47835  nnsum4primesoddALTV  47836  pgnbgreunbgrlem2lem1  48153  gpg5edgnedg  48169  ackval1012  48730