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Theorem 1p2e3 12362
Description: 1 + 2 = 3. For a shorter proof using addcomli 11413, see 1p2e3ALT 12363. (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
StepHypRef Expression
1 df-2 12282 . . 3 2 = (1 + 1)
21oveq2i 7423 . 2 (1 + 2) = (1 + (1 + 1))
3 ax-1cn 11174 . . 3 1 ∈ ℂ
43, 3, 3addassi 11231 . 2 ((1 + 1) + 1) = (1 + (1 + 1))
5 1p1e2 12344 . . . 4 (1 + 1) = 2
65oveq1i 7422 . . 3 ((1 + 1) + 1) = (2 + 1)
7 2p1e3 12361 . . 3 (2 + 1) = 3
86, 7eqtri 2759 . 2 ((1 + 1) + 1) = 3
92, 4, 83eqtr2i 2765 1 (1 + 2) = 3
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  (class class class)co 7412  1c1 11117   + caddc 11119  2c2 12274  3c3 12275
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-1cn 11174  ax-addass 11181
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-2 12282  df-3 12283
This theorem is referenced by:  fzo1to4tp  13727  binom3  14194  3lcm2e6woprm  16559  prmgaplem7  16997  2exp16  17031  prmlem1a  17047  23prm  17059  prmlem2  17060  83prm  17063  139prm  17064  163prm  17065  317prm  17066  631prm  17067  1259lem4  17074  1259prm  17076  2503lem2  17078  2503lem3  17079  4001lem2  17082  quart1lem  26700  log2ublem3  26793  log2ub  26794  pntibndlem2  27436  1kp2ke3k  30131  ex-ind-dvds  30146  fib4  33866  2np3bcnp1  41426  2xp3dxp2ge1d  41488  ex-decpmul  41668  sn-0ne2  41741  3cubeslem3r  41887  rabren3dioph  42015  fmtno4nprmfac193  46700  139prmALT  46722  127prm  46725  nnsum4primesodd  46922  nnsum4primesoddALTV  46923  ackval1012  47537
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