ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  6p3e9 GIF version

Theorem 6p3e9 8627
Description: 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
Assertion
Ref Expression
6p3e9 (6 + 3) = 9

Proof of Theorem 6p3e9
StepHypRef Expression
1 df-3 8543 . . . 4 3 = (2 + 1)
21oveq2i 5677 . . 3 (6 + 3) = (6 + (2 + 1))
3 6cn 8565 . . . 4 6 ∈ ℂ
4 2cn 8554 . . . 4 2 ∈ ℂ
5 ax-1cn 7499 . . . 4 1 ∈ ℂ
63, 4, 5addassi 7557 . . 3 ((6 + 2) + 1) = (6 + (2 + 1))
72, 6eqtr4i 2112 . 2 (6 + 3) = ((6 + 2) + 1)
8 df-9 8549 . . 3 9 = (8 + 1)
9 6p2e8 8626 . . . 4 (6 + 2) = 8
109oveq1i 5676 . . 3 ((6 + 2) + 1) = (8 + 1)
118, 10eqtr4i 2112 . 2 9 = ((6 + 2) + 1)
127, 11eqtr4i 2112 1 (6 + 3) = 9
Colors of variables: wff set class
Syntax hints:   = wceq 1290  (class class class)co 5666  1c1 7412   + caddc 7414  2c2 8534  3c3 8535  6c6 8538  8c8 8540  9c9 8541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-addrcl 7503  ax-addass 7508
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-iota 4993  df-fv 5036  df-ov 5669  df-2 8542  df-3 8543  df-4 8544  df-5 8545  df-6 8546  df-7 8547  df-8 8548  df-9 8549
This theorem is referenced by:  3t3e9  8634  6p4e10  9009  ex-gcd  11931
  Copyright terms: Public domain W3C validator