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Theorem 4t3lem 9246
Description: Lemma for 4t3e12 9247 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
Hypotheses
Ref Expression
4t3lem.1  |-  A  e. 
NN0
4t3lem.2  |-  B  e. 
NN0
4t3lem.3  |-  C  =  ( B  +  1 )
4t3lem.4  |-  ( A  x.  B )  =  D
4t3lem.5  |-  ( D  +  A )  =  E
Assertion
Ref Expression
4t3lem  |-  ( A  x.  C )  =  E

Proof of Theorem 4t3lem
StepHypRef Expression
1 4t3lem.3 . . 3  |-  C  =  ( B  +  1 )
21oveq2i 5753 . 2  |-  ( A  x.  C )  =  ( A  x.  ( B  +  1 ) )
3 4t3lem.1 . . . . . 6  |-  A  e. 
NN0
43nn0cni 8957 . . . . 5  |-  A  e.  CC
5 4t3lem.2 . . . . . 6  |-  B  e. 
NN0
65nn0cni 8957 . . . . 5  |-  B  e.  CC
7 ax-1cn 7681 . . . . 5  |-  1  e.  CC
84, 6, 7adddii 7744 . . . 4  |-  ( A  x.  ( B  + 
1 ) )  =  ( ( A  x.  B )  +  ( A  x.  1 ) )
9 4t3lem.4 . . . . 5  |-  ( A  x.  B )  =  D
104mulid1i 7736 . . . . 5  |-  ( A  x.  1 )  =  A
119, 10oveq12i 5754 . . . 4  |-  ( ( A  x.  B )  +  ( A  x.  1 ) )  =  ( D  +  A
)
128, 11eqtri 2138 . . 3  |-  ( A  x.  ( B  + 
1 ) )  =  ( D  +  A
)
13 4t3lem.5 . . 3  |-  ( D  +  A )  =  E
1412, 13eqtri 2138 . 2  |-  ( A  x.  ( B  + 
1 ) )  =  E
152, 14eqtri 2138 1  |-  ( A  x.  C )  =  E
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465  (class class class)co 5742   1c1 7589    + caddc 7591    x. cmul 7593   NN0cn0 8945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulcom 7689  ax-mulass 7691  ax-distr 7692  ax-1rid 7695  ax-rnegex 7697  ax-cnre 7699
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745  df-inn 8689  df-n0 8946
This theorem is referenced by:  4t3e12  9247  4t4e16  9248  5t2e10  9249  5t3e15  9250  5t4e20  9251  5t5e25  9252  6t3e18  9254  6t4e24  9255  6t5e30  9256  6t6e36  9257  7t3e21  9259  7t4e28  9260  7t5e35  9261  7t6e42  9262  7t7e49  9263  8t3e24  9265  8t4e32  9266  8t5e40  9267  8t6e48  9268  8t7e56  9269  8t8e64  9270  9t3e27  9272  9t4e36  9273  9t5e45  9274  9t6e54  9275  9t7e63  9276  9t8e72  9277  9t9e81  9278
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