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Theorem 4t3lem 8438
Description: Lemma for 4t3e12 8439 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 5523 . 2  |-  ( A  x.  C )  =  ( A  x.  ( B  +  1 ) )
3 4t3lem.1 . . . . . 6  |-  A  e. 
NN0
43nn0cni 8193 . . . . 5  |-  A  e.  CC
5 4t3lem.2 . . . . . 6  |-  B  e. 
NN0
65nn0cni 8193 . . . . 5  |-  B  e.  CC
7 ax-1cn 6977 . . . . 5  |-  1  e.  CC
84, 6, 7adddii 7037 . . . 4  |-  ( A  x.  ( B  + 
1 ) )  =  ( ( A  x.  B )  +  ( A  x.  1 ) )
9 4t3lem.4 . . . . 5  |-  ( A  x.  B )  =  D
104mulid1i 7029 . . . . 5  |-  ( A  x.  1 )  =  A
119, 10oveq12i 5524 . . . 4  |-  ( ( A  x.  B )  +  ( A  x.  1 ) )  =  ( D  +  A
)
128, 11eqtri 2060 . . 3  |-  ( A  x.  ( B  + 
1 ) )  =  ( D  +  A
)
13 4t3lem.5 . . 3  |-  ( D  +  A )  =  E
1412, 13eqtri 2060 . 2  |-  ( A  x.  ( B  + 
1 ) )  =  E
152, 14eqtri 2060 1  |-  ( A  x.  C )  =  E
Colors of variables: wff set class
Syntax hints:    = wceq 1243    e. wcel 1393  (class class class)co 5512   1c1 6890    + caddc 6892    x. cmul 6894   NN0cn0 8181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulcom 6985  ax-mulass 6987  ax-distr 6988  ax-1rid 6991  ax-rnegex 6993  ax-cnre 6995
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515  df-inn 7915  df-n0 8182
This theorem is referenced by:  4t3e12  8439  4t4e16  8440  5t3e15  8441  5t4e20  8442  5t5e25  8443  6t3e18  8445  6t4e24  8446  6t5e30  8447  6t6e36  8448  7t3e21  8450  7t4e28  8451  7t5e35  8452  7t6e42  8453  7t7e49  8454  8t3e24  8456  8t4e32  8457  8t5e40  8458  8t6e48  8459  8t7e56  8460  8t8e64  8461  9t3e27  8463  9t4e36  8464  9t5e45  8465  9t6e54  8466  9t7e63  8467  9t8e72  8468  9t9e81  8469
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