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Theorem 4t3lem 9271
Description: Lemma for 4t3e12 9272 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 5778 . 2  |-  ( A  x.  C )  =  ( A  x.  ( B  +  1 ) )
3 4t3lem.1 . . . . . 6  |-  A  e. 
NN0
43nn0cni 8982 . . . . 5  |-  A  e.  CC
5 4t3lem.2 . . . . . 6  |-  B  e. 
NN0
65nn0cni 8982 . . . . 5  |-  B  e.  CC
7 ax-1cn 7706 . . . . 5  |-  1  e.  CC
84, 6, 7adddii 7769 . . . 4  |-  ( A  x.  ( B  + 
1 ) )  =  ( ( A  x.  B )  +  ( A  x.  1 ) )
9 4t3lem.4 . . . . 5  |-  ( A  x.  B )  =  D
104mulid1i 7761 . . . . 5  |-  ( A  x.  1 )  =  A
119, 10oveq12i 5779 . . . 4  |-  ( ( A  x.  B )  +  ( A  x.  1 ) )  =  ( D  +  A
)
128, 11eqtri 2158 . . 3  |-  ( A  x.  ( B  + 
1 ) )  =  ( D  +  A
)
13 4t3lem.5 . . 3  |-  ( D  +  A )  =  E
1412, 13eqtri 2158 . 2  |-  ( A  x.  ( B  + 
1 ) )  =  E
152, 14eqtri 2158 1  |-  ( A  x.  C )  =  E
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480  (class class class)co 5767   1c1 7614    + caddc 7616    x. cmul 7618   NN0cn0 8970
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulcom 7714  ax-mulass 7716  ax-distr 7717  ax-1rid 7720  ax-rnegex 7722  ax-cnre 7724
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770  df-inn 8714  df-n0 8971
This theorem is referenced by:  4t3e12  9272  4t4e16  9273  5t2e10  9274  5t3e15  9275  5t4e20  9276  5t5e25  9277  6t3e18  9279  6t4e24  9280  6t5e30  9281  6t6e36  9282  7t3e21  9284  7t4e28  9285  7t5e35  9286  7t6e42  9287  7t7e49  9288  8t3e24  9290  8t4e32  9291  8t5e40  9292  8t6e48  9293  8t7e56  9294  8t8e64  9295  9t3e27  9297  9t4e36  9298  9t5e45  9299  9t6e54  9300  9t7e63  9301  9t8e72  9302  9t9e81  9303
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