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Theorem 4t3lem 8523
Description: Lemma for 4t3e12 8524 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 5551 . 2  |-  ( A  x.  C )  =  ( A  x.  ( B  +  1 ) )
3 4t3lem.1 . . . . . 6  |-  A  e. 
NN0
43nn0cni 8251 . . . . 5  |-  A  e.  CC
5 4t3lem.2 . . . . . 6  |-  B  e. 
NN0
65nn0cni 8251 . . . . 5  |-  B  e.  CC
7 ax-1cn 7035 . . . . 5  |-  1  e.  CC
84, 6, 7adddii 7095 . . . 4  |-  ( A  x.  ( B  + 
1 ) )  =  ( ( A  x.  B )  +  ( A  x.  1 ) )
9 4t3lem.4 . . . . 5  |-  ( A  x.  B )  =  D
104mulid1i 7087 . . . . 5  |-  ( A  x.  1 )  =  A
119, 10oveq12i 5552 . . . 4  |-  ( ( A  x.  B )  +  ( A  x.  1 ) )  =  ( D  +  A
)
128, 11eqtri 2076 . . 3  |-  ( A  x.  ( B  + 
1 ) )  =  ( D  +  A
)
13 4t3lem.5 . . 3  |-  ( D  +  A )  =  E
1412, 13eqtri 2076 . 2  |-  ( A  x.  ( B  + 
1 ) )  =  E
152, 14eqtri 2076 1  |-  ( A  x.  C )  =  E
Colors of variables: wff set class
Syntax hints:    = wceq 1259    e. wcel 1409  (class class class)co 5540   1c1 6948    + caddc 6950    x. cmul 6952   NN0cn0 8239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulcom 7043  ax-mulass 7045  ax-distr 7046  ax-1rid 7049  ax-rnegex 7051  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543  df-inn 7991  df-n0 8240
This theorem is referenced by:  4t3e12  8524  4t4e16  8525  5t2e10  8526  5t3e15  8527  5t4e20  8528  5t5e25  8529  6t3e18  8531  6t4e24  8532  6t5e30  8533  6t6e36  8534  7t3e21  8536  7t4e28  8537  7t5e35  8538  7t6e42  8539  7t7e49  8540  8t3e24  8542  8t4e32  8543  8t5e40  8544  8t6e48  8545  8t7e56  8546  8t8e64  8547  9t3e27  8549  9t4e36  8550  9t5e45  8551  9t6e54  8552  9t7e63  8553  9t8e72  8554  9t9e81  8555
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