Theorem 4t3lem 9805

Description: Lemma for 4t3e12 9806 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

Step	Hyp	Ref	Expression
1		4t3lem.3	. . 3 |- C = ( B + 1 )
2	1	oveq2i 6061	. 2 |- ( A x. C ) = ( A x. ( B + 1 ) )
3		4t3lem.1	. . . . . 6 |- A e. NN0
4	3	nn0cni 9508	. . . . 5 |- A e. CC
5		4t3lem.2	. . . . . 6 |- B e. NN0
6	5	nn0cni 9508	. . . . 5 |- B e. CC
7		ax-1cn 8220	. . . . 5 |- 1 e. CC
8	4, 6, 7	adddii 8284	. . . 4 |- ( A x. ( B + 1 ) ) = ( ( A x. B ) + ( A x. 1 ) )
9		4t3lem.4	. . . . 5 |- ( A x. B ) = D
10	4	mulridi 8276	. . . . 5 |- ( A x. 1 ) = A
11	9, 10	oveq12i 6062	. . . 4 |- ( ( A x. B ) + ( A x. 1 ) ) = ( D + A )
12	8, 11	eqtri 2253	. . 3 |- ( A x. ( B + 1 ) ) = ( D + A )
13		4t3lem.5	. . 3 |- ( D + A ) = E
14	12, 13	eqtri 2253	. 2 |- ( A x. ( B + 1 ) ) = E
15	2, 14	eqtri 2253	1 |- ( A x. C ) = E

Syntax hints:   = wceq 1398   e. wcel 2203  (class class class)co 6050   1c1 8128   + caddc 8130   x. cmul 8132   NN0cn0 9496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-sep 4228  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulcom 8228  ax-mulass 8230  ax-distr 8231  ax-1rid 8234  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053  df-inn 9238  df-n0 9497

This theorem is referenced by:  4t3e12  9806  4t4e16  9807  5t2e10  9808  5t3e15  9809  5t4e20  9810  5t5e25  9811  6t3e18  9813  6t4e24  9814  6t5e30  9815  6t6e36  9816  7t3e21  9818  7t4e28  9819  7t5e35  9820  7t6e42  9821  7t7e49  9822  8t3e24  9824  8t4e32  9825  8t5e40  9826  8t6e48  9827  8t7e56  9828  8t8e64  9829  9t3e27  9831  9t4e36  9832  9t5e45  9833  9t6e54  9834  9t7e63  9835  9t8e72  9836  9t9e81  9837