Theorem 4t3lem 9553

Description: Lemma for 4t3e12 9554 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 5933	. 2 |- ( A x. C ) = ( A x. ( B + 1 ) )
3		4t3lem.1	. . . . . 6 |- A e. NN0
4	3	nn0cni 9261	. . . . 5 |- A e. CC
5		4t3lem.2	. . . . . 6 |- B e. NN0
6	5	nn0cni 9261	. . . . 5 |- B e. CC
7		ax-1cn 7972	. . . . 5 |- 1 e. CC
8	4, 6, 7	adddii 8036	. . . 4 |- ( A x. ( B + 1 ) ) = ( ( A x. B ) + ( A x. 1 ) )
9		4t3lem.4	. . . . 5 |- ( A x. B ) = D
10	4	mulridi 8028	. . . . 5 |- ( A x. 1 ) = A
11	9, 10	oveq12i 5934	. . . 4 |- ( ( A x. B ) + ( A x. 1 ) ) = ( D + A )
12	8, 11	eqtri 2217	. . 3 |- ( A x. ( B + 1 ) ) = ( D + A )
13		4t3lem.5	. . 3 |- ( D + A ) = E
14	12, 13	eqtri 2217	. 2 |- ( A x. ( B + 1 ) ) = E
15	2, 14	eqtri 2217	1 |- ( A x. C ) = E

Syntax hints:   = wceq 1364   e. wcel 2167  (class class class)co 5922   1c1 7880   + caddc 7882   x. cmul 7884   NN0cn0 9249

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-sep 4151  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulcom 7980  ax-mulass 7982  ax-distr 7983  ax-1rid 7986  ax-rnegex 7988  ax-cnre 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925  df-inn 8991  df-n0 9250

This theorem is referenced by:  4t3e12  9554  4t4e16  9555  5t2e10  9556  5t3e15  9557  5t4e20  9558  5t5e25  9559  6t3e18  9561  6t4e24  9562  6t5e30  9563  6t6e36  9564  7t3e21  9566  7t4e28  9567  7t5e35  9568  7t6e42  9569  7t7e49  9570  8t3e24  9572  8t4e32  9573  8t5e40  9574  8t6e48  9575  8t7e56  9576  8t8e64  9577  9t3e27  9579  9t4e36  9580  9t5e45  9581  9t6e54  9582  9t7e63  9583  9t8e72  9584  9t9e81  9585