Theorem 4t3lem 9570

Description: Lemma for 4t3e12 9571 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 5936	. 2 |- ( A x. C ) = ( A x. ( B + 1 ) )
3		4t3lem.1	. . . . . 6 |- A e. NN0
4	3	nn0cni 9278	. . . . 5 |- A e. CC
5		4t3lem.2	. . . . . 6 |- B e. NN0
6	5	nn0cni 9278	. . . . 5 |- B e. CC
7		ax-1cn 7989	. . . . 5 |- 1 e. CC
8	4, 6, 7	adddii 8053	. . . 4 |- ( A x. ( B + 1 ) ) = ( ( A x. B ) + ( A x. 1 ) )
9		4t3lem.4	. . . . 5 |- ( A x. B ) = D
10	4	mulridi 8045	. . . . 5 |- ( A x. 1 ) = A
11	9, 10	oveq12i 5937	. . . 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 5925   1c1 7897   + caddc 7899   x. cmul 7901   NN0cn0 9266

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 4152  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulcom 7997  ax-mulass 7999  ax-distr 8000  ax-1rid 8003  ax-rnegex 8005  ax-cnre 8007

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 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-iota 5220  df-fv 5267  df-ov 5928  df-inn 9008  df-n0 9267

This theorem is referenced by:  4t3e12  9571  4t4e16  9572  5t2e10  9573  5t3e15  9574  5t4e20  9575  5t5e25  9576  6t3e18  9578  6t4e24  9579  6t5e30  9580  6t6e36  9581  7t3e21  9583  7t4e28  9584  7t5e35  9585  7t6e42  9586  7t7e49  9587  8t3e24  9589  8t4e32  9590  8t5e40  9591  8t6e48  9592  8t7e56  9593  8t8e64  9594  9t3e27  9596  9t4e36  9597  9t5e45  9598  9t6e54  9599  9t7e63  9600  9t8e72  9601  9t9e81  9602