Theorem 4t3lem 9509

Description: Lemma for 4t3e12 9510 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 5906	. 2 |- ( A x. C ) = ( A x. ( B + 1 ) )
3		4t3lem.1	. . . . . 6 |- A e. NN0
4	3	nn0cni 9217	. . . . 5 |- A e. CC
5		4t3lem.2	. . . . . 6 |- B e. NN0
6	5	nn0cni 9217	. . . . 5 |- B e. CC
7		ax-1cn 7933	. . . . 5 |- 1 e. CC
8	4, 6, 7	adddii 7996	. . . 4 |- ( A x. ( B + 1 ) ) = ( ( A x. B ) + ( A x. 1 ) )
9		4t3lem.4	. . . . 5 |- ( A x. B ) = D
10	4	mulid1i 7988	. . . . 5 |- ( A x. 1 ) = A
11	9, 10	oveq12i 5907	. . . 4 |- ( ( A x. B ) + ( A x. 1 ) ) = ( D + A )
12	8, 11	eqtri 2210	. . 3 |- ( A x. ( B + 1 ) ) = ( D + A )
13		4t3lem.5	. . 3 |- ( D + A ) = E
14	12, 13	eqtri 2210	. 2 |- ( A x. ( B + 1 ) ) = E
15	2, 14	eqtri 2210	1 |- ( A x. C ) = E

Syntax hints:   = wceq 1364   e. wcel 2160  (class class class)co 5895   1c1 7841   + caddc 7843   x. cmul 7845   NN0cn0 9205

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4136  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulcom 7941  ax-mulass 7943  ax-distr 7944  ax-1rid 7947  ax-rnegex 7949  ax-cnre 7951

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5898  df-inn 8949  df-n0 9206

This theorem is referenced by:  4t3e12  9510  4t4e16  9511  5t2e10  9512  5t3e15  9513  5t4e20  9514  5t5e25  9515  6t3e18  9517  6t4e24  9518  6t5e30  9519  6t6e36  9520  7t3e21  9522  7t4e28  9523  7t5e35  9524  7t6e42  9525  7t7e49  9526  8t3e24  9528  8t4e32  9529  8t5e40  9530  8t6e48  9531  8t7e56  9532  8t8e64  9533  9t3e27  9535  9t4e36  9536  9t5e45  9537  9t6e54  9538  9t7e63  9539  9t8e72  9540  9t9e81  9541