Theorem 4t3lem 9883

Description: Lemma for 4t3e12 9884 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 6096	. 2 |- ( A x. C ) = ( A x. ( B + 1 ) )
3		4t3lem.1	. . . . . 6 |- A e. NN0
4	3	nn0cni 9580	. . . . 5 |- A e. CC
5		4t3lem.2	. . . . . 6 |- B e. NN0
6	5	nn0cni 9580	. . . . 5 |- B e. CC
7		ax-1cn 8273	. . . . 5 |- 1 e. CC
8	4, 6, 7	adddii 8337	. . . 4 |- ( A x. ( B + 1 ) ) = ( ( A x. B ) + ( A x. 1 ) )
9		4t3lem.4	. . . . 5 |- ( A x. B ) = D
10	4	mulridi 8329	. . . . 5 |- ( A x. 1 ) = A
11	9, 10	oveq12i 6097	. . . 4 |- ( ( A x. B ) + ( A x. 1 ) ) = ( D + A )
12	8, 11	eqtri 2259	. . 3 |- ( A x. ( B + 1 ) ) = ( D + A )
13		4t3lem.5	. . 3 |- ( D + A ) = E
14	12, 13	eqtri 2259	. 2 |- ( A x. ( B + 1 ) ) = E
15	2, 14	eqtri 2259	1 |- ( A x. C ) = E

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209  (class class class)co 6085   1c1 8181   + caddc 8183   x. cmul 8185   NN0cn0 9568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4249  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulcom 8281  ax-mulass 8283  ax-distr 8284  ax-1rid 8287  ax-rnegex 8289  ax-cnre 8291

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088  df-inn 9308  df-n0 9569

This theorem is used by:  4t3e12  9884  4t4e16  9885  5t2e10  9886  5t3e15  9887  5t4e20  9888  5t5e25  9889  6t3e18  9891  6t4e24  9892  6t5e30  9893  6t6e36  9894  7t3e21  9896  7t4e28  9897  7t5e35  9898  7t6e42  9899  7t7e49  9900  8t3e24  9902  8t4e32  9903  8t5e40  9904  8t6e48  9905  8t7e56  9906  8t8e64  9907  9t3e27  9909  9t4e36  9910  9t5e45  9911  9t6e54  9912  9t7e63  9913  9t8e72  9914  9t9e81  9915