Theorem 4t3lem 9873

Description: Lemma for 4t3e12 9874 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 9575	. . . . 5 |- A e. CC
5		4t3lem.2	. . . . . 6 |- B e. NN0
6	5	nn0cni 9575	. . . . 5 |- B e. CC
7		ax-1cn 8272	. . . . 5 |- 1 e. CC
8	4, 6, 7	adddii 8336	. . . 4 |- ( A x. ( B + 1 ) ) = ( ( A x. B ) + ( A x. 1 ) )
9		4t3lem.4	. . . . 5 |- ( A x. B ) = D
10	4	mulridi 8328	. . . . 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 8180   + caddc 8182   x. cmul 8184   NN0cn0 9563

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 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulcom 8280  ax-mulass 8282  ax-distr 8283  ax-1rid 8286  ax-rnegex 8288  ax-cnre 8290

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 9305  df-n0 9564

This theorem is used by:  4t3e12  9874  4t4e16  9875  5t2e10  9876  5t3e15  9877  5t4e20  9878  5t5e25  9879  6t3e18  9881  6t4e24  9882  6t5e30  9883  6t6e36  9884  7t3e21  9886  7t4e28  9887  7t5e35  9888  7t6e42  9889  7t7e49  9890  8t3e24  9892  8t4e32  9893  8t5e40  9894  8t6e48  9895  8t7e56  9896  8t8e64  9897  9t3e27  9899  9t4e36  9900  9t5e45  9901  9t6e54  9902  9t7e63  9903  9t8e72  9904  9t9e81  9905