Theorem 4t3lem 9482

Description: Lemma for 4t3e12 9483 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 5888	. 2 |- ( A x. C ) = ( A x. ( B + 1 ) )
3		4t3lem.1	. . . . . 6 |- A e. NN0
4	3	nn0cni 9190	. . . . 5 |- A e. CC
5		4t3lem.2	. . . . . 6 |- B e. NN0
6	5	nn0cni 9190	. . . . 5 |- B e. CC
7		ax-1cn 7906	. . . . 5 |- 1 e. CC
8	4, 6, 7	adddii 7969	. . . 4 |- ( A x. ( B + 1 ) ) = ( ( A x. B ) + ( A x. 1 ) )
9		4t3lem.4	. . . . 5 |- ( A x. B ) = D
10	4	mulid1i 7961	. . . . 5 |- ( A x. 1 ) = A
11	9, 10	oveq12i 5889	. . . 4 |- ( ( A x. B ) + ( A x. 1 ) ) = ( D + A )
12	8, 11	eqtri 2198	. . 3 |- ( A x. ( B + 1 ) ) = ( D + A )
13		4t3lem.5	. . 3 |- ( D + A ) = E
14	12, 13	eqtri 2198	. 2 |- ( A x. ( B + 1 ) ) = E
15	2, 14	eqtri 2198	1 |- ( A x. C ) = E

Syntax hints:   = wceq 1353   e. wcel 2148  (class class class)co 5877   1c1 7814   + caddc 7816   x. cmul 7818   NN0cn0 9178

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4123  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulcom 7914  ax-mulass 7916  ax-distr 7917  ax-1rid 7920  ax-rnegex 7922  ax-cnre 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880  df-inn 8922  df-n0 9179

This theorem is referenced by:  4t3e12  9483  4t4e16  9484  5t2e10  9485  5t3e15  9486  5t4e20  9487  5t5e25  9488  6t3e18  9490  6t4e24  9491  6t5e30  9492  6t6e36  9493  7t3e21  9495  7t4e28  9496  7t5e35  9497  7t6e42  9498  7t7e49  9499  8t3e24  9501  8t4e32  9502  8t5e40  9503  8t6e48  9504  8t7e56  9505  8t8e64  9506  9t3e27  9508  9t4e36  9509  9t5e45  9510  9t6e54  9511  9t7e63  9512  9t8e72  9513  9t9e81  9514