Theorem 4t3lem 9674

Description: Lemma for 4t3e12 9675 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 6012	. 2 |- ( A x. C ) = ( A x. ( B + 1 ) )
3		4t3lem.1	. . . . . 6 |- A e. NN0
4	3	nn0cni 9381	. . . . 5 |- A e. CC
5		4t3lem.2	. . . . . 6 |- B e. NN0
6	5	nn0cni 9381	. . . . 5 |- B e. CC
7		ax-1cn 8092	. . . . 5 |- 1 e. CC
8	4, 6, 7	adddii 8156	. . . 4 |- ( A x. ( B + 1 ) ) = ( ( A x. B ) + ( A x. 1 ) )
9		4t3lem.4	. . . . 5 |- ( A x. B ) = D
10	4	mulridi 8148	. . . . 5 |- ( A x. 1 ) = A
11	9, 10	oveq12i 6013	. . . 4 |- ( ( A x. B ) + ( A x. 1 ) ) = ( D + A )
12	8, 11	eqtri 2250	. . 3 |- ( A x. ( B + 1 ) ) = ( D + A )
13		4t3lem.5	. . 3 |- ( D + A ) = E
14	12, 13	eqtri 2250	. 2 |- ( A x. ( B + 1 ) ) = E
15	2, 14	eqtri 2250	1 |- ( A x. C ) = E

Syntax hints:   = wceq 1395   e. wcel 2200  (class class class)co 6001   1c1 8000   + caddc 8002   x. cmul 8004   NN0cn0 9369

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-sep 4202  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulcom 8100  ax-mulass 8102  ax-distr 8103  ax-1rid 8106  ax-rnegex 8108  ax-cnre 8110

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6004  df-inn 9111  df-n0 9370

This theorem is referenced by:  4t3e12  9675  4t4e16  9676  5t2e10  9677  5t3e15  9678  5t4e20  9679  5t5e25  9680  6t3e18  9682  6t4e24  9683  6t5e30  9684  6t6e36  9685  7t3e21  9687  7t4e28  9688  7t5e35  9689  7t6e42  9690  7t7e49  9691  8t3e24  9693  8t4e32  9694  8t5e40  9695  8t6e48  9696  8t7e56  9697  8t8e64  9698  9t3e27  9700  9t4e36  9701  9t5e45  9702  9t6e54  9703  9t7e63  9704  9t8e72  9705  9t9e81  9706