Theorem 4t3lem 8573

Description: Lemma for 4t3e12 8574 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 5543	. 2 |- ( A x. C ) = ( A x. ( B + 1 ) )
3		4t3lem.1	. . . . . 6 |- A e. NN0
4	3	nn0cni 8300	. . . . 5 |- A e. CC
5		4t3lem.2	. . . . . 6 |- B e. NN0
6	5	nn0cni 8300	. . . . 5 |- B e. CC
7		ax-1cn 7069	. . . . 5 |- 1 e. CC
8	4, 6, 7	adddii 7129	. . . 4 |- ( A x. ( B + 1 ) ) = ( ( A x. B ) + ( A x. 1 ) )
9		4t3lem.4	. . . . 5 |- ( A x. B ) = D
10	4	mulid1i 7121	. . . . 5 |- ( A x. 1 ) = A
11	9, 10	oveq12i 5544	. . . 4 |- ( ( A x. B ) + ( A x. 1 ) ) = ( D + A )
12	8, 11	eqtri 2101	. . 3 |- ( A x. ( B + 1 ) ) = ( D + A )
13		4t3lem.5	. . 3 |- ( D + A ) = E
14	12, 13	eqtri 2101	. 2 |- ( A x. ( B + 1 ) ) = E
15	2, 14	eqtri 2101	1 |- ( A x. C ) = E

Syntax hints:   = wceq 1284   e. wcel 1433  (class class class)co 5532   1c1 6982   + caddc 6984   x. cmul 6986   NN0cn0 8288

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulcom 7077  ax-mulass 7079  ax-distr 7080  ax-1rid 7083  ax-rnegex 7085  ax-cnre 7087

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535  df-inn 8040  df-n0 8289

This theorem is referenced by:  4t3e12  8574  4t4e16  8575  5t2e10  8576  5t3e15  8577  5t4e20  8578  5t5e25  8579  6t3e18  8581  6t4e24  8582  6t5e30  8583  6t6e36  8584  7t3e21  8586  7t4e28  8587  7t5e35  8588  7t6e42  8589  7t7e49  8590  8t3e24  8592  8t4e32  8593  8t5e40  8594  8t6e48  8595  8t7e56  8596  8t8e64  8597  9t3e27  8599  9t4e36  8600  9t5e45  8601  9t6e54  8602  9t7e63  8603  9t8e72  8604  9t9e81  8605