Theorem 4t3lem 8942

Description: Lemma for 4t3e12 8943 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 5645	. 2 |- ( A x. C ) = ( A x. ( B + 1 ) )
3		4t3lem.1	. . . . . 6 |- A e. NN0
4	3	nn0cni 8655	. . . . 5 |- A e. CC
5		4t3lem.2	. . . . . 6 |- B e. NN0
6	5	nn0cni 8655	. . . . 5 |- B e. CC
7		ax-1cn 7417	. . . . 5 |- 1 e. CC
8	4, 6, 7	adddii 7477	. . . 4 |- ( A x. ( B + 1 ) ) = ( ( A x. B ) + ( A x. 1 ) )
9		4t3lem.4	. . . . 5 |- ( A x. B ) = D
10	4	mulid1i 7469	. . . . 5 |- ( A x. 1 ) = A
11	9, 10	oveq12i 5646	. . . 4 |- ( ( A x. B ) + ( A x. 1 ) ) = ( D + A )
12	8, 11	eqtri 2108	. . 3 |- ( A x. ( B + 1 ) ) = ( D + A )
13		4t3lem.5	. . 3 |- ( D + A ) = E
14	12, 13	eqtri 2108	. 2 |- ( A x. ( B + 1 ) ) = E
15	2, 14	eqtri 2108	1 |- ( A x. C ) = E

Syntax hints:   = wceq 1289   e. wcel 1438  (class class class)co 5634   1c1 7330   + caddc 7332   x. cmul 7334   NN0cn0 8643

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulcom 7425  ax-mulass 7427  ax-distr 7428  ax-1rid 7431  ax-rnegex 7433  ax-cnre 7435

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-iota 4967  df-fv 5010  df-ov 5637  df-inn 8395  df-n0 8644

This theorem is referenced by:  4t3e12  8943  4t4e16  8944  5t2e10  8945  5t3e15  8946  5t4e20  8947  5t5e25  8948  6t3e18  8950  6t4e24  8951  6t5e30  8952  6t6e36  8953  7t3e21  8955  7t4e28  8956  7t5e35  8957  7t6e42  8958  7t7e49  8959  8t3e24  8961  8t4e32  8962  8t5e40  8963  8t6e48  8964  8t7e56  8965  8t8e64  8966  9t3e27  8968  9t4e36  8969  9t5e45  8970  9t6e54  8971  9t7e63  8972  9t8e72  8973  9t9e81  8974