Theorem 4t3lem 9768

Description: Lemma for 4t3e12 9769 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 6039	. 2 |- ( A x. C ) = ( A x. ( B + 1 ) )
3		4t3lem.1	. . . . . 6 |- A e. NN0
4	3	nn0cni 9473	. . . . 5 |- A e. CC
5		4t3lem.2	. . . . . 6 |- B e. NN0
6	5	nn0cni 9473	. . . . 5 |- B e. CC
7		ax-1cn 8185	. . . . 5 |- 1 e. CC
8	4, 6, 7	adddii 8249	. . . 4 |- ( A x. ( B + 1 ) ) = ( ( A x. B ) + ( A x. 1 ) )
9		4t3lem.4	. . . . 5 |- ( A x. B ) = D
10	4	mulridi 8241	. . . . 5 |- ( A x. 1 ) = A
11	9, 10	oveq12i 6040	. . . 4 |- ( ( A x. B ) + ( A x. 1 ) ) = ( D + A )
12	8, 11	eqtri 2252	. . 3 |- ( A x. ( B + 1 ) ) = ( D + A )
13		4t3lem.5	. . 3 |- ( D + A ) = E
14	12, 13	eqtri 2252	. 2 |- ( A x. ( B + 1 ) ) = E
15	2, 14	eqtri 2252	1 |- ( A x. C ) = E

Syntax hints:   = wceq 1398   e. wcel 2202  (class class class)co 6028   1c1 8093   + caddc 8095   x. cmul 8097   NN0cn0 9461

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-sep 4212  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulcom 8193  ax-mulass 8195  ax-distr 8196  ax-1rid 8199  ax-rnegex 8201  ax-cnre 8203

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031  df-inn 9203  df-n0 9462

This theorem is referenced by:  4t3e12  9769  4t4e16  9770  5t2e10  9771  5t3e15  9772  5t4e20  9773  5t5e25  9774  6t3e18  9776  6t4e24  9777  6t5e30  9778  6t6e36  9779  7t3e21  9781  7t4e28  9782  7t5e35  9783  7t6e42  9784  7t7e49  9785  8t3e24  9787  8t4e32  9788  8t5e40  9789  8t6e48  9790  8t7e56  9791  8t8e64  9792  9t3e27  9794  9t4e36  9795  9t5e45  9796  9t6e54  9797  9t7e63  9798  9t8e72  9799  9t9e81  9800