Theorem 4t3lem 9889

Description: Lemma for 4t3e12 9890 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 6099	. 2 |- ( A x. C ) = ( A x. ( B + 1 ) )
3		4t3lem.1	. . . . . 6 |- A e. NN0
4	3	nn0cni 9586	. . . . 5 |- A e. CC
5		4t3lem.2	. . . . . 6 |- B e. NN0
6	5	nn0cni 9586	. . . . 5 |- B e. CC
7		ax-1cn 8278	. . . . 5 |- 1 e. CC
8	4, 6, 7	adddii 8342	. . . 4 |- ( A x. ( B + 1 ) ) = ( ( A x. B ) + ( A x. 1 ) )
9		4t3lem.4	. . . . 5 |- ( A x. B ) = D
10	4	mulridi 8334	. . . . 5 |- ( A x. 1 ) = A
11	9, 10	oveq12i 6100	. . . 4 |- ( ( A x. B ) + ( A x. 1 ) ) = ( D + A )
12	8, 11	eqtri 2259	. . 3 |- ( A x. ( B + 1 ) ) = ( D + A )
13		4t3lem.5	. . 3 |- ( D + A ) = E
14	12, 13	eqtri 2259	. 2 |- ( A x. ( B + 1 ) ) = E
15	2, 14	eqtri 2259	1 |- ( A x. C ) = E

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209  (class class class)co 6088   1c1 8186   + caddc 8188   x. cmul 8190   NN0cn0 9574

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4251  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulcom 8286  ax-mulass 8288  ax-distr 8289  ax-1rid 8292  ax-rnegex 8294  ax-cnre 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-iota 5340  df-fv 5388  df-ov 6091  df-inn 9314  df-n0 9575

This theorem is used by:  4t3e12  9890  4t4e16  9891  5t2e10  9892  5t3e15  9893  5t4e20  9894  5t5e25  9895  6t3e18  9897  6t4e24  9898  6t5e30  9899  6t6e36  9900  7t3e21  9902  7t4e28  9903  7t5e35  9904  7t6e42  9905  7t7e49  9906  8t3e24  9908  8t4e32  9909  8t5e40  9910  8t6e48  9911  8t7e56  9912  8t8e64  9913  9t3e27  9915  9t4e36  9916  9t5e45  9917  9t6e54  9918  9t7e63  9919  9t8e72  9920  9t9e81  9921