Theorem 4t3lem 9547

Description: Lemma for 4t3e12 9548 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 5930	. 2 |- ( A x. C ) = ( A x. ( B + 1 ) )
3		4t3lem.1	. . . . . 6 |- A e. NN0
4	3	nn0cni 9255	. . . . 5 |- A e. CC
5		4t3lem.2	. . . . . 6 |- B e. NN0
6	5	nn0cni 9255	. . . . 5 |- B e. CC
7		ax-1cn 7967	. . . . 5 |- 1 e. CC
8	4, 6, 7	adddii 8031	. . . 4 |- ( A x. ( B + 1 ) ) = ( ( A x. B ) + ( A x. 1 ) )
9		4t3lem.4	. . . . 5 |- ( A x. B ) = D
10	4	mulid1i 8023	. . . . 5 |- ( A x. 1 ) = A
11	9, 10	oveq12i 5931	. . . 4 |- ( ( A x. B ) + ( A x. 1 ) ) = ( D + A )
12	8, 11	eqtri 2214	. . 3 |- ( A x. ( B + 1 ) ) = ( D + A )
13		4t3lem.5	. . 3 |- ( D + A ) = E
14	12, 13	eqtri 2214	. 2 |- ( A x. ( B + 1 ) ) = E
15	2, 14	eqtri 2214	1 |- ( A x. C ) = E

Syntax hints:   = wceq 1364   e. wcel 2164  (class class class)co 5919   1c1 7875   + caddc 7877   x. cmul 7879   NN0cn0 9243

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-sep 4148  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulcom 7975  ax-mulass 7977  ax-distr 7978  ax-1rid 7981  ax-rnegex 7983  ax-cnre 7985

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-iota 5216  df-fv 5263  df-ov 5922  df-inn 8985  df-n0 9244

This theorem is referenced by:  4t3e12  9548  4t4e16  9549  5t2e10  9550  5t3e15  9551  5t4e20  9552  5t5e25  9553  6t3e18  9555  6t4e24  9556  6t5e30  9557  6t6e36  9558  7t3e21  9560  7t4e28  9561  7t5e35  9562  7t6e42  9563  7t7e49  9564  8t3e24  9566  8t4e32  9567  8t5e40  9568  8t6e48  9569  8t7e56  9570  8t8e64  9571  9t3e27  9573  9t4e36  9574  9t5e45  9575  9t6e54  9576  9t7e63  9577  9t8e72  9578  9t9e81  9579