Theorem 4t3lem 9544

Description: Lemma for 4t3e12 9545 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 5929	. 2 |- ( A x. C ) = ( A x. ( B + 1 ) )
3		4t3lem.1	. . . . . 6 |- A e. NN0
4	3	nn0cni 9252	. . . . 5 |- A e. CC
5		4t3lem.2	. . . . . 6 |- B e. NN0
6	5	nn0cni 9252	. . . . 5 |- B e. CC
7		ax-1cn 7965	. . . . 5 |- 1 e. CC
8	4, 6, 7	adddii 8029	. . . 4 |- ( A x. ( B + 1 ) ) = ( ( A x. B ) + ( A x. 1 ) )
9		4t3lem.4	. . . . 5 |- ( A x. B ) = D
10	4	mulid1i 8021	. . . . 5 |- ( A x. 1 ) = A
11	9, 10	oveq12i 5930	. . . 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 5918   1c1 7873   + caddc 7875   x. cmul 7877   NN0cn0 9240

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 4147  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulcom 7973  ax-mulass 7975  ax-distr 7976  ax-1rid 7979  ax-rnegex 7981  ax-cnre 7983

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 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921  df-inn 8983  df-n0 9241

This theorem is referenced by:  4t3e12  9545  4t4e16  9546  5t2e10  9547  5t3e15  9548  5t4e20  9549  5t5e25  9550  6t3e18  9552  6t4e24  9553  6t5e30  9554  6t6e36  9555  7t3e21  9557  7t4e28  9558  7t5e35  9559  7t6e42  9560  7t7e49  9561  8t3e24  9563  8t4e32  9564  8t5e40  9565  8t6e48  9566  8t7e56  9567  8t8e64  9568  9t3e27  9570  9t4e36  9571  9t5e45  9572  9t6e54  9573  9t7e63  9574  9t8e72  9575  9t9e81  9576