Theorem decma 8405

Description: Perform a multiply-add of two numerals M and N against a fixed multiplicand P (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Hypotheses

Ref	Expression
decma.1	|- A e. NN0
decma.2	|- B e. NN0
decma.3	|- C e. NN0
decma.4	|- D e. NN0
decma.5	|- M = ; A B
decma.6	|- N = ; C D
decma.7	|- P e. NN0
decma.8	|- ( ( A x. P ) + C ) = E
decma.9	|- ( ( B x. P ) + D ) = F

Assertion

Ref	Expression
decma	|- ( ( M x. P ) + N ) = ; E F

Proof of Theorem decma

Step	Hyp	Ref	Expression
1		10nn0 8206	. . 3 |- 10 e. NN0
2		decma.1	. . 3 |- A e. NN0
3		decma.2	. . 3 |- B e. NN0
4		decma.3	. . 3 |- C e. NN0
5		decma.4	. . 3 |- D e. NN0
6		decma.5	. . . 4 |- M = ; A B
7		df-dec 8369	. . . 4 |- ; A B = ( ( 10 x. A ) + B )
8	6, 7	eqtri 2060	. . 3 |- M = ( ( 10 x. A ) + B )
9		decma.6	. . . 4 |- N = ; C D
10		df-dec 8369	. . . 4 |- ; C D = ( ( 10 x. C ) + D )
11	9, 10	eqtri 2060	. . 3 |- N = ( ( 10 x. C ) + D )
12		decma.7	. . 3 |- P e. NN0
13		decma.8	. . 3 |- ( ( A x. P ) + C ) = E
14		decma.9	. . 3 |- ( ( B x. P ) + D ) = F
15	1, 2, 3, 4, 5, 8, 11, 12, 13, 14	numma 8398	. 2 |- ( ( M x. P ) + N ) = ( ( 10 x. E ) + F )
16		df-dec 8369	. 2 |- ; E F = ( ( 10 x. E ) + F )
17	15, 16	eqtr4i 2063	1 |- ( ( M x. P ) + N ) = ; E F

Syntax hints:   = wceq 1243   e. wcel 1393  (class class class)co 5512   + caddc 6892   x. cmul 6894   10c10 7972   NN0cn0 8181  ;cdc 8368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-cnex 6975  ax-resscn 6976  ax-1re 6978  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-rnegex 6993

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515  df-inn 7915  df-2 7973  df-3 7974  df-4 7975  df-5 7976  df-6 7977  df-7 7978  df-8 7979  df-9 7980  df-10 7981  df-n0 8182  df-dec 8369

This theorem is referenced by: (None)