Theorem decmac 9807

Description: Perform a multiply-add of two numerals M and N against a fixed multiplicand P (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)

Hypotheses

Ref	Expression
decma.a	|- A e. NN0
decma.b	|- B e. NN0
decma.c	|- C e. NN0
decma.d	|- D e. NN0
decma.m	|- M = ; A B
decma.n	|- N = ; C D
decmac.p	|- P e. NN0
decmac.f	|- F e. NN0
decmac.g	|- G e. NN0
decmac.e	|- ( ( A x. P ) + ( C + G ) ) = E
decmac.2	|- ( ( B x. P ) + D ) = ; G F

Assertion

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

Proof of Theorem decmac

Step	Hyp	Ref	Expression
1		10nn0 9773	. . 3 |- ; 1 0 e. NN0
2		decma.a	. . 3 |- A e. NN0
3		decma.b	. . 3 |- B e. NN0
4		decma.c	. . 3 |- C e. NN0
5		decma.d	. . 3 |- D e. NN0
6		decma.m	. . . 4 |- M = ; A B
7		dfdec10 9759	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
8	6, 7	eqtri 2259	. . 3 |- M = ( (; 1 0 x. A ) + B )
9		decma.n	. . . 4 |- N = ; C D
10		dfdec10 9759	. . . 4 |- ; C D = ( (; 1 0 x. C ) + D )
11	9, 10	eqtri 2259	. . 3 |- N = ( (; 1 0 x. C ) + D )
12		decmac.p	. . 3 |- P e. NN0
13		decmac.f	. . 3 |- F e. NN0
14		decmac.g	. . 3 |- G e. NN0
15		decmac.e	. . 3 |- ( ( A x. P ) + ( C + G ) ) = E
16		decmac.2	. . . 4 |- ( ( B x. P ) + D ) = ; G F
17		dfdec10 9759	. . . 4 |- ; G F = ( (; 1 0 x. G ) + F )
18	16, 17	eqtri 2259	. . 3 |- ( ( B x. P ) + D ) = ( (; 1 0 x. G ) + F )
19	1, 2, 3, 4, 5, 8, 11, 12, 13, 14, 15, 18	nummac 9800	. 2 |- ( ( M x. P ) + N ) = ( (; 1 0 x. E ) + F )
20		dfdec10 9759	. 2 |- ; E F = ( (; 1 0 x. E ) + F )
21	19, 20	eqtr4i 2262	1 |- ( ( M x. P ) + N ) = ; E F

Syntax hints:   = wceq 1402   e. wcel 2209  (class class class)co 6075   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   NN0cn0 9542  ;cdc 9756

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-in1 623  ax-in2 624  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-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757

This theorem is referenced by:  decrmac  9813  2exp16  13194