Theorem decmac 9763

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 9729	. . 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 9715	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
8	6, 7	eqtri 2255	. . 3 |- M = ( (; 1 0 x. A ) + B )
9		decma.n	. . . 4 |- N = ; C D
10		dfdec10 9715	. . . 4 |- ; C D = ( (; 1 0 x. C ) + D )
11	9, 10	eqtri 2255	. . 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 9715	. . . 4 |- ; G F = ( (; 1 0 x. G ) + F )
18	16, 17	eqtri 2255	. . 3 |- ( ( B x. P ) + D ) = ( (; 1 0 x. G ) + F )
19	1, 2, 3, 4, 5, 8, 11, 12, 13, 14, 15, 18	nummac 9756	. 2 |- ( ( M x. P ) + N ) = ( (; 1 0 x. E ) + F )
20		dfdec10 9715	. 2 |- ; E F = ( (; 1 0 x. E ) + F )
21	19, 20	eqtr4i 2258	1 |- ( ( M x. P ) + N ) = ; E F

Syntax hints:   = wceq 1398   e. wcel 2205  (class class class)co 6052   0cc0 8129   1c1 8130   + caddc 8132   x. cmul 8134   NN0cn0 9498  ;cdc 9712

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-sub 8448  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-dec 9713

This theorem is referenced by:  decrmac  9769  2exp16  13139