Theorem decma2c 9641

Description: Perform a multiply-add of two numerals M and N against a fixed multiplier 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
decma2c.p	|- P e. NN0
decma2c.f	|- F e. NN0
decma2c.g	|- G e. NN0
decma2c.e	|- ( ( P x. A ) + ( C + G ) ) = E
decma2c.2	|- ( ( P x. B ) + D ) = ; G F

Assertion

Ref	Expression
decma2c	|- ( ( P x. M ) + N ) = ; E F

Proof of Theorem decma2c

Step	Hyp	Ref	Expression
1		10nn0 9606	. . 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 9592	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
8	6, 7	eqtri 2250	. . 3 |- M = ( (; 1 0 x. A ) + B )
9		decma.n	. . . 4 |- N = ; C D
10		dfdec10 9592	. . . 4 |- ; C D = ( (; 1 0 x. C ) + D )
11	9, 10	eqtri 2250	. . 3 |- N = ( (; 1 0 x. C ) + D )
12		decma2c.p	. . 3 |- P e. NN0
13		decma2c.f	. . 3 |- F e. NN0
14		decma2c.g	. . 3 |- G e. NN0
15		decma2c.e	. . 3 |- ( ( P x. A ) + ( C + G ) ) = E
16		decma2c.2	. . . 4 |- ( ( P x. B ) + D ) = ; G F
17		dfdec10 9592	. . . 4 |- ; G F = ( (; 1 0 x. G ) + F )
18	16, 17	eqtri 2250	. . 3 |- ( ( P x. B ) + D ) = ( (; 1 0 x. G ) + F )
19	1, 2, 3, 4, 5, 8, 11, 12, 13, 14, 15, 18	numma2c 9634	. 2 |- ( ( P x. M ) + N ) = ( (; 1 0 x. E ) + F )
20		dfdec10 9592	. 2 |- ; E F = ( (; 1 0 x. E ) + F )
21	19, 20	eqtr4i 2253	1 |- ( ( P x. M ) + N ) = ; E F

Syntax hints:   = wceq 1395   e. wcel 2200  (class class class)co 6007   0cc0 8010   1c1 8011   + caddc 8013   x. cmul 8015   NN0cn0 9380  ;cdc 9589

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-sub 8330  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-dec 9590

This theorem is referenced by:  2exp16  12975