Theorem decma2c 9761

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 9726	. . 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 9712	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
8	6, 7	eqtri 2253	. . 3 |- M = ( (; 1 0 x. A ) + B )
9		decma.n	. . . 4 |- N = ; C D
10		dfdec10 9712	. . . 4 |- ; C D = ( (; 1 0 x. C ) + D )
11	9, 10	eqtri 2253	. . 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 9712	. . . 4 |- ; G F = ( (; 1 0 x. G ) + F )
18	16, 17	eqtri 2253	. . 3 |- ( ( P x. B ) + D ) = ( (; 1 0 x. G ) + F )
19	1, 2, 3, 4, 5, 8, 11, 12, 13, 14, 15, 18	numma2c 9754	. 2 |- ( ( P x. M ) + N ) = ( (; 1 0 x. E ) + F )
20		dfdec10 9712	. 2 |- ; E F = ( (; 1 0 x. E ) + F )
21	19, 20	eqtr4i 2256	1 |- ( ( P x. M ) + N ) = ; E F

Syntax hints:   = wceq 1398   e. wcel 2203  (class class class)co 6050   0cc0 8127   1c1 8128   + caddc 8130   x. cmul 8132   NN0cn0 9496  ;cdc 9709

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-sub 8446  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-dec 9710

This theorem is referenced by:  2exp16  13135