Theorem decrmac 9635

Description: Perform a multiply-add of two numerals M and N against a fixed multiplicand P (with carry). (Contributed by AV, 16-Sep-2021.)

Hypotheses

Ref	Expression
decrmanc.a	|- A e. NN0
decrmanc.b	|- B e. NN0
decrmanc.n	|- N e. NN0
decrmanc.m	|- M = ; A B
decrmanc.p	|- P e. NN0
decrmac.f	|- F e. NN0
decrmac.g	|- G e. NN0
decrmac.e	|- ( ( A x. P ) + G ) = E
decrmac.2	|- ( ( B x. P ) + N ) = ; G F

Assertion

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

Proof of Theorem decrmac

Step	Hyp	Ref	Expression
1		decrmanc.a	. 2 |- A e. NN0
2		decrmanc.b	. 2 |- B e. NN0
3		0nn0 9384	. 2 |- 0 e. NN0
4		decrmanc.n	. 2 |- N e. NN0
5		decrmanc.m	. 2 |- M = ; A B
6	4	dec0h 9599	. 2 |- N = ; 0 N
7		decrmanc.p	. 2 |- P e. NN0
8		decrmac.f	. 2 |- F e. NN0
9		decrmac.g	. 2 |- G e. NN0
10	9	nn0cni 9381	. . . . 5 |- G e. CC
11	10	addlidi 8289	. . . 4 |- ( 0 + G ) = G
12	11	oveq2i 6012	. . 3 |- ( ( A x. P ) + ( 0 + G ) ) = ( ( A x. P ) + G )
13		decrmac.e	. . 3 |- ( ( A x. P ) + G ) = E
14	12, 13	eqtri 2250	. 2 |- ( ( A x. P ) + ( 0 + G ) ) = E
15		decrmac.2	. 2 |- ( ( B x. P ) + N ) = ; G F
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15	decmac 9629	1 |- ( ( M x. P ) + N ) = ; E F

Syntax hints:   = wceq 1395   e. wcel 2200  (class class class)co 6001   0cc0 7999   + caddc 8002   x. cmul 8004   NN0cn0 9369  ;cdc 9578

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 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110

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 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-sub 8319  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176  df-n0 9370  df-dec 9579

This theorem is referenced by:  2exp16  12960