Theorem decrmac 9816

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 9560	. 2 |- 0 e. NN0
4		decrmanc.n	. 2 |- N e. NN0
5		decrmanc.m	. 2 |- M = ; A B
6	4	dec0h 9780	. 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 9557	. . . . 5 |- G e. CC
11	10	addlidi 8462	. . . 4 |- ( 0 + G ) = G
12	11	oveq2i 6089	. . 3 |- ( ( A x. P ) + ( 0 + G ) ) = ( ( A x. P ) + G )
13		decrmac.e	. . 3 |- ( ( A x. P ) + G ) = E
14	12, 13	eqtri 2259	. 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 9810	1 |- ( ( M x. P ) + N ) = ; E F

Syntax hints:   = wceq 1402   e. wcel 2209  (class class class)co 6078   0cc0 8172   + caddc 8175   x. cmul 8177   NN0cn0 9545  ;cdc 9759

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 4247  ax-pow 4309  ax-pr 4344  ax-setind 4682  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-addcom 8272  ax-mulcom 8273  ax-addass 8274  ax-mulass 8275  ax-distr 8276  ax-i2m1 8277  ax-1rid 8279  ax-0id 8280  ax-rnegex 8281  ax-cnre 8283

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 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fv 5383  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-sub 8492  df-inn 9287  df-2 9345  df-3 9346  df-4 9347  df-5 9348  df-6 9349  df-7 9350  df-8 9351  df-9 9352  df-n0 9546  df-dec 9760

This theorem is referenced by:  2exp16  13197