Theorem decmul1 9819

Description: The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)

Hypotheses

Ref	Expression
decmul1.p	|- P e. NN0
decmul1.a	|- A e. NN0
decmul1.b	|- B e. NN0
decmul1.n	|- N = ; A B
decmul1.0	|- D e. NN0
decmul1.c	|- ( A x. P ) = C
decmul1.d	|- ( B x. P ) = D

Assertion

Ref	Expression
decmul1	|- ( N x. P ) = ; C D

Proof of Theorem decmul1

Step	Hyp	Ref	Expression
1		10nn0 9773	. . 3 |- ; 1 0 e. NN0
2		decmul1.p	. . 3 |- P e. NN0
3		decmul1.a	. . 3 |- A e. NN0
4		decmul1.b	. . 3 |- B e. NN0
5		decmul1.n	. . . 4 |- N = ; A B
6		dfdec10 9759	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
7	5, 6	eqtri 2259	. . 3 |- N = ( (; 1 0 x. A ) + B )
8		decmul1.0	. . 3 |- D e. NN0
9		0nn0 9557	. . 3 |- 0 e. NN0
10	3, 2	nn0mulcli 9580	. . . . . 6 |- ( A x. P ) e. NN0
11	10	nn0cni 9554	. . . . 5 |- ( A x. P ) e. CC
12	11	addridi 8458	. . . 4 |- ( ( A x. P ) + 0 ) = ( A x. P )
13		decmul1.c	. . . 4 |- ( A x. P ) = C
14	12, 13	eqtri 2259	. . 3 |- ( ( A x. P ) + 0 ) = C
15		decmul1.d	. . . . 5 |- ( B x. P ) = D
16	15	oveq2i 6086	. . . 4 |- ( 0 + ( B x. P ) ) = ( 0 + D )
17	4, 2	nn0mulcli 9580	. . . . . 6 |- ( B x. P ) e. NN0
18	17	nn0cni 9554	. . . . 5 |- ( B x. P ) e. CC
19	18	addlidi 8459	. . . 4 |- ( 0 + ( B x. P ) ) = ( B x. P )
20	1	nn0cni 9554	. . . . . . 7 |- ; 1 0 e. CC
21	20	mul01i 8708	. . . . . 6 |- (; 1 0 x. 0 ) = 0
22	21	eqcomi 2242	. . . . 5 |- 0 = (; 1 0 x. 0 )
23	22	oveq1i 6085	. . . 4 |- ( 0 + D ) = ( (; 1 0 x. 0 ) + D )
24	16, 19, 23	3eqtr3i 2267	. . 3 |- ( B x. P ) = ( (; 1 0 x. 0 ) + D )
25	1, 2, 3, 4, 7, 8, 9, 14, 24	nummul1c 9804	. 2 |- ( N x. P ) = ( (; 1 0 x. C ) + D )
26		dfdec10 9759	. 2 |- ; C D = ( (; 1 0 x. C ) + D )
27	25, 26	eqtr4i 2262	1 |- ( N x. P ) = ; C D

Syntax hints:   = wceq 1402   e. wcel 2209  (class class class)co 6075   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   NN0cn0 9542  ;cdc 9756

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 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757

This theorem is referenced by:  sq10  11128  2exp7  13191