Theorem decmul1 9718

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 9672	. . 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 9658	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
7	5, 6	eqtri 2252	. . 3 |- N = ( (; 1 0 x. A ) + B )
8		decmul1.0	. . 3 |- D e. NN0
9		0nn0 9459	. . 3 |- 0 e. NN0
10	3, 2	nn0mulcli 9482	. . . . . 6 |- ( A x. P ) e. NN0
11	10	nn0cni 9456	. . . . 5 |- ( A x. P ) e. CC
12	11	addridi 8363	. . . 4 |- ( ( A x. P ) + 0 ) = ( A x. P )
13		decmul1.c	. . . 4 |- ( A x. P ) = C
14	12, 13	eqtri 2252	. . 3 |- ( ( A x. P ) + 0 ) = C
15		decmul1.d	. . . . 5 |- ( B x. P ) = D
16	15	oveq2i 6039	. . . 4 |- ( 0 + ( B x. P ) ) = ( 0 + D )
17	4, 2	nn0mulcli 9482	. . . . . 6 |- ( B x. P ) e. NN0
18	17	nn0cni 9456	. . . . 5 |- ( B x. P ) e. CC
19	18	addlidi 8364	. . . 4 |- ( 0 + ( B x. P ) ) = ( B x. P )
20	1	nn0cni 9456	. . . . . . 7 |- ; 1 0 e. CC
21	20	mul01i 8612	. . . . . 6 |- (; 1 0 x. 0 ) = 0
22	21	eqcomi 2235	. . . . 5 |- 0 = (; 1 0 x. 0 )
23	22	oveq1i 6038	. . . 4 |- ( 0 + D ) = ( (; 1 0 x. 0 ) + D )
24	16, 19, 23	3eqtr3i 2260	. . 3 |- ( B x. P ) = ( (; 1 0 x. 0 ) + D )
25	1, 2, 3, 4, 7, 8, 9, 14, 24	nummul1c 9703	. 2 |- ( N x. P ) = ( (; 1 0 x. C ) + D )
26		dfdec10 9658	. 2 |- ; C D = ( (; 1 0 x. C ) + D )
27	25, 26	eqtr4i 2255	1 |- ( N x. P ) = ; C D

Syntax hints:   = wceq 1398   e. wcel 2202  (class class class)co 6028   0cc0 8075   1c1 8076   + caddc 8078   x. cmul 8080   NN0cn0 9444  ;cdc 9655

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-sub 8394  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-dec 9656

This theorem is referenced by:  sq10  11020  2exp7  13070