Theorem decmul10add 9250

Description: A multiplication of a number and a numeral expressed as addition with first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)

Hypotheses

Ref	Expression
decmul10add.1	|- A e. NN0
decmul10add.2	|- B e. NN0
decmul10add.3	|- M e. NN0
decmul10add.4	|- E = ( M x. A )
decmul10add.5	|- F = ( M x. B )

Assertion

Ref	Expression
decmul10add	|- ( M x. ; A B ) = (; E 0 + F )

Proof of Theorem decmul10add

Step	Hyp	Ref	Expression
1		dfdec10 9185	. . 3 |- ; A B = ( (; 1 0 x. A ) + B )
2	1	oveq2i 5785	. 2 |- ( M x. ; A B ) = ( M x. ( (; 1 0 x. A ) + B ) )
3		decmul10add.3	. . . 4 |- M e. NN0
4	3	nn0cni 8989	. . 3 |- M e. CC
5		10nn0 9199	. . . . 5 |- ; 1 0 e. NN0
6		decmul10add.1	. . . . 5 |- A e. NN0
7	5, 6	nn0mulcli 9015	. . . 4 |- (; 1 0 x. A ) e. NN0
8	7	nn0cni 8989	. . 3 |- (; 1 0 x. A ) e. CC
9		decmul10add.2	. . . 4 |- B e. NN0
10	9	nn0cni 8989	. . 3 |- B e. CC
11	4, 8, 10	adddii 7776	. 2 |- ( M x. ( (; 1 0 x. A ) + B ) ) = ( ( M x. (; 1 0 x. A ) ) + ( M x. B ) )
12	5	nn0cni 8989	. . . . 5 |- ; 1 0 e. CC
13	6	nn0cni 8989	. . . . 5 |- A e. CC
14	4, 12, 13	mul12i 7908	. . . 4 |- ( M x. (; 1 0 x. A ) ) = (; 1 0 x. ( M x. A ) )
15	3, 6	nn0mulcli 9015	. . . . 5 |- ( M x. A ) e. NN0
16	15	dec0u 9202	. . . 4 |- (; 1 0 x. ( M x. A ) ) = ; ( M x. A ) 0
17		decmul10add.4	. . . . . 6 |- E = ( M x. A )
18	17	eqcomi 2143	. . . . 5 |- ( M x. A ) = E
19	18	deceq1i 9188	. . . 4 |- ; ( M x. A ) 0 = ; E 0
20	14, 16, 19	3eqtri 2164	. . 3 |- ( M x. (; 1 0 x. A ) ) = ; E 0
21		decmul10add.5	. . . 4 |- F = ( M x. B )
22	21	eqcomi 2143	. . 3 |- ( M x. B ) = F
23	20, 22	oveq12i 5786	. 2 |- ( ( M x. (; 1 0 x. A ) ) + ( M x. B ) ) = (; E 0 + F )
24	2, 11, 23	3eqtri 2164	1 |- ( M x. ; A B ) = (; E 0 + F )

Syntax hints:   = wceq 1331   e. wcel 1480  (class class class)co 5774   0cc0 7620   1c1 7621   + caddc 7623   x. cmul 7625   NN0cn0 8977  ;cdc 9182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7935  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-5 8782  df-6 8783  df-7 8784  df-8 8785  df-9 8786  df-n0 8978  df-dec 9183

This theorem is referenced by: (None)