Theorem decmul10add 9454

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 9389	. . 3 |- ; A B = ( (; 1 0 x. A ) + B )
2	1	oveq2i 5888	. 2 |- ( M x. ; A B ) = ( M x. ( (; 1 0 x. A ) + B ) )
3		decmul10add.3	. . . 4 |- M e. NN0
4	3	nn0cni 9190	. . 3 |- M e. CC
5		10nn0 9403	. . . . 5 |- ; 1 0 e. NN0
6		decmul10add.1	. . . . 5 |- A e. NN0
7	5, 6	nn0mulcli 9216	. . . 4 |- (; 1 0 x. A ) e. NN0
8	7	nn0cni 9190	. . 3 |- (; 1 0 x. A ) e. CC
9		decmul10add.2	. . . 4 |- B e. NN0
10	9	nn0cni 9190	. . 3 |- B e. CC
11	4, 8, 10	adddii 7969	. 2 |- ( M x. ( (; 1 0 x. A ) + B ) ) = ( ( M x. (; 1 0 x. A ) ) + ( M x. B ) )
12	5	nn0cni 9190	. . . . 5 |- ; 1 0 e. CC
13	6	nn0cni 9190	. . . . 5 |- A e. CC
14	4, 12, 13	mul12i 8105	. . . 4 |- ( M x. (; 1 0 x. A ) ) = (; 1 0 x. ( M x. A ) )
15	3, 6	nn0mulcli 9216	. . . . 5 |- ( M x. A ) e. NN0
16	15	dec0u 9406	. . . 4 |- (; 1 0 x. ( M x. A ) ) = ; ( M x. A ) 0
17		decmul10add.4	. . . . . 6 |- E = ( M x. A )
18	17	eqcomi 2181	. . . . 5 |- ( M x. A ) = E
19	18	deceq1i 9392	. . . 4 |- ; ( M x. A ) 0 = ; E 0
20	14, 16, 19	3eqtri 2202	. . 3 |- ( M x. (; 1 0 x. A ) ) = ; E 0
21		decmul10add.5	. . . 4 |- F = ( M x. B )
22	21	eqcomi 2181	. . 3 |- ( M x. B ) = F
23	20, 22	oveq12i 5889	. 2 |- ( ( M x. (; 1 0 x. A ) ) + ( M x. B ) ) = (; E 0 + F )
24	2, 11, 23	3eqtri 2202	1 |- ( M x. ; A B ) = (; E 0 + F )

Syntax hints:   = wceq 1353   e. wcel 2148  (class class class)co 5877   0cc0 7813   1c1 7814   + caddc 7816   x. cmul 7818   NN0cn0 9178  ;cdc 9386

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-sub 8132  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-6 8984  df-7 8985  df-8 8986  df-9 8987  df-n0 9179  df-dec 9387

This theorem is referenced by: (None)