Theorem decmul10add 9102

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 9037	. . 3 |- ; A B = ( (; 1 0 x. A ) + B )
2	1	oveq2i 5717	. 2 |- ( M x. ; A B ) = ( M x. ( (; 1 0 x. A ) + B ) )
3		decmul10add.3	. . . 4 |- M e. NN0
4	3	nn0cni 8841	. . 3 |- M e. CC
5		10nn0 9051	. . . . 5 |- ; 1 0 e. NN0
6		decmul10add.1	. . . . 5 |- A e. NN0
7	5, 6	nn0mulcli 8867	. . . 4 |- (; 1 0 x. A ) e. NN0
8	7	nn0cni 8841	. . 3 |- (; 1 0 x. A ) e. CC
9		decmul10add.2	. . . 4 |- B e. NN0
10	9	nn0cni 8841	. . 3 |- B e. CC
11	4, 8, 10	adddii 7648	. 2 |- ( M x. ( (; 1 0 x. A ) + B ) ) = ( ( M x. (; 1 0 x. A ) ) + ( M x. B ) )
12	5	nn0cni 8841	. . . . 5 |- ; 1 0 e. CC
13	6	nn0cni 8841	. . . . 5 |- A e. CC
14	4, 12, 13	mul12i 7779	. . . 4 |- ( M x. (; 1 0 x. A ) ) = (; 1 0 x. ( M x. A ) )
15	3, 6	nn0mulcli 8867	. . . . 5 |- ( M x. A ) e. NN0
16	15	dec0u 9054	. . . 4 |- (; 1 0 x. ( M x. A ) ) = ; ( M x. A ) 0
17		decmul10add.4	. . . . . 6 |- E = ( M x. A )
18	17	eqcomi 2104	. . . . 5 |- ( M x. A ) = E
19	18	deceq1i 9040	. . . 4 |- ; ( M x. A ) 0 = ; E 0
20	14, 16, 19	3eqtri 2124	. . 3 |- ( M x. (; 1 0 x. A ) ) = ; E 0
21		decmul10add.5	. . . 4 |- F = ( M x. B )
22	21	eqcomi 2104	. . 3 |- ( M x. B ) = F
23	20, 22	oveq12i 5718	. 2 |- ( ( M x. (; 1 0 x. A ) ) + ( M x. B ) ) = (; E 0 + F )
24	2, 11, 23	3eqtri 2124	1 |- ( M x. ; A B ) = (; E 0 + F )

Syntax hints:   = wceq 1299   e. wcel 1448  (class class class)co 5706   0cc0 7500   1c1 7501   + caddc 7503   x. cmul 7505   NN0cn0 8829  ;cdc 9034

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-sub 7806  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-5 8640  df-6 8641  df-7 8642  df-8 8643  df-9 8644  df-n0 8830  df-dec 9035

This theorem is referenced by: (None)