Theorem decmul10add 9398

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 9333	. . 3 |- ; A B = ( (; 1 0 x. A ) + B )
2	1	oveq2i 5861	. 2 |- ( M x. ; A B ) = ( M x. ( (; 1 0 x. A ) + B ) )
3		decmul10add.3	. . . 4 |- M e. NN0
4	3	nn0cni 9134	. . 3 |- M e. CC
5		10nn0 9347	. . . . 5 |- ; 1 0 e. NN0
6		decmul10add.1	. . . . 5 |- A e. NN0
7	5, 6	nn0mulcli 9160	. . . 4 |- (; 1 0 x. A ) e. NN0
8	7	nn0cni 9134	. . 3 |- (; 1 0 x. A ) e. CC
9		decmul10add.2	. . . 4 |- B e. NN0
10	9	nn0cni 9134	. . 3 |- B e. CC
11	4, 8, 10	adddii 7917	. 2 |- ( M x. ( (; 1 0 x. A ) + B ) ) = ( ( M x. (; 1 0 x. A ) ) + ( M x. B ) )
12	5	nn0cni 9134	. . . . 5 |- ; 1 0 e. CC
13	6	nn0cni 9134	. . . . 5 |- A e. CC
14	4, 12, 13	mul12i 8052	. . . 4 |- ( M x. (; 1 0 x. A ) ) = (; 1 0 x. ( M x. A ) )
15	3, 6	nn0mulcli 9160	. . . . 5 |- ( M x. A ) e. NN0
16	15	dec0u 9350	. . . 4 |- (; 1 0 x. ( M x. A ) ) = ; ( M x. A ) 0
17		decmul10add.4	. . . . . 6 |- E = ( M x. A )
18	17	eqcomi 2174	. . . . 5 |- ( M x. A ) = E
19	18	deceq1i 9336	. . . 4 |- ; ( M x. A ) 0 = ; E 0
20	14, 16, 19	3eqtri 2195	. . 3 |- ( M x. (; 1 0 x. A ) ) = ; E 0
21		decmul10add.5	. . . 4 |- F = ( M x. B )
22	21	eqcomi 2174	. . 3 |- ( M x. B ) = F
23	20, 22	oveq12i 5862	. 2 |- ( ( M x. (; 1 0 x. A ) ) + ( M x. B ) ) = (; E 0 + F )
24	2, 11, 23	3eqtri 2195	1 |- ( M x. ; A B ) = (; E 0 + F )

Syntax hints:   = wceq 1348   e. wcel 2141  (class class class)co 5850   0cc0 7761   1c1 7762   + caddc 7764   x. cmul 7766   NN0cn0 9122  ;cdc 9330

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-cnre 7872

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-sub 8079  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-5 8927  df-6 8928  df-7 8929  df-8 8930  df-9 8931  df-n0 9123  df-dec 9331

This theorem is referenced by: (None)