Theorem nummul1c 9522

Description: The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Hypotheses

Ref	Expression
nummul1c.1	|- T e. NN0
nummul1c.2	|- P e. NN0
nummul1c.3	|- A e. NN0
nummul1c.4	|- B e. NN0
nummul1c.5	|- N = ( ( T x. A ) + B )
nummul1c.6	|- D e. NN0
nummul1c.7	|- E e. NN0
nummul1c.8	|- ( ( A x. P ) + E ) = C
nummul1c.9	|- ( B x. P ) = ( ( T x. E ) + D )

Assertion

Ref	Expression
nummul1c	|- ( N x. P ) = ( ( T x. C ) + D )

Proof of Theorem nummul1c

Step	Hyp	Ref	Expression
1		nummul1c.5	. . . 4 |- N = ( ( T x. A ) + B )
2		nummul1c.1	. . . . 5 |- T e. NN0
3		nummul1c.3	. . . . 5 |- A e. NN0
4		nummul1c.4	. . . . 5 |- B e. NN0
5	2, 3, 4	numcl 9486	. . . 4 |- ( ( T x. A ) + B ) e. NN0
6	1, 5	eqeltri 2269	. . 3 |- N e. NN0
7		nummul1c.2	. . 3 |- P e. NN0
8	6, 7	num0u 9484	. 2 |- ( N x. P ) = ( ( N x. P ) + 0 )
9		0nn0 9281	. . 3 |- 0 e. NN0
10	2, 9	num0h 9485	. . 3 |- 0 = ( ( T x. 0 ) + 0 )
11		nummul1c.6	. . 3 |- D e. NN0
12		nummul1c.7	. . 3 |- E e. NN0
13	12	nn0cni 9278	. . . . . 6 |- E e. CC
14	13	addlidi 8186	. . . . 5 |- ( 0 + E ) = E
15	14	oveq2i 5936	. . . 4 |- ( ( A x. P ) + ( 0 + E ) ) = ( ( A x. P ) + E )
16		nummul1c.8	. . . 4 |- ( ( A x. P ) + E ) = C
17	15, 16	eqtri 2217	. . 3 |- ( ( A x. P ) + ( 0 + E ) ) = C
18	4, 7	num0u 9484	. . . 4 |- ( B x. P ) = ( ( B x. P ) + 0 )
19		nummul1c.9	. . . 4 |- ( B x. P ) = ( ( T x. E ) + D )
20	18, 19	eqtr3i 2219	. . 3 |- ( ( B x. P ) + 0 ) = ( ( T x. E ) + D )
21	2, 3, 4, 9, 9, 1, 10, 7, 11, 12, 17, 20	nummac 9518	. 2 |- ( ( N x. P ) + 0 ) = ( ( T x. C ) + D )
22	8, 21	eqtri 2217	1 |- ( N x. P ) = ( ( T x. C ) + D )

Syntax hints:   = wceq 1364   e. wcel 2167  (class class class)co 5925   0cc0 7896   + caddc 7899   x. cmul 7901   NN0cn0 9266

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-sub 8216  df-inn 9008  df-n0 9267

This theorem is referenced by:  nummul2c  9523  decmul1  9537  decmul1c  9538