Theorem nummul1c 9825

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 9789	. . . 4 |- ( ( T x. A ) + B ) e. NN0
6	1, 5	eqeltri 2311	. . 3 |- N e. NN0
7		nummul1c.2	. . 3 |- P e. NN0
8	6, 7	num0u 9787	. 2 |- ( N x. P ) = ( ( N x. P ) + 0 )
9		0nn0 9578	. . 3 |- 0 e. NN0
10	2, 9	num0h 9788	. . 3 |- 0 = ( ( T x. 0 ) + 0 )
11		nummul1c.6	. . 3 |- D e. NN0
12		nummul1c.7	. . 3 |- E e. NN0
13	12	nn0cni 9575	. . . . . 6 |- E e. CC
14	13	addlidi 8469	. . . . 5 |- ( 0 + E ) = E
15	14	oveq2i 6096	. . . 4 |- ( ( A x. P ) + ( 0 + E ) ) = ( ( A x. P ) + E )
16		nummul1c.8	. . . 4 |- ( ( A x. P ) + E ) = C
17	15, 16	eqtri 2259	. . 3 |- ( ( A x. P ) + ( 0 + E ) ) = C
18	4, 7	num0u 9787	. . . 4 |- ( B x. P ) = ( ( B x. P ) + 0 )
19		nummul1c.9	. . . 4 |- ( B x. P ) = ( ( T x. E ) + D )
20	18, 19	eqtr3i 2261	. . 3 |- ( ( B x. P ) + 0 ) = ( ( T x. E ) + D )
21	2, 3, 4, 9, 9, 1, 10, 7, 11, 12, 17, 20	nummac 9821	. 2 |- ( ( N x. P ) + 0 ) = ( ( T x. C ) + D )
22	8, 21	eqtri 2259	1 |- ( N x. P ) = ( ( T x. C ) + D )

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209  (class class class)co 6085   0cc0 8179   + caddc 8182   x. cmul 8184   NN0cn0 9563

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-sub 8499  df-inn 9305  df-n0 9564

This theorem is used by:  nummul2c  9826  decmul1  9840  decmul1c  9841