Theorem nummul1c 9658

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 9622	. . . 4 |- ( ( T x. A ) + B ) e. NN0
6	1, 5	eqeltri 2304	. . 3 |- N e. NN0
7		nummul1c.2	. . 3 |- P e. NN0
8	6, 7	num0u 9620	. 2 |- ( N x. P ) = ( ( N x. P ) + 0 )
9		0nn0 9416	. . 3 |- 0 e. NN0
10	2, 9	num0h 9621	. . 3 |- 0 = ( ( T x. 0 ) + 0 )
11		nummul1c.6	. . 3 |- D e. NN0
12		nummul1c.7	. . 3 |- E e. NN0
13	12	nn0cni 9413	. . . . . 6 |- E e. CC
14	13	addlidi 8321	. . . . 5 |- ( 0 + E ) = E
15	14	oveq2i 6028	. . . 4 |- ( ( A x. P ) + ( 0 + E ) ) = ( ( A x. P ) + E )
16		nummul1c.8	. . . 4 |- ( ( A x. P ) + E ) = C
17	15, 16	eqtri 2252	. . 3 |- ( ( A x. P ) + ( 0 + E ) ) = C
18	4, 7	num0u 9620	. . . 4 |- ( B x. P ) = ( ( B x. P ) + 0 )
19		nummul1c.9	. . . 4 |- ( B x. P ) = ( ( T x. E ) + D )
20	18, 19	eqtr3i 2254	. . 3 |- ( ( B x. P ) + 0 ) = ( ( T x. E ) + D )
21	2, 3, 4, 9, 9, 1, 10, 7, 11, 12, 17, 20	nummac 9654	. 2 |- ( ( N x. P ) + 0 ) = ( ( T x. C ) + D )
22	8, 21	eqtri 2252	1 |- ( N x. P ) = ( ( T x. C ) + D )

Syntax hints:   = wceq 1397   e. wcel 2202  (class class class)co 6017   0cc0 8031   + caddc 8034   x. cmul 8036   NN0cn0 9401

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-sub 8351  df-inn 9143  df-n0 9402

This theorem is referenced by:  nummul2c  9659  decmul1  9673  decmul1c  9674