Theorem nummul2c 9776

Description: The product of a decimal integer with a number (with carry). (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
nummul2c.7	|- ( ( P x. A ) + E ) = C
nummul2c.8	|- ( P x. B ) = ( ( T x. E ) + D )

Assertion

Ref	Expression
nummul2c	|- ( P x. N ) = ( ( T x. C ) + D )

Proof of Theorem nummul2c

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 9739	. . . 4 |- ( ( T x. A ) + B ) e. NN0
6	1, 5	eqeltri 2307	. . 3 |- N e. NN0
7	6	nn0cni 9525	. 2 |- N e. CC
8		nummul1c.2	. . 3 |- P e. NN0
9	8	nn0cni 9525	. 2 |- P e. CC
10		nummul1c.6	. . 3 |- D e. NN0
11		nummul1c.7	. . 3 |- E e. NN0
12	3	nn0cni 9525	. . . . . 6 |- A e. CC
13	12, 9	mulcomi 8296	. . . . 5 |- ( A x. P ) = ( P x. A )
14	13	oveq1i 6068	. . . 4 |- ( ( A x. P ) + E ) = ( ( P x. A ) + E )
15		nummul2c.7	. . . 4 |- ( ( P x. A ) + E ) = C
16	14, 15	eqtri 2255	. . 3 |- ( ( A x. P ) + E ) = C
17	4	nn0cni 9525	. . . 4 |- B e. CC
18		nummul2c.8	. . . 4 |- ( P x. B ) = ( ( T x. E ) + D )
19	9, 17, 18	mulcomli 8297	. . 3 |- ( B x. P ) = ( ( T x. E ) + D )
20	2, 8, 3, 4, 1, 10, 11, 16, 19	nummul1c 9775	. 2 |- ( N x. P ) = ( ( T x. C ) + D )
21	7, 9, 20	mulcomli 8297	1 |- ( P x. N ) = ( ( T x. C ) + D )

Syntax hints:   = wceq 1398   e. wcel 2205  (class class class)co 6058   + caddc 8146   x. cmul 8148   NN0cn0 9513

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-sub 8462  df-inn 9255  df-n0 9514

This theorem is referenced by:  decmul2c  9792