Theorem nummul2c 9842

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 9800	. . . 4 |- ( ( T x. A ) + B ) e. NN0
6	1, 5	eqeltri 2311	. . 3 |- N e. NN0
7	6	nn0cni 9586	. 2 |- N e. CC
8		nummul1c.2	. . 3 |- P e. NN0
9	8	nn0cni 9586	. 2 |- P e. CC
10		nummul1c.6	. . 3 |- D e. NN0
11		nummul1c.7	. . 3 |- E e. NN0
12	3	nn0cni 9586	. . . . . 6 |- A e. CC
13	12, 9	mulcomi 8338	. . . . 5 |- ( A x. P ) = ( P x. A )
14	13	oveq1i 6098	. . . 4 |- ( ( A x. P ) + E ) = ( ( P x. A ) + E )
15		nummul2c.7	. . . 4 |- ( ( P x. A ) + E ) = C
16	14, 15	eqtri 2259	. . 3 |- ( ( A x. P ) + E ) = C
17	4	nn0cni 9586	. . . 4 |- B e. CC
18		nummul2c.8	. . . 4 |- ( P x. B ) = ( ( T x. E ) + D )
19	9, 17, 18	mulcomli 8339	. . 3 |- ( B x. P ) = ( ( T x. E ) + D )
20	2, 8, 3, 4, 1, 10, 11, 16, 19	nummul1c 9841	. 2 |- ( N x. P ) = ( ( T x. C ) + D )
21	7, 9, 20	mulcomli 8339	1 |- ( P x. N ) = ( ( T x. C ) + D )

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209  (class class class)co 6088   + caddc 8188   x. cmul 8190   NN0cn0 9574

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 4251  ax-pow 4313  ax-pr 4348  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296

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 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-iota 5340  df-fun 5382  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-sub 8506  df-inn 9314  df-n0 9575

This theorem is used by:  decmul2c  9858