Theorem numaddc 9369

Description: Add two decimal integers M and N (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Hypotheses

Ref	Expression
numma.1	|- T e. NN0
numma.2	|- A e. NN0
numma.3	|- B e. NN0
numma.4	|- C e. NN0
numma.5	|- D e. NN0
numma.6	|- M = ( ( T x. A ) + B )
numma.7	|- N = ( ( T x. C ) + D )
numaddc.8	|- F e. NN0
numaddc.9	|- ( ( A + C ) + 1 ) = E
numaddc.10	|- ( B + D ) = ( ( T x. 1 ) + F )

Assertion

Ref	Expression
numaddc	|- ( M + N ) = ( ( T x. E ) + F )

Proof of Theorem numaddc

Step	Hyp	Ref	Expression
1		numma.6	. . . . . 6 |- M = ( ( T x. A ) + B )
2		numma.1	. . . . . . 7 |- T e. NN0
3		numma.2	. . . . . . 7 |- A e. NN0
4		numma.3	. . . . . . 7 |- B e. NN0
5	2, 3, 4	numcl 9334	. . . . . 6 |- ( ( T x. A ) + B ) e. NN0
6	1, 5	eqeltri 2239	. . . . 5 |- M e. NN0
7	6	nn0cni 9126	. . . 4 |- M e. CC
8	7	mulid1i 7901	. . 3 |- ( M x. 1 ) = M
9	8	oveq1i 5852	. 2 |- ( ( M x. 1 ) + N ) = ( M + N )
10		numma.4	. . 3 |- C e. NN0
11		numma.5	. . 3 |- D e. NN0
12		numma.7	. . 3 |- N = ( ( T x. C ) + D )
13		1nn0 9130	. . 3 |- 1 e. NN0
14		numaddc.8	. . 3 |- F e. NN0
15	3	nn0cni 9126	. . . . . 6 |- A e. CC
16	15	mulid1i 7901	. . . . 5 |- ( A x. 1 ) = A
17	16	oveq1i 5852	. . . 4 |- ( ( A x. 1 ) + ( C + 1 ) ) = ( A + ( C + 1 ) )
18	10	nn0cni 9126	. . . . 5 |- C e. CC
19		ax-1cn 7846	. . . . 5 |- 1 e. CC
20	15, 18, 19	addassi 7907	. . . 4 |- ( ( A + C ) + 1 ) = ( A + ( C + 1 ) )
21		numaddc.9	. . . 4 |- ( ( A + C ) + 1 ) = E
22	17, 20, 21	3eqtr2i 2192	. . 3 |- ( ( A x. 1 ) + ( C + 1 ) ) = E
23	4	nn0cni 9126	. . . . . 6 |- B e. CC
24	23	mulid1i 7901	. . . . 5 |- ( B x. 1 ) = B
25	24	oveq1i 5852	. . . 4 |- ( ( B x. 1 ) + D ) = ( B + D )
26		numaddc.10	. . . 4 |- ( B + D ) = ( ( T x. 1 ) + F )
27	25, 26	eqtri 2186	. . 3 |- ( ( B x. 1 ) + D ) = ( ( T x. 1 ) + F )
28	2, 3, 4, 10, 11, 1, 12, 13, 14, 13, 22, 27	nummac 9366	. 2 |- ( ( M x. 1 ) + N ) = ( ( T x. E ) + F )
29	9, 28	eqtr3i 2188	1 |- ( M + N ) = ( ( T x. E ) + F )

Syntax hints:   = wceq 1343   e. wcel 2136  (class class class)co 5842   1c1 7754   + caddc 7756   x. cmul 7758   NN0cn0 9114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-sub 8071  df-inn 8858  df-n0 9115

This theorem is referenced by:  decaddc  9376