Theorem numaddc 9390

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 9355	. . . . . 6 |- ( ( T x. A ) + B ) e. NN0
6	1, 5	eqeltri 2243	. . . . 5 |- M e. NN0
7	6	nn0cni 9147	. . . 4 |- M e. CC
8	7	mulid1i 7922	. . 3 |- ( M x. 1 ) = M
9	8	oveq1i 5863	. 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 9151	. . 3 |- 1 e. NN0
14		numaddc.8	. . 3 |- F e. NN0
15	3	nn0cni 9147	. . . . . 6 |- A e. CC
16	15	mulid1i 7922	. . . . 5 |- ( A x. 1 ) = A
17	16	oveq1i 5863	. . . 4 |- ( ( A x. 1 ) + ( C + 1 ) ) = ( A + ( C + 1 ) )
18	10	nn0cni 9147	. . . . 5 |- C e. CC
19		ax-1cn 7867	. . . . 5 |- 1 e. CC
20	15, 18, 19	addassi 7928	. . . 4 |- ( ( A + C ) + 1 ) = ( A + ( C + 1 ) )
21		numaddc.9	. . . 4 |- ( ( A + C ) + 1 ) = E
22	17, 20, 21	3eqtr2i 2197	. . 3 |- ( ( A x. 1 ) + ( C + 1 ) ) = E
23	4	nn0cni 9147	. . . . . 6 |- B e. CC
24	23	mulid1i 7922	. . . . 5 |- ( B x. 1 ) = B
25	24	oveq1i 5863	. . . 4 |- ( ( B x. 1 ) + D ) = ( B + D )
26		numaddc.10	. . . 4 |- ( B + D ) = ( ( T x. 1 ) + F )
27	25, 26	eqtri 2191	. . 3 |- ( ( B x. 1 ) + D ) = ( ( T x. 1 ) + F )
28	2, 3, 4, 10, 11, 1, 12, 13, 14, 13, 22, 27	nummac 9387	. 2 |- ( ( M x. 1 ) + N ) = ( ( T x. E ) + F )
29	9, 28	eqtr3i 2193	1 |- ( M + N ) = ( ( T x. E ) + F )

Syntax hints:   = wceq 1348   e. wcel 2141  (class class class)co 5853   1c1 7775   + caddc 7777   x. cmul 7779   NN0cn0 9135

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sub 8092  df-inn 8879  df-n0 9136

This theorem is referenced by:  decaddc  9397