Theorem numaddc 9803

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 9768	. . . . . 6 |- ( ( T x. A ) + B ) e. NN0
6	1, 5	eqeltri 2311	. . . . 5 |- M e. NN0
7	6	nn0cni 9554	. . . 4 |- M e. CC
8	7	mulridi 8318	. . 3 |- ( M x. 1 ) = M
9	8	oveq1i 6085	. 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 9558	. . 3 |- 1 e. NN0
14		numaddc.8	. . 3 |- F e. NN0
15	3	nn0cni 9554	. . . . . 6 |- A e. CC
16	15	mulridi 8318	. . . . 5 |- ( A x. 1 ) = A
17	16	oveq1i 6085	. . . 4 |- ( ( A x. 1 ) + ( C + 1 ) ) = ( A + ( C + 1 ) )
18	10	nn0cni 9554	. . . . 5 |- C e. CC
19		ax-1cn 8262	. . . . 5 |- 1 e. CC
20	15, 18, 19	addassi 8324	. . . 4 |- ( ( A + C ) + 1 ) = ( A + ( C + 1 ) )
21		numaddc.9	. . . 4 |- ( ( A + C ) + 1 ) = E
22	17, 20, 21	3eqtr2i 2265	. . 3 |- ( ( A x. 1 ) + ( C + 1 ) ) = E
23	4	nn0cni 9554	. . . . . 6 |- B e. CC
24	23	mulridi 8318	. . . . 5 |- ( B x. 1 ) = B
25	24	oveq1i 6085	. . . 4 |- ( ( B x. 1 ) + D ) = ( B + D )
26		numaddc.10	. . . 4 |- ( B + D ) = ( ( T x. 1 ) + F )
27	25, 26	eqtri 2259	. . 3 |- ( ( B x. 1 ) + D ) = ( ( T x. 1 ) + F )
28	2, 3, 4, 10, 11, 1, 12, 13, 14, 13, 22, 27	nummac 9800	. 2 |- ( ( M x. 1 ) + N ) = ( ( T x. E ) + F )
29	9, 28	eqtr3i 2261	1 |- ( M + N ) = ( ( T x. E ) + F )

Syntax hints:   = wceq 1402   e. wcel 2209  (class class class)co 6075   1c1 8170   + caddc 8172   x. cmul 8174   NN0cn0 9542

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 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 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-sub 8489  df-inn 9284  df-n0 9543

This theorem is referenced by:  decaddc  9810