Theorem numadd 9620

Description: Add two decimal integers M and N (no 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 )
numadd.8	|- ( A + C ) = E
numadd.9	|- ( B + D ) = F

Assertion

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

Proof of Theorem numadd

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 9586	. . . . . 6 |- ( ( T x. A ) + B ) e. NN0
6	1, 5	eqeltri 2302	. . . . 5 |- M e. NN0
7	6	nn0cni 9377	. . . 4 |- M e. CC
8	7	mulridi 8144	. . 3 |- ( M x. 1 ) = M
9	8	oveq1i 6010	. 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 9381	. . 3 |- 1 e. NN0
14	3	nn0cni 9377	. . . . . 6 |- A e. CC
15	14	mulridi 8144	. . . . 5 |- ( A x. 1 ) = A
16	15	oveq1i 6010	. . . 4 |- ( ( A x. 1 ) + C ) = ( A + C )
17		numadd.8	. . . 4 |- ( A + C ) = E
18	16, 17	eqtri 2250	. . 3 |- ( ( A x. 1 ) + C ) = E
19	4	nn0cni 9377	. . . . . 6 |- B e. CC
20	19	mulridi 8144	. . . . 5 |- ( B x. 1 ) = B
21	20	oveq1i 6010	. . . 4 |- ( ( B x. 1 ) + D ) = ( B + D )
22		numadd.9	. . . 4 |- ( B + D ) = F
23	21, 22	eqtri 2250	. . 3 |- ( ( B x. 1 ) + D ) = F
24	2, 3, 4, 10, 11, 1, 12, 13, 18, 23	numma 9617	. 2 |- ( ( M x. 1 ) + N ) = ( ( T x. E ) + F )
25	9, 24	eqtr3i 2252	1 |- ( M + N ) = ( ( T x. E ) + F )

Syntax hints:   = wceq 1395   e. wcel 2200  (class class class)co 6000   1c1 7996   + caddc 7998   x. cmul 8000   NN0cn0 9365

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-sub 8315  df-inn 9107  df-n0 9366

This theorem is referenced by:  decadd  9627