Theorem numsucc 9748

Description: The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Hypotheses

Ref	Expression
numsucc.1	|- Y e. NN0
numsucc.2	|- T = ( Y + 1 )
numsucc.3	|- A e. NN0
numsucc.4	|- ( A + 1 ) = B
numsucc.5	|- N = ( ( T x. A ) + Y )

Assertion

Ref	Expression
numsucc	|- ( N + 1 ) = ( ( T x. B ) + 0 )

Proof of Theorem numsucc

Step	Hyp	Ref	Expression
1		numsucc.2	. . . . . . 7 |- T = ( Y + 1 )
2		numsucc.1	. . . . . . . 8 |- Y e. NN0
3		1nn0 9512	. . . . . . . 8 |- 1 e. NN0
4	2, 3	nn0addcli 9533	. . . . . . 7 |- ( Y + 1 ) e. NN0
5	1, 4	eqeltri 2305	. . . . . 6 |- T e. NN0
6	5	nn0cni 9508	. . . . 5 |- T e. CC
7	6	mulridi 8276	. . . 4 |- ( T x. 1 ) = T
8	7	oveq2i 6061	. . 3 |- ( ( T x. A ) + ( T x. 1 ) ) = ( ( T x. A ) + T )
9		numsucc.3	. . . . 5 |- A e. NN0
10	9	nn0cni 9508	. . . 4 |- A e. CC
11		ax-1cn 8220	. . . 4 |- 1 e. CC
12	6, 10, 11	adddii 8284	. . 3 |- ( T x. ( A + 1 ) ) = ( ( T x. A ) + ( T x. 1 ) )
13	1	eqcomi 2236	. . . 4 |- ( Y + 1 ) = T
14		numsucc.5	. . . 4 |- N = ( ( T x. A ) + Y )
15	5, 9, 2, 13, 14	numsuc 9722	. . 3 |- ( N + 1 ) = ( ( T x. A ) + T )
16	8, 12, 15	3eqtr4ri 2264	. 2 |- ( N + 1 ) = ( T x. ( A + 1 ) )
17		numsucc.4	. . 3 |- ( A + 1 ) = B
18	17	oveq2i 6061	. 2 |- ( T x. ( A + 1 ) ) = ( T x. B )
19	9, 3	nn0addcli 9533	. . . 4 |- ( A + 1 ) e. NN0
20	17, 19	eqeltrri 2306	. . 3 |- B e. NN0
21	5, 20	num0u 9719	. 2 |- ( T x. B ) = ( ( T x. B ) + 0 )
22	16, 18, 21	3eqtri 2257	1 |- ( N + 1 ) = ( ( T x. B ) + 0 )

Syntax hints:   = wceq 1398   e. wcel 2203  (class class class)co 6050   0cc0 8127   1c1 8128   + caddc 8130   x. cmul 8132   NN0cn0 9496

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-sub 8446  df-inn 9238  df-n0 9497

This theorem is referenced by:  decsucc  9749