Theorem numltc 9440

Description: Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)

Hypotheses

Ref	Expression
numlt.1	|- T e. NN
numlt.2	|- A e. NN0
numlt.3	|- B e. NN0
numltc.3	|- C e. NN0
numltc.4	|- D e. NN0
numltc.5	|- C < T
numltc.6	|- A < B

Assertion

Ref	Expression
numltc	|- ( ( T x. A ) + C ) < ( ( T x. B ) + D )

Proof of Theorem numltc

Step	Hyp	Ref	Expression
1		numlt.1	. . . . 5 |- T e. NN
2		numlt.2	. . . . 5 |- A e. NN0
3		numltc.3	. . . . 5 |- C e. NN0
4		numltc.5	. . . . 5 |- C < T
5	1, 2, 3, 1, 4	numlt 9439	. . . 4 |- ( ( T x. A ) + C ) < ( ( T x. A ) + T )
6	1	nnrei 8959	. . . . . . 7 |- T e. RR
7	6	recni 8000	. . . . . 6 |- T e. CC
8	2	nn0rei 9218	. . . . . . 7 |- A e. RR
9	8	recni 8000	. . . . . 6 |- A e. CC
10		ax-1cn 7935	. . . . . 6 |- 1 e. CC
11	7, 9, 10	adddii 7998	. . . . 5 |- ( T x. ( A + 1 ) ) = ( ( T x. A ) + ( T x. 1 ) )
12	7	mulid1i 7990	. . . . . 6 |- ( T x. 1 ) = T
13	12	oveq2i 5908	. . . . 5 |- ( ( T x. A ) + ( T x. 1 ) ) = ( ( T x. A ) + T )
14	11, 13	eqtri 2210	. . . 4 |- ( T x. ( A + 1 ) ) = ( ( T x. A ) + T )
15	5, 14	breqtrri 4045	. . 3 |- ( ( T x. A ) + C ) < ( T x. ( A + 1 ) )
16		numltc.6	. . . . 5 |- A < B
17		numlt.3	. . . . . 6 |- B e. NN0
18		nn0ltp1le 9346	. . . . . 6 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A < B <-> ( A + 1 ) <_ B ) )
19	2, 17, 18	mp2an 426	. . . . 5 |- ( A < B <-> ( A + 1 ) <_ B )
20	16, 19	mpbi 145	. . . 4 |- ( A + 1 ) <_ B
21	1	nngt0i 8980	. . . . 5 |- 0 < T
22		peano2re 8124	. . . . . . 7 |- ( A e. RR -> ( A + 1 ) e. RR )
23	8, 22	ax-mp 5	. . . . . 6 |- ( A + 1 ) e. RR
24	17	nn0rei 9218	. . . . . 6 |- B e. RR
25	23, 24, 6	lemul2i 8913	. . . . 5 |- ( 0 < T -> ( ( A + 1 ) <_ B <-> ( T x. ( A + 1 ) ) <_ ( T x. B ) ) )
26	21, 25	ax-mp 5	. . . 4 |- ( ( A + 1 ) <_ B <-> ( T x. ( A + 1 ) ) <_ ( T x. B ) )
27	20, 26	mpbi 145	. . 3 |- ( T x. ( A + 1 ) ) <_ ( T x. B )
28	6, 8	remulcli 8002	. . . . 5 |- ( T x. A ) e. RR
29	3	nn0rei 9218	. . . . 5 |- C e. RR
30	28, 29	readdcli 8001	. . . 4 |- ( ( T x. A ) + C ) e. RR
31	6, 23	remulcli 8002	. . . 4 |- ( T x. ( A + 1 ) ) e. RR
32	6, 24	remulcli 8002	. . . 4 |- ( T x. B ) e. RR
33	30, 31, 32	ltletri 8095	. . 3 |- ( ( ( ( T x. A ) + C ) < ( T x. ( A + 1 ) ) /\ ( T x. ( A + 1 ) ) <_ ( T x. B ) ) -> ( ( T x. A ) + C ) < ( T x. B ) )
34	15, 27, 33	mp2an 426	. 2 |- ( ( T x. A ) + C ) < ( T x. B )
35		numltc.4	. . 3 |- D e. NN0
36	32, 35	nn0addge1i 9255	. 2 |- ( T x. B ) <_ ( ( T x. B ) + D )
37	35	nn0rei 9218	. . . 4 |- D e. RR
38	32, 37	readdcli 8001	. . 3 |- ( ( T x. B ) + D ) e. RR
39	30, 32, 38	ltletri 8095	. 2 |- ( ( ( ( T x. A ) + C ) < ( T x. B ) /\ ( T x. B ) <_ ( ( T x. B ) + D ) ) -> ( ( T x. A ) + C ) < ( ( T x. B ) + D ) )
40	34, 36, 39	mp2an 426	1 |- ( ( T x. A ) + C ) < ( ( T x. B ) + D )

Syntax hints:   <-> wb 105   e. wcel 2160   class class class wbr 4018  (class class class)co 5897   RRcr 7841   0cc0 7842   1c1 7843   + caddc 7845   x. cmul 7847   < clt 8023   <_ cle 8024   NNcn 8950   NN0cn0 9207

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-ltadd 7958  ax-pre-mulgt0 7959

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-inn 8951  df-n0 9208  df-z 9285

This theorem is referenced by:  decltc  9443  numlti  9451