Theorem numltc 9482

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 9481	. . . 4 |- ( ( T x. A ) + C ) < ( ( T x. A ) + T )
6	1	nnrei 8999	. . . . . . 7 |- T e. RR
7	6	recni 8038	. . . . . 6 |- T e. CC
8	2	nn0rei 9260	. . . . . . 7 |- A e. RR
9	8	recni 8038	. . . . . 6 |- A e. CC
10		ax-1cn 7972	. . . . . 6 |- 1 e. CC
11	7, 9, 10	adddii 8036	. . . . 5 |- ( T x. ( A + 1 ) ) = ( ( T x. A ) + ( T x. 1 ) )
12	7	mulridi 8028	. . . . . 6 |- ( T x. 1 ) = T
13	12	oveq2i 5933	. . . . 5 |- ( ( T x. A ) + ( T x. 1 ) ) = ( ( T x. A ) + T )
14	11, 13	eqtri 2217	. . . 4 |- ( T x. ( A + 1 ) ) = ( ( T x. A ) + T )
15	5, 14	breqtrri 4060	. . 3 |- ( ( T x. A ) + C ) < ( T x. ( A + 1 ) )
16		numltc.6	. . . . 5 |- A < B
17		numlt.3	. . . . . 6 |- B e. NN0
18		nn0ltp1le 9388	. . . . . 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 9020	. . . . 5 |- 0 < T
22		peano2re 8162	. . . . . . 7 |- ( A e. RR -> ( A + 1 ) e. RR )
23	8, 22	ax-mp 5	. . . . . 6 |- ( A + 1 ) e. RR
24	17	nn0rei 9260	. . . . . 6 |- B e. RR
25	23, 24, 6	lemul2i 8952	. . . . 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 8040	. . . . 5 |- ( T x. A ) e. RR
29	3	nn0rei 9260	. . . . 5 |- C e. RR
30	28, 29	readdcli 8039	. . . 4 |- ( ( T x. A ) + C ) e. RR
31	6, 23	remulcli 8040	. . . 4 |- ( T x. ( A + 1 ) ) e. RR
32	6, 24	remulcli 8040	. . . 4 |- ( T x. B ) e. RR
33	30, 31, 32	ltletri 8133	. . 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 9297	. 2 |- ( T x. B ) <_ ( ( T x. B ) + D )
37	35	nn0rei 9260	. . . 4 |- D e. RR
38	32, 37	readdcli 8039	. . 3 |- ( ( T x. B ) + D ) e. RR
39	30, 32, 38	ltletri 8133	. 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 2167   class class class wbr 4033  (class class class)co 5922   RRcr 7878   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   < clt 8061   <_ cle 8062   NNcn 8990   NN0cn0 9249

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995  ax-pre-mulgt0 7996

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327

This theorem is referenced by:  decltc  9485  numlti  9493