Theorem numltc 9001

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 9000	. . . 4 |- ( ( T x. A ) + C ) < ( ( T x. A ) + T )
6	1	nnrei 8529	. . . . . . 7 |- T e. RR
7	6	recni 7597	. . . . . 6 |- T e. CC
8	2	nn0rei 8782	. . . . . . 7 |- A e. RR
9	8	recni 7597	. . . . . 6 |- A e. CC
10		ax-1cn 7535	. . . . . 6 |- 1 e. CC
11	7, 9, 10	adddii 7595	. . . . 5 |- ( T x. ( A + 1 ) ) = ( ( T x. A ) + ( T x. 1 ) )
12	7	mulid1i 7587	. . . . . 6 |- ( T x. 1 ) = T
13	12	oveq2i 5701	. . . . 5 |- ( ( T x. A ) + ( T x. 1 ) ) = ( ( T x. A ) + T )
14	11, 13	eqtri 2115	. . . 4 |- ( T x. ( A + 1 ) ) = ( ( T x. A ) + T )
15	5, 14	breqtrri 3892	. . 3 |- ( ( T x. A ) + C ) < ( T x. ( A + 1 ) )
16		numltc.6	. . . . 5 |- A < B
17		numlt.3	. . . . . 6 |- B e. NN0
18		nn0ltp1le 8910	. . . . . 6 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A < B <-> ( A + 1 ) <_ B ) )
19	2, 17, 18	mp2an 418	. . . . 5 |- ( A < B <-> ( A + 1 ) <_ B )
20	16, 19	mpbi 144	. . . 4 |- ( A + 1 ) <_ B
21	1	nngt0i 8550	. . . . 5 |- 0 < T
22		peano2re 7715	. . . . . . 7 |- ( A e. RR -> ( A + 1 ) e. RR )
23	8, 22	ax-mp 7	. . . . . 6 |- ( A + 1 ) e. RR
24	17	nn0rei 8782	. . . . . 6 |- B e. RR
25	23, 24, 6	lemul2i 8483	. . . . 5 |- ( 0 < T -> ( ( A + 1 ) <_ B <-> ( T x. ( A + 1 ) ) <_ ( T x. B ) ) )
26	21, 25	ax-mp 7	. . . 4 |- ( ( A + 1 ) <_ B <-> ( T x. ( A + 1 ) ) <_ ( T x. B ) )
27	20, 26	mpbi 144	. . 3 |- ( T x. ( A + 1 ) ) <_ ( T x. B )
28	6, 8	remulcli 7599	. . . . 5 |- ( T x. A ) e. RR
29	3	nn0rei 8782	. . . . 5 |- C e. RR
30	28, 29	readdcli 7598	. . . 4 |- ( ( T x. A ) + C ) e. RR
31	6, 23	remulcli 7599	. . . 4 |- ( T x. ( A + 1 ) ) e. RR
32	6, 24	remulcli 7599	. . . 4 |- ( T x. B ) e. RR
33	30, 31, 32	ltletri 7688	. . 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 418	. 2 |- ( ( T x. A ) + C ) < ( T x. B )
35		numltc.4	. . 3 |- D e. NN0
36	32, 35	nn0addge1i 8819	. 2 |- ( T x. B ) <_ ( ( T x. B ) + D )
37	35	nn0rei 8782	. . . 4 |- D e. RR
38	32, 37	readdcli 7598	. . 3 |- ( ( T x. B ) + D ) e. RR
39	30, 32, 38	ltletri 7688	. 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 418	1 |- ( ( T x. A ) + C ) < ( ( T x. B ) + D )

Syntax hints:   <-> wb 104   e. wcel 1445   class class class wbr 3867  (class class class)co 5690   RRcr 7446   0cc0 7447   1c1 7448   + caddc 7450   x. cmul 7452   < clt 7619   <_ cle 7620   NNcn 8520   NN0cn0 8771

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-ltadd 7558  ax-pre-mulgt0 7559

This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-inn 8521  df-n0 8772  df-z 8849

This theorem is referenced by:  decltc  9004  numlti  9012