Theorem numltc 9355

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 9354	. . . 4 |- ( ( T x. A ) + C ) < ( ( T x. A ) + T )
6	1	nnrei 8874	. . . . . . 7 |- T e. RR
7	6	recni 7919	. . . . . 6 |- T e. CC
8	2	nn0rei 9133	. . . . . . 7 |- A e. RR
9	8	recni 7919	. . . . . 6 |- A e. CC
10		ax-1cn 7854	. . . . . 6 |- 1 e. CC
11	7, 9, 10	adddii 7917	. . . . 5 |- ( T x. ( A + 1 ) ) = ( ( T x. A ) + ( T x. 1 ) )
12	7	mulid1i 7909	. . . . . 6 |- ( T x. 1 ) = T
13	12	oveq2i 5861	. . . . 5 |- ( ( T x. A ) + ( T x. 1 ) ) = ( ( T x. A ) + T )
14	11, 13	eqtri 2191	. . . 4 |- ( T x. ( A + 1 ) ) = ( ( T x. A ) + T )
15	5, 14	breqtrri 4014	. . 3 |- ( ( T x. A ) + C ) < ( T x. ( A + 1 ) )
16		numltc.6	. . . . 5 |- A < B
17		numlt.3	. . . . . 6 |- B e. NN0
18		nn0ltp1le 9261	. . . . . 6 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A < B <-> ( A + 1 ) <_ B ) )
19	2, 17, 18	mp2an 424	. . . . 5 |- ( A < B <-> ( A + 1 ) <_ B )
20	16, 19	mpbi 144	. . . 4 |- ( A + 1 ) <_ B
21	1	nngt0i 8895	. . . . 5 |- 0 < T
22		peano2re 8042	. . . . . . 7 |- ( A e. RR -> ( A + 1 ) e. RR )
23	8, 22	ax-mp 5	. . . . . 6 |- ( A + 1 ) e. RR
24	17	nn0rei 9133	. . . . . 6 |- B e. RR
25	23, 24, 6	lemul2i 8828	. . . . 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 144	. . 3 |- ( T x. ( A + 1 ) ) <_ ( T x. B )
28	6, 8	remulcli 7921	. . . . 5 |- ( T x. A ) e. RR
29	3	nn0rei 9133	. . . . 5 |- C e. RR
30	28, 29	readdcli 7920	. . . 4 |- ( ( T x. A ) + C ) e. RR
31	6, 23	remulcli 7921	. . . 4 |- ( T x. ( A + 1 ) ) e. RR
32	6, 24	remulcli 7921	. . . 4 |- ( T x. B ) e. RR
33	30, 31, 32	ltletri 8013	. . 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 424	. 2 |- ( ( T x. A ) + C ) < ( T x. B )
35		numltc.4	. . 3 |- D e. NN0
36	32, 35	nn0addge1i 9170	. 2 |- ( T x. B ) <_ ( ( T x. B ) + D )
37	35	nn0rei 9133	. . . 4 |- D e. RR
38	32, 37	readdcli 7920	. . 3 |- ( ( T x. B ) + D ) e. RR
39	30, 32, 38	ltletri 8013	. 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 424	1 |- ( ( T x. A ) + C ) < ( ( T x. B ) + D )

Syntax hints:   <-> wb 104   e. wcel 2141   class class class wbr 3987  (class class class)co 5850   RRcr 7760   0cc0 7761   1c1 7762   + caddc 7764   x. cmul 7766   < clt 7941   <_ cle 7942   NNcn 8865   NN0cn0 9122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-ltadd 7877  ax-pre-mulgt0 7878

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-inn 8866  df-n0 9123  df-z 9200

This theorem is referenced by:  decltc  9358  numlti  9366