Theorem numltc 9697

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 9696	. . . 4 |- ( ( T x. A ) + C ) < ( ( T x. A ) + T )
6	1	nnrei 9211	. . . . . . 7 |- T e. RR
7	6	recni 8251	. . . . . 6 |- T e. CC
8	2	nn0rei 9472	. . . . . . 7 |- A e. RR
9	8	recni 8251	. . . . . 6 |- A e. CC
10		ax-1cn 8185	. . . . . 6 |- 1 e. CC
11	7, 9, 10	adddii 8249	. . . . 5 |- ( T x. ( A + 1 ) ) = ( ( T x. A ) + ( T x. 1 ) )
12	7	mulridi 8241	. . . . . 6 |- ( T x. 1 ) = T
13	12	oveq2i 6039	. . . . 5 |- ( ( T x. A ) + ( T x. 1 ) ) = ( ( T x. A ) + T )
14	11, 13	eqtri 2252	. . . 4 |- ( T x. ( A + 1 ) ) = ( ( T x. A ) + T )
15	5, 14	breqtrri 4120	. . 3 |- ( ( T x. A ) + C ) < ( T x. ( A + 1 ) )
16		numltc.6	. . . . 5 |- A < B
17		numlt.3	. . . . . 6 |- B e. NN0
18		nn0ltp1le 9603	. . . . . 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 9232	. . . . 5 |- 0 < T
22		peano2re 8374	. . . . . . 7 |- ( A e. RR -> ( A + 1 ) e. RR )
23	8, 22	ax-mp 5	. . . . . 6 |- ( A + 1 ) e. RR
24	17	nn0rei 9472	. . . . . 6 |- B e. RR
25	23, 24, 6	lemul2i 9164	. . . . 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 8253	. . . . 5 |- ( T x. A ) e. RR
29	3	nn0rei 9472	. . . . 5 |- C e. RR
30	28, 29	readdcli 8252	. . . 4 |- ( ( T x. A ) + C ) e. RR
31	6, 23	remulcli 8253	. . . 4 |- ( T x. ( A + 1 ) ) e. RR
32	6, 24	remulcli 8253	. . . 4 |- ( T x. B ) e. RR
33	30, 31, 32	ltletri 8345	. . 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 9509	. 2 |- ( T x. B ) <_ ( ( T x. B ) + D )
37	35	nn0rei 9472	. . . 4 |- D e. RR
38	32, 37	readdcli 8252	. . 3 |- ( ( T x. B ) + D ) e. RR
39	30, 32, 38	ltletri 8345	. 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 2202   class class class wbr 4093  (class class class)co 6028   RRcr 8091   0cc0 8092   1c1 8093   + caddc 8095   x. cmul 8097   < clt 8273   <_ cle 8274   NNcn 9202   NN0cn0 9461

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-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208  ax-pre-mulgt0 8209

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-n0 9462  df-z 9541

This theorem is referenced by:  decltc  9700  numlti  9708