Theorem divalglemnqt 11261

Description: Lemma for divalg 11265. The Q < T case involved in showing uniqueness. (Contributed by Jim Kingdon, 4-Dec-2021.)

Hypotheses

Ref	Expression
divalglemnqt.d	|- ( ph -> D e. NN )
divalglemnqt.r	|- ( ph -> R e. ZZ )
divalglemnqt.s	|- ( ph -> S e. ZZ )
divalglemnqt.q	|- ( ph -> Q e. ZZ )
divalglemnqt.t	|- ( ph -> T e. ZZ )
divalglemnqt.0s	|- ( ph -> 0 <_ S )
divalglemnqt.rd	|- ( ph -> R < D )
divalglemnqt.eq	|- ( ph -> ( ( Q x. D ) + R ) = ( ( T x. D ) + S ) )

Assertion

Ref	Expression
divalglemnqt	|- ( ph -> -. Q < T )

Proof of Theorem divalglemnqt

Step	Hyp	Ref	Expression
1		divalglemnqt.rd	. . 3 |- ( ph -> R < D )
2	1	adantr 271	. 2 |- ( ( ph /\ Q < T ) -> R < D )
3		divalglemnqt.d	. . . . 5 |- ( ph -> D e. NN )
4	3	adantr 271	. . . 4 |- ( ( ph /\ Q < T ) -> D e. NN )
5	4	nnred 8498	. . 3 |- ( ( ph /\ Q < T ) -> D e. RR )
6		divalglemnqt.r	. . . . 5 |- ( ph -> R e. ZZ )
7	6	adantr 271	. . . 4 |- ( ( ph /\ Q < T ) -> R e. ZZ )
8	7	zred 8931	. . 3 |- ( ( ph /\ Q < T ) -> R e. RR )
9		divalglemnqt.s	. . . . . . 7 |- ( ph -> S e. ZZ )
10	9	adantr 271	. . . . . 6 |- ( ( ph /\ Q < T ) -> S e. ZZ )
11	10	zred 8931	. . . . 5 |- ( ( ph /\ Q < T ) -> S e. RR )
12	5, 11	readdcld 7580	. . . 4 |- ( ( ph /\ Q < T ) -> ( D + S ) e. RR )
13		divalglemnqt.0s	. . . . . 6 |- ( ph -> 0 <_ S )
14	13	adantr 271	. . . . 5 |- ( ( ph /\ Q < T ) -> 0 <_ S )
15	5, 11	addge01d 8073	. . . . 5 |- ( ( ph /\ Q < T ) -> ( 0 <_ S <-> D <_ ( D + S ) ) )
16	14, 15	mpbid 146	. . . 4 |- ( ( ph /\ Q < T ) -> D <_ ( D + S ) )
17		divalglemnqt.q	. . . . . . . . . . 11 |- ( ph -> Q e. ZZ )
18	17	adantr 271	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> Q e. ZZ )
19	18	zred 8931	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> Q e. RR )
20	19	recnd 7579	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> Q e. CC )
21	5	recnd 7579	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> D e. CC )
22	20, 21	mulcld 7571	. . . . . . 7 |- ( ( ph /\ Q < T ) -> ( Q x. D ) e. CC )
23	11	recnd 7579	. . . . . . 7 |- ( ( ph /\ Q < T ) -> S e. CC )
24	22, 21, 23	addassd 7573	. . . . . 6 |- ( ( ph /\ Q < T ) -> ( ( ( Q x. D ) + D ) + S ) = ( ( Q x. D ) + ( D + S ) ) )
25	19, 5	remulcld 7581	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> ( Q x. D ) e. RR )
26	25, 5	readdcld 7580	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> ( ( Q x. D ) + D ) e. RR )
27		divalglemnqt.t	. . . . . . . . . . 11 |- ( ph -> T e. ZZ )
28	27	adantr 271	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> T e. ZZ )
29	28	zred 8931	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> T e. RR )
30	29, 5	remulcld 7581	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> ( T x. D ) e. RR )
31	20, 21	adddirp1d 7577	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> ( ( Q + 1 ) x. D ) = ( ( Q x. D ) + D ) )
32		peano2re 7681	. . . . . . . . . . 11 |- ( Q e. RR -> ( Q + 1 ) e. RR )
33	19, 32	syl 14	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> ( Q + 1 ) e. RR )
34	4	nnnn0d 8789	. . . . . . . . . . 11 |- ( ( ph /\ Q < T ) -> D e. NN0 )
35	34	nn0ge0d 8792	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> 0 <_ D )
36		simpr 109	. . . . . . . . . . 11 |- ( ( ph /\ Q < T ) -> Q < T )
37		zltp1le 8867	. . . . . . . . . . . 12 |- ( ( Q e. ZZ /\ T e. ZZ ) -> ( Q < T <-> ( Q + 1 ) <_ T ) )
38	17, 28, 37	syl2an2r 563	. . . . . . . . . . 11 |- ( ( ph /\ Q < T ) -> ( Q < T <-> ( Q + 1 ) <_ T ) )
39	36, 38	mpbid 146	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> ( Q + 1 ) <_ T )
40	33, 29, 5, 35, 39	lemul1ad 8463	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> ( ( Q + 1 ) x. D ) <_ ( T x. D ) )
41	31, 40	eqbrtrrd 3875	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> ( ( Q x. D ) + D ) <_ ( T x. D ) )
42	26, 30, 11, 41	leadd1dd 8099	. . . . . . 7 |- ( ( ph /\ Q < T ) -> ( ( ( Q x. D ) + D ) + S ) <_ ( ( T x. D ) + S ) )
43		divalglemnqt.eq	. . . . . . . 8 |- ( ph -> ( ( Q x. D ) + R ) = ( ( T x. D ) + S ) )
44	43	adantr 271	. . . . . . 7 |- ( ( ph /\ Q < T ) -> ( ( Q x. D ) + R ) = ( ( T x. D ) + S ) )
45	42, 44	breqtrrd 3879	. . . . . 6 |- ( ( ph /\ Q < T ) -> ( ( ( Q x. D ) + D ) + S ) <_ ( ( Q x. D ) + R ) )
46	24, 45	eqbrtrrd 3875	. . . . 5 |- ( ( ph /\ Q < T ) -> ( ( Q x. D ) + ( D + S ) ) <_ ( ( Q x. D ) + R ) )
47	12, 8, 25	leadd2d 8080	. . . . 5 |- ( ( ph /\ Q < T ) -> ( ( D + S ) <_ R <-> ( ( Q x. D ) + ( D + S ) ) <_ ( ( Q x. D ) + R ) ) )
48	46, 47	mpbird 166	. . . 4 |- ( ( ph /\ Q < T ) -> ( D + S ) <_ R )
49	5, 12, 8, 16, 48	letrd 7670	. . 3 |- ( ( ph /\ Q < T ) -> D <_ R )
50	5, 8, 49	lensymd 7668	. 2 |- ( ( ph /\ Q < T ) -> -. R < D )
51	2, 50	pm2.65da 623	1 |- ( ph -> -. Q < T )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1290   e. wcel 1439   class class class wbr 3853  (class class class)co 5668   RRcr 7412   0cc0 7413   1c1 7414   + caddc 7416   x. cmul 7418   < clt 7585   <_ cle 7586   NNcn 8485   ZZcz 8813

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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-mulrcl 7507  ax-addcom 7508  ax-mulcom 7509  ax-addass 7510  ax-mulass 7511  ax-distr 7512  ax-i2m1 7513  ax-0lt1 7514  ax-1rid 7515  ax-0id 7516  ax-rnegex 7517  ax-precex 7518  ax-cnre 7519  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522  ax-pre-apti 7523  ax-pre-ltadd 7524  ax-pre-mulgt0 7525  ax-pre-mulext 7526

This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2624  df-sbc 2844  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-br 3854  df-opab 3908  df-id 4131  df-po 4134  df-iso 4135  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-iota 4995  df-fun 5032  df-fv 5038  df-riota 5624  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-sub 7718  df-neg 7719  df-reap 8115  df-ap 8122  df-inn 8486  df-n0 8737  df-z 8814

This theorem is referenced by:  divalglemeunn  11262  divalglemeuneg  11264