Theorem divalglemnqt 11617

Description: Lemma for divalg 11621. 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 274	. 2 |- ( ( ph /\ Q < T ) -> R < D )
3		divalglemnqt.d	. . . . 5 |- ( ph -> D e. NN )
4	3	adantr 274	. . . 4 |- ( ( ph /\ Q < T ) -> D e. NN )
5	4	nnred 8733	. . 3 |- ( ( ph /\ Q < T ) -> D e. RR )
6		divalglemnqt.r	. . . . 5 |- ( ph -> R e. ZZ )
7	6	adantr 274	. . . 4 |- ( ( ph /\ Q < T ) -> R e. ZZ )
8	7	zred 9173	. . 3 |- ( ( ph /\ Q < T ) -> R e. RR )
9		divalglemnqt.s	. . . . . . 7 |- ( ph -> S e. ZZ )
10	9	adantr 274	. . . . . 6 |- ( ( ph /\ Q < T ) -> S e. ZZ )
11	10	zred 9173	. . . . 5 |- ( ( ph /\ Q < T ) -> S e. RR )
12	5, 11	readdcld 7795	. . . 4 |- ( ( ph /\ Q < T ) -> ( D + S ) e. RR )
13		divalglemnqt.0s	. . . . . 6 |- ( ph -> 0 <_ S )
14	13	adantr 274	. . . . 5 |- ( ( ph /\ Q < T ) -> 0 <_ S )
15	5, 11	addge01d 8295	. . . . 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 274	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> Q e. ZZ )
19	18	zred 9173	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> Q e. RR )
20	19	recnd 7794	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> Q e. CC )
21	5	recnd 7794	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> D e. CC )
22	20, 21	mulcld 7786	. . . . . . 7 |- ( ( ph /\ Q < T ) -> ( Q x. D ) e. CC )
23	11	recnd 7794	. . . . . . 7 |- ( ( ph /\ Q < T ) -> S e. CC )
24	22, 21, 23	addassd 7788	. . . . . 6 |- ( ( ph /\ Q < T ) -> ( ( ( Q x. D ) + D ) + S ) = ( ( Q x. D ) + ( D + S ) ) )
25	19, 5	remulcld 7796	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> ( Q x. D ) e. RR )
26	25, 5	readdcld 7795	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> ( ( Q x. D ) + D ) e. RR )
27		divalglemnqt.t	. . . . . . . . . . 11 |- ( ph -> T e. ZZ )
28	27	adantr 274	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> T e. ZZ )
29	28	zred 9173	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> T e. RR )
30	29, 5	remulcld 7796	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> ( T x. D ) e. RR )
31	20, 21	adddirp1d 7792	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> ( ( Q + 1 ) x. D ) = ( ( Q x. D ) + D ) )
32		peano2re 7898	. . . . . . . . . . 11 |- ( Q e. RR -> ( Q + 1 ) e. RR )
33	19, 32	syl 14	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> ( Q + 1 ) e. RR )
34	4	nnnn0d 9030	. . . . . . . . . . 11 |- ( ( ph /\ Q < T ) -> D e. NN0 )
35	34	nn0ge0d 9033	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> 0 <_ D )
36		simpr 109	. . . . . . . . . . 11 |- ( ( ph /\ Q < T ) -> Q < T )
37		zltp1le 9108	. . . . . . . . . . . 12 |- ( ( Q e. ZZ /\ T e. ZZ ) -> ( Q < T <-> ( Q + 1 ) <_ T ) )
38	17, 28, 37	syl2an2r 584	. . . . . . . . . . 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 8697	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> ( ( Q + 1 ) x. D ) <_ ( T x. D ) )
41	31, 40	eqbrtrrd 3952	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> ( ( Q x. D ) + D ) <_ ( T x. D ) )
42	26, 30, 11, 41	leadd1dd 8321	. . . . . . 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 274	. . . . . . 7 |- ( ( ph /\ Q < T ) -> ( ( Q x. D ) + R ) = ( ( T x. D ) + S ) )
45	42, 44	breqtrrd 3956	. . . . . 6 |- ( ( ph /\ Q < T ) -> ( ( ( Q x. D ) + D ) + S ) <_ ( ( Q x. D ) + R ) )
46	24, 45	eqbrtrrd 3952	. . . . 5 |- ( ( ph /\ Q < T ) -> ( ( Q x. D ) + ( D + S ) ) <_ ( ( Q x. D ) + R ) )
47	12, 8, 25	leadd2d 8302	. . . . 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 7886	. . 3 |- ( ( ph /\ Q < T ) -> D <_ R )
50	5, 8, 49	lensymd 7884	. 2 |- ( ( ph /\ Q < T ) -> -. R < D )
51	2, 50	pm2.65da 650	1 |- ( ph -> -. Q < T )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   class class class wbr 3929  (class class class)co 5774   RRcr 7619   0cc0 7620   1c1 7621   + caddc 7623   x. cmul 7625   < clt 7800   <_ cle 7801   NNcn 8720   ZZcz 9054

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-inn 8721  df-n0 8978  df-z 9055

This theorem is referenced by:  divalglemeunn  11618  divalglemeuneg  11620