Theorem divalglemnqt 11879

Description: Lemma for divalg 11883. 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 8891	. . 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 9334	. . 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 9334	. . . . 5 |- ( ( ph /\ Q < T ) -> S e. RR )
12	5, 11	readdcld 7949	. . . 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 8452	. . . . 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 9334	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> Q e. RR )
20	19	recnd 7948	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> Q e. CC )
21	5	recnd 7948	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> D e. CC )
22	20, 21	mulcld 7940	. . . . . . 7 |- ( ( ph /\ Q < T ) -> ( Q x. D ) e. CC )
23	11	recnd 7948	. . . . . . 7 |- ( ( ph /\ Q < T ) -> S e. CC )
24	22, 21, 23	addassd 7942	. . . . . 6 |- ( ( ph /\ Q < T ) -> ( ( ( Q x. D ) + D ) + S ) = ( ( Q x. D ) + ( D + S ) ) )
25	19, 5	remulcld 7950	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> ( Q x. D ) e. RR )
26	25, 5	readdcld 7949	. . . . . . . 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 9334	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> T e. RR )
30	29, 5	remulcld 7950	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> ( T x. D ) e. RR )
31	20, 21	adddirp1d 7946	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> ( ( Q + 1 ) x. D ) = ( ( Q x. D ) + D ) )
32		peano2re 8055	. . . . . . . . . . 11 |- ( Q e. RR -> ( Q + 1 ) e. RR )
33	19, 32	syl 14	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> ( Q + 1 ) e. RR )
34	4	nnnn0d 9188	. . . . . . . . . . 11 |- ( ( ph /\ Q < T ) -> D e. NN0 )
35	34	nn0ge0d 9191	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> 0 <_ D )
36		simpr 109	. . . . . . . . . . 11 |- ( ( ph /\ Q < T ) -> Q < T )
37		zltp1le 9266	. . . . . . . . . . . 12 |- ( ( Q e. ZZ /\ T e. ZZ ) -> ( Q < T <-> ( Q + 1 ) <_ T ) )
38	17, 28, 37	syl2an2r 590	. . . . . . . . . . 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 8855	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> ( ( Q + 1 ) x. D ) <_ ( T x. D ) )
41	31, 40	eqbrtrrd 4013	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> ( ( Q x. D ) + D ) <_ ( T x. D ) )
42	26, 30, 11, 41	leadd1dd 8478	. . . . . . 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 4017	. . . . . 6 |- ( ( ph /\ Q < T ) -> ( ( ( Q x. D ) + D ) + S ) <_ ( ( Q x. D ) + R ) )
46	24, 45	eqbrtrrd 4013	. . . . 5 |- ( ( ph /\ Q < T ) -> ( ( Q x. D ) + ( D + S ) ) <_ ( ( Q x. D ) + R ) )
47	12, 8, 25	leadd2d 8459	. . . . 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 8043	. . 3 |- ( ( ph /\ Q < T ) -> D <_ R )
50	5, 8, 49	lensymd 8041	. 2 |- ( ( ph /\ Q < T ) -> -. R < D )
51	2, 50	pm2.65da 656	1 |- ( ph -> -. Q < T )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   < clt 7954   <_ cle 7955   NNcn 8878   ZZcz 9212

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 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

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 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-inn 8879  df-n0 9136  df-z 9213

This theorem is referenced by:  divalglemeunn  11880  divalglemeuneg  11882