Theorem divalglemnqt 12561

Description: Lemma for divalg 12565. 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 276	. 2 |- ( ( ph /\ Q < T ) -> R < D )
3		divalglemnqt.d	. . . . 5 |- ( ph -> D e. NN )
4	3	adantr 276	. . . 4 |- ( ( ph /\ Q < T ) -> D e. NN )
5	4	nnred 9215	. . 3 |- ( ( ph /\ Q < T ) -> D e. RR )
6		divalglemnqt.r	. . . . 5 |- ( ph -> R e. ZZ )
7	6	adantr 276	. . . 4 |- ( ( ph /\ Q < T ) -> R e. ZZ )
8	7	zred 9663	. . 3 |- ( ( ph /\ Q < T ) -> R e. RR )
9		divalglemnqt.s	. . . . . . 7 |- ( ph -> S e. ZZ )
10	9	adantr 276	. . . . . 6 |- ( ( ph /\ Q < T ) -> S e. ZZ )
11	10	zred 9663	. . . . 5 |- ( ( ph /\ Q < T ) -> S e. RR )
12	5, 11	readdcld 8268	. . . 4 |- ( ( ph /\ Q < T ) -> ( D + S ) e. RR )
13		divalglemnqt.0s	. . . . . 6 |- ( ph -> 0 <_ S )
14	13	adantr 276	. . . . 5 |- ( ( ph /\ Q < T ) -> 0 <_ S )
15	5, 11	addge01d 8772	. . . . 5 |- ( ( ph /\ Q < T ) -> ( 0 <_ S <-> D <_ ( D + S ) ) )
16	14, 15	mpbid 147	. . . 4 |- ( ( ph /\ Q < T ) -> D <_ ( D + S ) )
17		divalglemnqt.q	. . . . . . . . . . 11 |- ( ph -> Q e. ZZ )
18	17	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> Q e. ZZ )
19	18	zred 9663	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> Q e. RR )
20	19	recnd 8267	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> Q e. CC )
21	5	recnd 8267	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> D e. CC )
22	20, 21	mulcld 8259	. . . . . . 7 |- ( ( ph /\ Q < T ) -> ( Q x. D ) e. CC )
23	11	recnd 8267	. . . . . . 7 |- ( ( ph /\ Q < T ) -> S e. CC )
24	22, 21, 23	addassd 8261	. . . . . 6 |- ( ( ph /\ Q < T ) -> ( ( ( Q x. D ) + D ) + S ) = ( ( Q x. D ) + ( D + S ) ) )
25	19, 5	remulcld 8269	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> ( Q x. D ) e. RR )
26	25, 5	readdcld 8268	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> ( ( Q x. D ) + D ) e. RR )
27		divalglemnqt.t	. . . . . . . . . . 11 |- ( ph -> T e. ZZ )
28	27	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> T e. ZZ )
29	28	zred 9663	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> T e. RR )
30	29, 5	remulcld 8269	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> ( T x. D ) e. RR )
31	20, 21	adddirp1d 8265	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> ( ( Q + 1 ) x. D ) = ( ( Q x. D ) + D ) )
32		peano2re 8374	. . . . . . . . . . 11 |- ( Q e. RR -> ( Q + 1 ) e. RR )
33	19, 32	syl 14	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> ( Q + 1 ) e. RR )
34	4	nnnn0d 9516	. . . . . . . . . . 11 |- ( ( ph /\ Q < T ) -> D e. NN0 )
35	34	nn0ge0d 9519	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> 0 <_ D )
36		simpr 110	. . . . . . . . . . 11 |- ( ( ph /\ Q < T ) -> Q < T )
37		zltp1le 9595	. . . . . . . . . . . 12 |- ( ( Q e. ZZ /\ T e. ZZ ) -> ( Q < T <-> ( Q + 1 ) <_ T ) )
38	17, 28, 37	syl2an2r 599	. . . . . . . . . . 11 |- ( ( ph /\ Q < T ) -> ( Q < T <-> ( Q + 1 ) <_ T ) )
39	36, 38	mpbid 147	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> ( Q + 1 ) <_ T )
40	33, 29, 5, 35, 39	lemul1ad 9178	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> ( ( Q + 1 ) x. D ) <_ ( T x. D ) )
41	31, 40	eqbrtrrd 4117	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> ( ( Q x. D ) + D ) <_ ( T x. D ) )
42	26, 30, 11, 41	leadd1dd 8798	. . . . . . 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 276	. . . . . . 7 |- ( ( ph /\ Q < T ) -> ( ( Q x. D ) + R ) = ( ( T x. D ) + S ) )
45	42, 44	breqtrrd 4121	. . . . . 6 |- ( ( ph /\ Q < T ) -> ( ( ( Q x. D ) + D ) + S ) <_ ( ( Q x. D ) + R ) )
46	24, 45	eqbrtrrd 4117	. . . . 5 |- ( ( ph /\ Q < T ) -> ( ( Q x. D ) + ( D + S ) ) <_ ( ( Q x. D ) + R ) )
47	12, 8, 25	leadd2d 8779	. . . . 5 |- ( ( ph /\ Q < T ) -> ( ( D + S ) <_ R <-> ( ( Q x. D ) + ( D + S ) ) <_ ( ( Q x. D ) + R ) ) )
48	46, 47	mpbird 167	. . . 4 |- ( ( ph /\ Q < T ) -> ( D + S ) <_ R )
49	5, 12, 8, 16, 48	letrd 8362	. . 3 |- ( ( ph /\ Q < T ) -> D <_ R )
50	5, 8, 49	lensymd 8360	. 2 |- ( ( ph /\ Q < T ) -> -. R < D )
51	2, 50	pm2.65da 667	1 |- ( ph -> -. Q < T )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   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   ZZcz 9540

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-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210

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-po 4399  df-iso 4400  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-reap 8814  df-ap 8821  df-inn 9203  df-n0 9462  df-z 9541

This theorem is referenced by:  divalglemeunn  12562  divalglemeuneg  12564