Theorem divalglemnqt 12665

Description: Lemma for divalg 12669. 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 9296	. . 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 9747	. . 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 9747	. . . . 5 |- ( ( ph /\ Q < T ) -> S e. RR )
12	5, 11	readdcld 8345	. . . 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 8851	. . . . 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 9747	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> Q e. RR )
20	19	recnd 8344	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> Q e. CC )
21	5	recnd 8344	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> D e. CC )
22	20, 21	mulcld 8336	. . . . . . 7 |- ( ( ph /\ Q < T ) -> ( Q x. D ) e. CC )
23	11	recnd 8344	. . . . . . 7 |- ( ( ph /\ Q < T ) -> S e. CC )
24	22, 21, 23	addassd 8338	. . . . . 6 |- ( ( ph /\ Q < T ) -> ( ( ( Q x. D ) + D ) + S ) = ( ( Q x. D ) + ( D + S ) ) )
25	19, 5	remulcld 8346	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> ( Q x. D ) e. RR )
26	25, 5	readdcld 8345	. . . . . . . 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 9747	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> T e. RR )
30	29, 5	remulcld 8346	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> ( T x. D ) e. RR )
31	20, 21	adddirp1d 8342	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> ( ( Q + 1 ) x. D ) = ( ( Q x. D ) + D ) )
32		peano2re 8452	. . . . . . . . . . 11 |- ( Q e. RR -> ( Q + 1 ) e. RR )
33	19, 32	syl 14	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> ( Q + 1 ) e. RR )
34	4	nnnn0d 9599	. . . . . . . . . . 11 |- ( ( ph /\ Q < T ) -> D e. NN0 )
35	34	nn0ge0d 9602	. . . . . . . . . 10 |- ( ( ph /\ Q < T ) -> 0 <_ D )
36		simpr 110	. . . . . . . . . . 11 |- ( ( ph /\ Q < T ) -> Q < T )
37		zltp1le 9678	. . . . . . . . . . . 12 |- ( ( Q e. ZZ /\ T e. ZZ ) -> ( Q < T <-> ( Q + 1 ) <_ T ) )
38	17, 28, 37	syl2an2r 603	. . . . . . . . . . 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 9259	. . . . . . . . 9 |- ( ( ph /\ Q < T ) -> ( ( Q + 1 ) x. D ) <_ ( T x. D ) )
41	31, 40	eqbrtrrd 4149	. . . . . . . 8 |- ( ( ph /\ Q < T ) -> ( ( Q x. D ) + D ) <_ ( T x. D ) )
42	26, 30, 11, 41	leadd1dd 8877	. . . . . . 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 4153	. . . . . 6 |- ( ( ph /\ Q < T ) -> ( ( ( Q x. D ) + D ) + S ) <_ ( ( Q x. D ) + R ) )
46	24, 45	eqbrtrrd 4149	. . . . 5 |- ( ( ph /\ Q < T ) -> ( ( Q x. D ) + ( D + S ) ) <_ ( ( Q x. D ) + R ) )
47	12, 8, 25	leadd2d 8858	. . . . 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 8440	. . 3 |- ( ( ph /\ Q < T ) -> D <_ R )
50	5, 8, 49	lensymd 8438	. 2 |- ( ( ph /\ Q < T ) -> -. R < D )
51	2, 50	pm2.65da 671	1 |- ( ph -> -. Q < T )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   RRcr 8168   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   < clt 8350   <_ cle 8351   NNcn 9283   ZZcz 9623

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-inn 9284  df-n0 9543  df-z 9624

This theorem is referenced by:  divalglemeunn  12666  divalglemeuneg  12668