Theorem divalglem2 15118

Description: Lemma for divalg 15126. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by AV, 2-Oct-2020.)

Hypotheses

Ref	Expression
divalglem0.1	|- N e. ZZ
divalglem0.2	|- D e. ZZ
divalglem1.3	|- D =/= 0
divalglem2.4	|- S = { r e. NN0 | D || ( N - r ) }

Assertion

Ref	Expression
divalglem2	|- inf ( S , RR , < ) e. S

Distinct variable groups:   D, r   N, r

Allowed substitution hint:   S( r)

Proof of Theorem divalglem2

Step	Hyp	Ref	Expression
1		divalglem2.4	. . . 4 |- S = { r e. NN0 | D || ( N - r ) }
2		ssrab2 3687	. . . 4 |- { r e. NN0 | D || ( N - r ) } C_ NN0
3	1, 2	eqsstri 3635	. . 3 |- S C_ NN0
4		nn0uz 11722	. . 3 |- NN0 = ( ZZ>= ` 0 )
5	3, 4	sseqtri 3637	. 2 |- S C_ ( ZZ>= ` 0 )
6		divalglem0.1	. . . . . 6 |- N e. ZZ
7		divalglem0.2	. . . . . . . . 9 |- D e. ZZ
8		zmulcl 11426	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. ZZ ) -> ( N x. D ) e. ZZ )
9	6, 7, 8	mp2an 708	. . . . . . . 8 |- ( N x. D ) e. ZZ
10		nn0abscl 14052	. . . . . . . 8 |- ( ( N x. D ) e. ZZ -> ( abs ` ( N x. D ) ) e. NN0 )
11	9, 10	ax-mp 5	. . . . . . 7 |- ( abs ` ( N x. D ) ) e. NN0
12	11	nn0zi 11402	. . . . . 6 |- ( abs ` ( N x. D ) ) e. ZZ
13		zaddcl 11417	. . . . . 6 |- ( ( N e. ZZ /\ ( abs ` ( N x. D ) ) e. ZZ ) -> ( N + ( abs ` ( N x. D ) ) ) e. ZZ )
14	6, 12, 13	mp2an 708	. . . . 5 |- ( N + ( abs ` ( N x. D ) ) ) e. ZZ
15		divalglem1.3	. . . . . 6 |- D =/= 0
16	6, 7, 15	divalglem1 15117	. . . . 5 |- 0 <_ ( N + ( abs ` ( N x. D ) ) )
17		elnn0z 11390	. . . . 5 |- ( ( N + ( abs ` ( N x. D ) ) ) e. NN0 <-> ( ( N + ( abs ` ( N x. D ) ) ) e. ZZ /\ 0 <_ ( N + ( abs ` ( N x. D ) ) ) ) )
18	14, 16, 17	mpbir2an 955	. . . 4 |- ( N + ( abs ` ( N x. D ) ) ) e. NN0
19		iddvds 14995	. . . . . . . 8 |- ( D e. ZZ -> D || D )
20		dvdsabsb 15001	. . . . . . . . 9 |- ( ( D e. ZZ /\ D e. ZZ ) -> ( D || D <-> D || ( abs ` D ) ) )
21	20	anidms 677	. . . . . . . 8 |- ( D e. ZZ -> ( D || D <-> D || ( abs ` D ) ) )
22	19, 21	mpbid 222	. . . . . . 7 |- ( D e. ZZ -> D || ( abs ` D ) )
23	7, 22	ax-mp 5	. . . . . 6 |- D || ( abs ` D )
24		nn0abscl 14052	. . . . . . . . 9 |- ( N e. ZZ -> ( abs ` N ) e. NN0 )
25	6, 24	ax-mp 5	. . . . . . . 8 |- ( abs ` N ) e. NN0
26	25	nn0negzi 11416	. . . . . . 7 |- -u ( abs ` N ) e. ZZ
27		nn0abscl 14052	. . . . . . . . 9 |- ( D e. ZZ -> ( abs ` D ) e. NN0 )
28	7, 27	ax-mp 5	. . . . . . . 8 |- ( abs ` D ) e. NN0
29	28	nn0zi 11402	. . . . . . 7 |- ( abs ` D ) e. ZZ
30		dvdsmultr2 15021	. . . . . . 7 |- ( ( D e. ZZ /\ -u ( abs ` N ) e. ZZ /\ ( abs ` D ) e. ZZ ) -> ( D || ( abs ` D ) -> D || ( -u ( abs ` N ) x. ( abs ` D ) ) ) )
31	7, 26, 29, 30	mp3an 1424	. . . . . 6 |- ( D || ( abs ` D ) -> D || ( -u ( abs ` N ) x. ( abs ` D ) ) )
32	23, 31	ax-mp 5	. . . . 5 |- D || ( -u ( abs ` N ) x. ( abs ` D ) )
33		zcn 11382	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
34	6, 33	ax-mp 5	. . . . . . . 8 |- N e. CC
35		zcn 11382	. . . . . . . . 9 |- ( D e. ZZ -> D e. CC )
36	7, 35	ax-mp 5	. . . . . . . 8 |- D e. CC
37	34, 36	absmuli 14143	. . . . . . 7 |- ( abs ` ( N x. D ) ) = ( ( abs ` N ) x. ( abs ` D ) )
38	37	negeqi 10274	. . . . . 6 |- -u ( abs ` ( N x. D ) ) = -u ( ( abs ` N ) x. ( abs ` D ) )
39		df-neg 10269	. . . . . . 7 |- -u ( abs ` ( N x. D ) ) = ( 0 - ( abs ` ( N x. D ) ) )
40	34	subidi 10352	. . . . . . . 8 |- ( N - N ) = 0
41	40	oveq1i 6660	. . . . . . 7 |- ( ( N - N ) - ( abs ` ( N x. D ) ) ) = ( 0 - ( abs ` ( N x. D ) ) )
42	11	nn0cni 11304	. . . . . . . 8 |- ( abs ` ( N x. D ) ) e. CC
43		subsub4 10314	. . . . . . . 8 |- ( ( N e. CC /\ N e. CC /\ ( abs ` ( N x. D ) ) e. CC ) -> ( ( N - N ) - ( abs ` ( N x. D ) ) ) = ( N - ( N + ( abs ` ( N x. D ) ) ) ) )
44	34, 34, 42, 43	mp3an 1424	. . . . . . 7 |- ( ( N - N ) - ( abs ` ( N x. D ) ) ) = ( N - ( N + ( abs ` ( N x. D ) ) ) )
45	39, 41, 44	3eqtr2ri 2651	. . . . . 6 |- ( N - ( N + ( abs ` ( N x. D ) ) ) ) = -u ( abs ` ( N x. D ) )
46	34	abscli 14134	. . . . . . . 8 |- ( abs ` N ) e. RR
47	46	recni 10052	. . . . . . 7 |- ( abs ` N ) e. CC
48	36	abscli 14134	. . . . . . . 8 |- ( abs ` D ) e. RR
49	48	recni 10052	. . . . . . 7 |- ( abs ` D ) e. CC
50	47, 49	mulneg1i 10476	. . . . . 6 |- ( -u ( abs ` N ) x. ( abs ` D ) ) = -u ( ( abs ` N ) x. ( abs ` D ) )
51	38, 45, 50	3eqtr4i 2654	. . . . 5 |- ( N - ( N + ( abs ` ( N x. D ) ) ) ) = ( -u ( abs ` N ) x. ( abs ` D ) )
52	32, 51	breqtrri 4680	. . . 4 |- D || ( N - ( N + ( abs ` ( N x. D ) ) ) )
53		oveq2 6658	. . . . . 6 |- ( r = ( N + ( abs ` ( N x. D ) ) ) -> ( N - r ) = ( N - ( N + ( abs ` ( N x. D ) ) ) ) )
54	53	breq2d 4665	. . . . 5 |- ( r = ( N + ( abs ` ( N x. D ) ) ) -> ( D || ( N - r ) <-> D || ( N - ( N + ( abs ` ( N x. D ) ) ) ) ) )
55	54, 1	elrab2 3366	. . . 4 |- ( ( N + ( abs ` ( N x. D ) ) ) e. S <-> ( ( N + ( abs ` ( N x. D ) ) ) e. NN0 /\ D || ( N - ( N + ( abs ` ( N x. D ) ) ) ) ) )
56	18, 52, 55	mpbir2an 955	. . 3 |- ( N + ( abs ` ( N x. D ) ) ) e. S
57	56	ne0ii 3923	. 2 |- S =/= (/)
58		infssuzcl 11772	. 2 |- ( ( S C_ ( ZZ>= ` 0 ) /\ S =/= (/) ) -> inf ( S , RR , < ) e. S )
59	5, 57, 58	mp2an 708	1 |- inf ( S , RR , < ) e. S

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650  infcinf 8347   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   abscabs 13974   || cdvds 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984

This theorem is referenced by:  divalglem5  15120  divalglem9  15124