Theorem divalglem10 13705

Description: Lemma for divalg 13706.

Hypotheses

Ref	Expression
divalglem8.1	|- N e. ZZ
divalglem8.2	|- D e. ZZ
divalglem8.3	|- D =/= 0
divalglem8.4	|- S = {r e. NN0 | D||(N - r)}

Assertion

Ref	Expression
divalglem10	|- E!r e. ZZ E.q e. ZZ (0 <_ r /\ r < (abs` D) /\ N = ((q x. D) + r))

Distinct variable groups:   D,q,r   N,q,r

Proof of Theorem divalglem10

Step	Hyp	Ref	Expression
1		divalglem8.1	. . . 4 |- N e. ZZ
2		divalglem8.2	. . . 4 |- D e. ZZ
3		divalglem8.3	. . . 4 |- D =/= 0
4		divalglem8.4	. . . 4 |- S = {r e. NN0 | D||(N - r)}
5		eqid 1884	. . . 4 |- sup(S, RR, `' < ) = sup(S, RR, `' < )
6	1, 2, 3, 4, 5	divalglem9 13704	. . 3 |- E!x e. S x < (abs` D)
7		opreq2 4890	. . . . . . . . . . . 12 |- (r = x -> ((q x. D) + r) = ((q x. D) + x))
8	7	eqeq2d 1895	. . . . . . . . . . 11 |- (r = x -> (N = ((q x. D) + r) <-> N = ((q x. D) + x)))
9	8	rexbidv 2124	. . . . . . . . . 10 |- (r = x -> (E.q e. ZZ N = ((q x. D) + r) <-> E.q e. ZZ N = ((q x. D) + x)))
10	1, 2, 3, 4	divalglem4 13699	. . . . . . . . . 10 |- S = {r e. NN0 | E.q e. ZZ N = ((q x. D) + r)}
11	9, 10	elrab2 2416	. . . . . . . . 9 |- (x e. S <-> (x e. NN0 /\ E.q e. ZZ N = ((q x. D) + x)))
12	11	anbi2i 538	. . . . . . . 8 |- ((x < (abs` D) /\ x e. S) <-> (x < (abs` D) /\ (x e. NN0 /\ E.q e. ZZ N = ((q x. D) + x))))
13		ancom 482	. . . . . . . 8 |- ((x e. S /\ x < (abs` D)) <-> (x < (abs` D) /\ x e. S))
14		anass 487	. . . . . . . 8 |- (((x < (abs` D) /\ x e. NN0) /\ E.q e. ZZ N = ((q x. D) + x)) <-> (x < (abs` D) /\ (x e. NN0 /\ E.q e. ZZ N = ((q x. D) + x))))
15	12, 13, 14	3bitr4i 200	. . . . . . 7 |- ((x e. S /\ x < (abs` D)) <-> ((x < (abs` D) /\ x e. NN0) /\ E.q e. ZZ N = ((q x. D) + x)))
16		elnn0z 7356	. . . . . . . . . 10 |- (x e. NN0 <-> (x e. ZZ /\ 0 <_ x))
17	16	anbi2i 538	. . . . . . . . 9 |- ((x < (abs` D) /\ x e. NN0) <-> (x < (abs` D) /\ (x e. ZZ /\ 0 <_ x)))
18		an12 542	. . . . . . . . 9 |- ((x < (abs` D) /\ (x e. ZZ /\ 0 <_ x)) <-> (x e. ZZ /\ (x < (abs` D) /\ 0 <_ x)))
19		ancom 482	. . . . . . . . . 10 |- ((x < (abs` D) /\ 0 <_ x) <-> (0 <_ x /\ x < (abs` D)))
20	19	anbi2i 538	. . . . . . . . 9 |- ((x e. ZZ /\ (x < (abs` D) /\ 0 <_ x)) <-> (x e. ZZ /\ (0 <_ x /\ x < (abs` D))))
21	17, 18, 20	3bitri 194	. . . . . . . 8 |- ((x < (abs` D) /\ x e. NN0) <-> (x e. ZZ /\ (0 <_ x /\ x < (abs` D))))
22	21	anbi1i 539	. . . . . . 7 |- (((x < (abs` D) /\ x e. NN0) /\ E.q e. ZZ N = ((q x. D) + x)) <-> ((x e. ZZ /\ (0 <_ x /\ x < (abs` D))) /\ E.q e. ZZ N = ((q x. D) + x)))
23		anass 487	. . . . . . 7 |- (((x e. ZZ /\ (0 <_ x /\ x < (abs` D))) /\ E.q e. ZZ N = ((q x. D) + x)) <-> (x e. ZZ /\ ((0 <_ x /\ x < (abs` D)) /\ E.q e. ZZ N = ((q x. D) + x))))
24	15, 22, 23	3bitri 194	. . . . . 6 |- ((x e. S /\ x < (abs` D)) <-> (x e. ZZ /\ ((0 <_ x /\ x < (abs` D)) /\ E.q e. ZZ N = ((q x. D) + x))))
25		df-3an 860	. . . . . . . . 9 |- ((0 <_ x /\ x < (abs` D) /\ N = ((q x. D) + x)) <-> ((0 <_ x /\ x < (abs` D)) /\ N = ((q x. D) + x)))
26	25	rexbii 2128	. . . . . . . 8 |- (E.q e. ZZ (0 <_ x /\ x < (abs` D) /\ N = ((q x. D) + x)) <-> E.q e. ZZ ((0 <_ x /\ x < (abs` D)) /\ N = ((q x. D) + x)))
27		r19.42v 2237	. . . . . . . 8 |- (E.q e. ZZ ((0 <_ x /\ x < (abs` D)) /\ N = ((q x. D) + x)) <-> ((0 <_ x /\ x < (abs` D)) /\ E.q e. ZZ N = ((q x. D) + x)))
28	26, 27	bitri 190	. . . . . . 7 |- (E.q e. ZZ (0 <_ x /\ x < (abs` D) /\ N = ((q x. D) + x)) <-> ((0 <_ x /\ x < (abs` D)) /\ E.q e. ZZ N = ((q x. D) + x)))
29	28	anbi2i 538	. . . . . 6 |- ((x e. ZZ /\ E.q e. ZZ (0 <_ x /\ x < (abs` D) /\ N = ((q x. D) + x))) <-> (x e. ZZ /\ ((0 <_ x /\ x < (abs` D)) /\ E.q e. ZZ N = ((q x. D) + x))))
30	24, 29	bitr4i 193	. . . . 5 |- ((x e. S /\ x < (abs` D)) <-> (x e. ZZ /\ E.q e. ZZ (0 <_ x /\ x < (abs` D) /\ N = ((q x. D) + x))))
31	30	eubii 1780	. . . 4 |- (E!x(x e. S /\ x < (abs` D)) <-> E!x(x e. ZZ /\ E.q e. ZZ (0 <_ x /\ x < (abs` D) /\ N = ((q x. D) + x))))
32		df-reu 2111	. . . 4 |- (E!x e. S x < (abs` D) <-> E!x(x e. S /\ x < (abs` D)))
33		df-reu 2111	. . . 4 |- (E!x e. ZZ E.q e. ZZ (0 <_ x /\ x < (abs` D) /\ N = ((q x. D) + x)) <-> E!x(x e. ZZ /\ E.q e. ZZ (0 <_ x /\ x < (abs` D) /\ N = ((q x. D) + x))))
34	31, 32, 33	3bitr4i 200	. . 3 |- (E!x e. S x < (abs` D) <-> E!x e. ZZ E.q e. ZZ (0 <_ x /\ x < (abs` D) /\ N = ((q x. D) + x)))
35	6, 34	mpbi 206	. 2 |- E!x e. ZZ E.q e. ZZ (0 <_ x /\ x < (abs` D) /\ N = ((q x. D) + x))
36		breq2 3342	. . . . 5 |- (x = r -> (0 <_ x <-> 0 <_ r))
37		breq1 3341	. . . . 5 |- (x = r -> (x < (abs` D) <-> r < (abs` D)))
38		opreq2 4890	. . . . . 6 |- (x = r -> ((q x. D) + x) = ((q x. D) + r))
39	38	eqeq2d 1895	. . . . 5 |- (x = r -> (N = ((q x. D) + x) <-> N = ((q x. D) + r)))
40	36, 37, 39	3anbi123d 1168	. . . 4 |- (x = r -> ((0 <_ x /\ x < (abs` D) /\ N = ((q x. D) + x)) <-> (0 <_ r /\ r < (abs` D) /\ N = ((q x. D) + r))))
41	40	rexbidv 2124	. . 3 |- (x = r -> (E.q e. ZZ (0 <_ x /\ x < (abs` D) /\ N = ((q x. D) + x)) <-> E.q e. ZZ (0 <_ r /\ r < (abs` D) /\ N = ((q x. D) + r))))
42	41	cbvreuv 2282	. 2 |- (E!x e. ZZ E.q e. ZZ (0 <_ x /\ x < (abs` D) /\ N = ((q x. D) + x)) <-> E!r e. ZZ E.q e. ZZ (0 <_ r /\ r < (abs` D) /\ N = ((q x. D) + r)))
43	35, 42	mpbi 206	1 |- E!r e. ZZ E.q e. ZZ (0 <_ r /\ r < (abs` D) /\ N = ((q x. D) + r))

Syntax hints:   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E!weu 1771   =/= wne 2017  E.wrex 2106  E!wreu 2107  {crab 2108   class class class wbr 3338  `'ccnv 3985  ` cfv 3998  (class class class)co 4884  supcsup 5663  RRcr 6385  0cc0 6386   + caddc 6389   x. cmul 6391   - cmin 6445   <_ cle 6448  NN0cn0 6450  ZZcz 6451   < clt 6653  abscabs 8000  ||cdivides 13662

This theorem is referenced by:  divalg 13706

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-divides 13663

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