Theorem divalglem9 13704

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)}
divalglem9.5	|- R = sup(S, RR, `' < )

Assertion

Ref	Expression
divalglem9	|- E!x e. S x < (abs` D)

Distinct variable groups:   D,r,x   N,r,x   x,S   x,R

Proof of Theorem divalglem9

Step	Hyp	Ref	Expression
1		breq1 3341	. . 3 |- (x = y -> (x < (abs` D) <-> y < (abs` D)))
2	1	reu4 2446	. 2 |- (E!x e. S x < (abs` D) <-> (E.x e. S x < (abs` D) /\ A.x e. S A.y e. S ((x < (abs` D) /\ y < (abs` D)) -> x = y)))
3		divalglem9.5	. . . 4 |- R = sup(S, RR, `' < )
4		divalglem8.1	. . . . 5 |- N e. ZZ
5		divalglem8.2	. . . . 5 |- D e. ZZ
6		divalglem8.3	. . . . 5 |- D =/= 0
7		divalglem8.4	. . . . 5 |- S = {r e. NN0 | D||(N - r)}
8	4, 5, 6, 7	divalglem2 13698	. . . 4 |- sup(S, RR, `' < ) e. S
9	3, 8	eqeltri 1967	. . 3 |- R e. S
10	4, 5, 6, 7, 3	divalglem5 13700	. . . 4 |- (0 <_ R /\ R < (abs` D))
11	10	simpri 351	. . 3 |- R < (abs` D)
12		breq1 3341	. . . 4 |- (x = R -> (x < (abs` D) <-> R < (abs` D)))
13	12	rcla4ev 2381	. . 3 |- ((R e. S /\ R < (abs` D)) -> E.x e. S x < (abs` D))
14	9, 11, 13	mp2an 761	. 2 |- E.x e. S x < (abs` D)
15		opreq2 4890	. . . . . . . . . . . . . 14 |- (r = x -> (N - r) = (N - x))
16	15	breq2d 3350	. . . . . . . . . . . . 13 |- (r = x -> (D||(N - r) <-> D||(N - x)))
17	16, 7	elrab2 2416	. . . . . . . . . . . 12 |- (x e. S <-> (x e. NN0 /\ D||(N - x)))
18	17	simprbi 353	. . . . . . . . . . 11 |- (x e. S -> D||(N - x))
19		opreq2 4890	. . . . . . . . . . . . . 14 |- (r = y -> (N - r) = (N - y))
20	19	breq2d 3350	. . . . . . . . . . . . 13 |- (r = y -> (D||(N - r) <-> D||(N - y)))
21	20, 7	elrab2 2416	. . . . . . . . . . . 12 |- (y e. S <-> (y e. NN0 /\ D||(N - y)))
22	21	simprbi 353	. . . . . . . . . . 11 |- (y e. S -> D||(N - y))
23	18, 22	anim12i 360	. . . . . . . . . 10 |- ((x e. S /\ y e. S) -> (D||(N - x) /\ D||(N - y)))
24	17	simplbi 349	. . . . . . . . . . . . . 14 |- (x e. S -> x e. NN0)
25		nn0z 7363	. . . . . . . . . . . . . 14 |- (x e. NN0 -> x e. ZZ)
26	24, 25	syl 12	. . . . . . . . . . . . 13 |- (x e. S -> x e. ZZ)
27	21	simplbi 349	. . . . . . . . . . . . . 14 |- (y e. S -> y e. NN0)
28		nn0z 7363	. . . . . . . . . . . . . 14 |- (y e. NN0 -> y e. ZZ)
29	27, 28	syl 12	. . . . . . . . . . . . 13 |- (y e. S -> y e. ZZ)
30	26, 29	anim12i 360	. . . . . . . . . . . 12 |- ((x e. S /\ y e. S) -> (x e. ZZ /\ y e. ZZ))
31		zsubcl 7377	. . . . . . . . . . . . . 14 |- ((N e. ZZ /\ x e. ZZ) -> (N - x) e. ZZ)
32	4, 31	mpan 759	. . . . . . . . . . . . 13 |- (x e. ZZ -> (N - x) e. ZZ)
33		zsubcl 7377	. . . . . . . . . . . . . 14 |- ((N e. ZZ /\ y e. ZZ) -> (N - y) e. ZZ)
34	4, 33	mpan 759	. . . . . . . . . . . . 13 |- (y e. ZZ -> (N - y) e. ZZ)
35	32, 34	anim12i 360	. . . . . . . . . . . 12 |- ((x e. ZZ /\ y e. ZZ) -> ((N - x) e. ZZ /\ (N - y) e. ZZ))
36	30, 35	syl 12	. . . . . . . . . . 11 |- ((x e. S /\ y e. S) -> ((N - x) e. ZZ /\ (N - y) e. ZZ))
37		dvds2sub 13686	. . . . . . . . . . . 12 |- ((D e. ZZ /\ (N - x) e. ZZ /\ (N - y) e. ZZ) -> ((D||(N - x) /\ D||(N - y)) -> D||((N - x) - (N - y))))
38	5, 37	mp3an1 1178	. . . . . . . . . . 11 |- (((N - x) e. ZZ /\ (N - y) e. ZZ) -> ((D||(N - x) /\ D||(N - y)) -> D||((N - x) - (N - y))))
39	36, 38	syl 12	. . . . . . . . . 10 |- ((x e. S /\ y e. S) -> ((D||(N - x) /\ D||(N - y)) -> D||((N - x) - (N - y))))
40	23, 39	mpd 29	. . . . . . . . 9 |- ((x e. S /\ y e. S) -> D||((N - x) - (N - y)))
41		0cn 6481	. . . . . . . . . . . . . . 15 |- 0 e. CC
42		subsub2 6626	. . . . . . . . . . . . . . 15 |- ((0 e. CC /\ x e. CC /\ y e. CC) -> (0 - (x - y)) = (0 + (y - x)))
43	41, 42	mp3an1 1178	. . . . . . . . . . . . . 14 |- ((x e. CC /\ y e. CC) -> (0 - (x - y)) = (0 + (y - x)))
44	4	zrei 7350	. . . . . . . . . . . . . . . . 17 |- N e. RR
45	44	recni 6467	. . . . . . . . . . . . . . . 16 |- N e. CC
46		subid 6555	. . . . . . . . . . . . . . . 16 |- (N e. CC -> (N - N) = 0)
47	45, 46	ax-mp 7	. . . . . . . . . . . . . . 15 |- (N - N) = 0
48	47	opreq1i 4892	. . . . . . . . . . . . . 14 |- ((N - N) - (x - y)) = (0 - (x - y))
49	43, 48	syl5eq 1940	. . . . . . . . . . . . 13 |- ((x e. CC /\ y e. CC) -> ((N - N) - (x - y)) = (0 + (y - x)))
50		sub4 6643	. . . . . . . . . . . . . 14 |- (((N e. CC /\ N e. CC) /\ (x e. CC /\ y e. CC)) -> ((N - N) - (x - y)) = ((N - x) - (N - y)))
51	45, 45, 50	mpanl12 773	. . . . . . . . . . . . 13 |- ((x e. CC /\ y e. CC) -> ((N - N) - (x - y)) = ((N - x) - (N - y)))
52		subcl 6524	. . . . . . . . . . . . . . 15 |- ((y e. CC /\ x e. CC) -> (y - x) e. CC)
53	52	ancoms 484	. . . . . . . . . . . . . 14 |- ((x e. CC /\ y e. CC) -> (y - x) e. CC)
54		addid2 6482	. . . . . . . . . . . . . 14 |- ((y - x) e. CC -> (0 + (y - x)) = (y - x))
55	53, 54	syl 12	. . . . . . . . . . . . 13 |- ((x e. CC /\ y e. CC) -> (0 + (y - x)) = (y - x))
56	49, 51, 55	3eqtr3d 1934	. . . . . . . . . . . 12 |- ((x e. CC /\ y e. CC) -> ((N - x) - (N - y)) = (y - x))
57		zcn 7349	. . . . . . . . . . . 12 |- (x e. ZZ -> x e. CC)
58		zcn 7349	. . . . . . . . . . . 12 |- (y e. ZZ -> y e. CC)
59	56, 57, 58	syl2an 503	. . . . . . . . . . 11 |- ((x e. ZZ /\ y e. ZZ) -> ((N - x) - (N - y)) = (y - x))
60	30, 59	syl 12	. . . . . . . . . 10 |- ((x e. S /\ y e. S) -> ((N - x) - (N - y)) = (y - x))
61	60	breq2d 3350	. . . . . . . . 9 |- ((x e. S /\ y e. S) -> (D||((N - x) - (N - y)) <-> D||(y - x)))
62	40, 61	mpbid 212	. . . . . . . 8 |- ((x e. S /\ y e. S) -> D||(y - x))
63		zsubcl 7377	. . . . . . . . . . 11 |- ((y e. ZZ /\ x e. ZZ) -> (y - x) e. ZZ)
64	63	ancoms 484	. . . . . . . . . 10 |- ((x e. ZZ /\ y e. ZZ) -> (y - x) e. ZZ)
65		absdvdsb 13673	. . . . . . . . . . 11 |- ((D e. ZZ /\ (y - x) e. ZZ) -> (D||(y - x) <-> (abs` D)||(y - x)))
66	5, 65	mpan 759	. . . . . . . . . 10 |- ((y - x) e. ZZ -> (D||(y - x) <-> (abs` D)||(y - x)))
67	64, 66	syl 12	. . . . . . . . 9 |- ((x e. ZZ /\ y e. ZZ) -> (D||(y - x) <-> (abs` D)||(y - x)))
68	30, 67	syl 12	. . . . . . . 8 |- ((x e. S /\ y e. S) -> (D||(y - x) <-> (abs` D)||(y - x)))
69	62, 68	mpbid 212	. . . . . . 7 |- ((x e. S /\ y e. S) -> (abs` D)||(y - x))
70		nnabscl 13601	. . . . . . . . . . . 12 |- ((D e. ZZ /\ D =/= 0) -> (abs` D) e. NN)
71	5, 6, 70	mp2an 761	. . . . . . . . . . 11 |- (abs` D) e. NN
72		nnz 7362	. . . . . . . . . . 11 |- ((abs` D) e. NN -> (abs` D) e. ZZ)
73	71, 72	ax-mp 7	. . . . . . . . . 10 |- (abs` D) e. ZZ
74		divides 13664	. . . . . . . . . 10 |- (((abs` D) e. ZZ /\ (y - x) e. ZZ) -> ((abs` D)||(y - x) <-> E.k e. ZZ (k x. (abs` D)) = (y - x)))
75	73, 74	mpan 759	. . . . . . . . 9 |- ((y - x) e. ZZ -> ((abs` D)||(y - x) <-> E.k e. ZZ (k x. (abs` D)) = (y - x)))
76	64, 75	syl 12	. . . . . . . 8 |- ((x e. ZZ /\ y e. ZZ) -> ((abs` D)||(y - x) <-> E.k e. ZZ (k x. (abs` D)) = (y - x)))
77	30, 76	syl 12	. . . . . . 7 |- ((x e. S /\ y e. S) -> ((abs` D)||(y - x) <-> E.k e. ZZ (k x. (abs` D)) = (y - x)))
78	69, 77	mpbid 212	. . . . . 6 |- ((x e. S /\ y e. S) -> E.k e. ZZ (k x. (abs` D)) = (y - x))
79	78	adantr 425	. . . . 5 |- (((x e. S /\ y e. S) /\ (x < (abs` D) /\ y < (abs` D))) -> E.k e. ZZ (k x. (abs` D)) = (y - x))
80	4, 5, 6, 7	divalglem8 13703	. . . . . 6 |- (((x e. S /\ y e. S) /\ (x < (abs` D) /\ y < (abs` D))) -> (k e. ZZ -> ((k x. (abs` D)) = (y - x) -> x = y)))
81	80	r19.23adv 2215	. . . . 5 |- (((x e. S /\ y e. S) /\ (x < (abs` D) /\ y < (abs` D))) -> (E.k e. ZZ (k x. (abs` D)) = (y - x) -> x = y))
82	79, 81	mpd 29	. . . 4 |- (((x e. S /\ y e. S) /\ (x < (abs` D) /\ y < (abs` D))) -> x = y)
83	82	ex 402	. . 3 |- ((x e. S /\ y e. S) -> ((x < (abs` D) /\ y < (abs` D)) -> x = y))
84	83	rgen2 2186	. 2 |- A.x e. S A.y e. S ((x < (abs` D) /\ y < (abs` D)) -> x = y)
85	2, 14, 84	mpbir2an 800	1 |- E!x e. S x < (abs` D)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  E!wreu 2107  {crab 2108   class class class wbr 3338  `'ccnv 3985  ` cfv 3998  (class class class)co 4884  supcsup 5663  CCcc 6384  RRcr 6385  0cc0 6386   + caddc 6389   x. cmul 6391   - cmin 6445   <_ cle 6448  NNcn 6449  NN0cn0 6450  ZZcz 6451   < clt 6653  abscabs 8000  ||cdivides 13662

This theorem is referenced by:  divalglem10 13705

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

Copyright terms: Public domain