Theorem cau3iri 8167

Description: A relationship used to derive two ways to express a Cauchy sequence. Normally Z and W are subsets of ZZ, and R is <_ or <. ph is ph(j, k, x).

Hypotheses

Ref	Expression
cau3ir.1	|- (k = m -> (ph <-> ps))
cau3ir.2	|- (j = k -> (ps <-> ch))
cau3ir.3	|- (x = (y / 2) -> ((ph /\ ps) <-> th))
cau3ir.4	|- (x = y -> (ch <-> ta))
cau3ir.5	|- (((et /\ y e. RR) /\ (j e. Z /\ k e. W /\ m e. W)) -> (th -> ta))

Assertion

Ref	Expression
cau3iri	|- (et -> (A.x e. RR (0 < x -> E.j e. Z A.k e. W (jRk -> ph)) -> A.x e. RR (0 < x -> E.m e. Z A.j e. W A.k e. W ((mRj /\ mRk) -> ph))))

Distinct variable groups:   j,k,m,x,y,R   j,W,k,m,x,y   j,Z,k,m,x,y   ch,j,y   et,j,k,m,y   ph,m,y   ps,k   th,x   ta,x

Proof of Theorem cau3iri

Step	Hyp	Ref	Expression
1		halfpos2 7223	. . . . . . . . 9 |- (y e. RR -> (0 < y <-> 0 < (y / 2)))
2	1	adantl 424	. . . . . . . 8 |- ((A.x e. RR (0 < x -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> (ph /\ ps))) /\ y e. RR) -> (0 < y <-> 0 < (y / 2)))
3		breq2 3342	. . . . . . . . . . 11 |- (x = (y / 2) -> (0 < x <-> 0 < (y / 2)))
4		cau3ir.3	. . . . . . . . . . . . . 14 |- (x = (y / 2) -> ((ph /\ ps) <-> th))
5	4	imbi2d 674	. . . . . . . . . . . . 13 |- (x = (y / 2) -> (((jRk /\ jRm) -> (ph /\ ps)) <-> ((jRk /\ jRm) -> th)))
6	5	2ralbidv 2140	. . . . . . . . . . . 12 |- (x = (y / 2) -> (A.k e. W A.m e. W ((jRk /\ jRm) -> (ph /\ ps)) <-> A.k e. W A.m e. W ((jRk /\ jRm) -> th)))
7	6	rexbidv 2124	. . . . . . . . . . 11 |- (x = (y / 2) -> (E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> (ph /\ ps)) <-> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> th)))
8	3, 7	imbi12d 688	. . . . . . . . . 10 |- (x = (y / 2) -> ((0 < x -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> (ph /\ ps))) <-> (0 < (y / 2) -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> th))))
9	8	rcla4cva 2379	. . . . . . . . 9 |- ((A.x e. RR (0 < x -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> (ph /\ ps))) /\ (y / 2) e. RR) -> (0 < (y / 2) -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> th)))
10		rehalfcl 7220	. . . . . . . . 9 |- (y e. RR -> (y / 2) e. RR)
11	9, 10	sylan2 500	. . . . . . . 8 |- ((A.x e. RR (0 < x -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> (ph /\ ps))) /\ y e. RR) -> (0 < (y / 2) -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> th)))
12	2, 11	sylbid 220	. . . . . . 7 |- ((A.x e. RR (0 < x -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> (ph /\ ps))) /\ y e. RR) -> (0 < y -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> th)))
13		raaanv 2977	. . . . . . . . . . . 12 |- (A.k e. W A.m e. W ((jRk -> ph) /\ (jRm -> ps)) <-> (A.k e. W (jRk -> ph) /\ A.m e. W (jRm -> ps)))
14		breq2 3342	. . . . . . . . . . . . . . 15 |- (k = m -> (jRk <-> jRm))
15		cau3ir.1	. . . . . . . . . . . . . . 15 |- (k = m -> (ph <-> ps))
16	14, 15	imbi12d 688	. . . . . . . . . . . . . 14 |- (k = m -> ((jRk -> ph) <-> (jRm -> ps)))
17	16	cbvralv 2280	. . . . . . . . . . . . 13 |- (A.k e. W (jRk -> ph) <-> A.m e. W (jRm -> ps))
18	17	anbi2i 538	. . . . . . . . . . . 12 |- ((A.k e. W (jRk -> ph) /\ A.k e. W (jRk -> ph)) <-> (A.k e. W (jRk -> ph) /\ A.m e. W (jRm -> ps)))
19		anidm 478	. . . . . . . . . . . 12 |- ((A.k e. W (jRk -> ph) /\ A.k e. W (jRk -> ph)) <-> A.k e. W (jRk -> ph))
20	13, 18, 19	3bitr2ri 197	. . . . . . . . . . 11 |- (A.k e. W (jRk -> ph) <-> A.k e. W A.m e. W ((jRk -> ph) /\ (jRm -> ps)))
21		prth 615	. . . . . . . . . . . . 13 |- (((jRk -> ph) /\ (jRm -> ps)) -> ((jRk /\ jRm) -> (ph /\ ps)))
22	21	ralimi 2168	. . . . . . . . . . . 12 |- (A.m e. W ((jRk -> ph) /\ (jRm -> ps)) -> A.m e. W ((jRk /\ jRm) -> (ph /\ ps)))
23	22	ralimi 2168	. . . . . . . . . . 11 |- (A.k e. W A.m e. W ((jRk -> ph) /\ (jRm -> ps)) -> A.k e. W A.m e. W ((jRk /\ jRm) -> (ph /\ ps)))
24	20, 23	sylbi 216	. . . . . . . . . 10 |- (A.k e. W (jRk -> ph) -> A.k e. W A.m e. W ((jRk /\ jRm) -> (ph /\ ps)))
25	24	reximi 2198	. . . . . . . . 9 |- (E.j e. Z A.k e. W (jRk -> ph) -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> (ph /\ ps)))
26	25	imim2i 11	. . . . . . . 8 |- ((0 < x -> E.j e. Z A.k e. W (jRk -> ph)) -> (0 < x -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> (ph /\ ps))))
27	26	ralimi 2168	. . . . . . 7 |- (A.x e. RR (0 < x -> E.j e. Z A.k e. W (jRk -> ph)) -> A.x e. RR (0 < x -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> (ph /\ ps))))
28	12, 27	sylan 497	. . . . . 6 |- ((A.x e. RR (0 < x -> E.j e. Z A.k e. W (jRk -> ph)) /\ y e. RR) -> (0 < y -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> th)))
29	28	adantl 424	. . . . 5 |- ((et /\ (A.x e. RR (0 < x -> E.j e. Z A.k e. W (jRk -> ph)) /\ y e. RR)) -> (0 < y -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> th)))
30		cau3ir.5	. . . . . . . . . . . 12 |- (((et /\ y e. RR) /\ (j e. Z /\ k e. W /\ m e. W)) -> (th -> ta))
31	30	3exp2 1086	. . . . . . . . . . 11 |- ((et /\ y e. RR) -> (j e. Z -> (k e. W -> (m e. W -> (th -> ta)))))
32	31	imp41 395	. . . . . . . . . 10 |- (((((et /\ y e. RR) /\ j e. Z) /\ k e. W) /\ m e. W) -> (th -> ta))
33	32	imim2d 28	. . . . . . . . 9 |- (((((et /\ y e. RR) /\ j e. Z) /\ k e. W) /\ m e. W) -> (((jRk /\ jRm) -> th) -> ((jRk /\ jRm) -> ta)))
34	33	ralimdvaa 2171	. . . . . . . 8 |- ((((et /\ y e. RR) /\ j e. Z) /\ k e. W) -> (A.m e. W ((jRk /\ jRm) -> th) -> A.m e. W ((jRk /\ jRm) -> ta)))
35	34	ralimdvaa 2171	. . . . . . 7 |- (((et /\ y e. RR) /\ j e. Z) -> (A.k e. W A.m e. W ((jRk /\ jRm) -> th) -> A.k e. W A.m e. W ((jRk /\ jRm) -> ta)))
36	35	reximdva 2203	. . . . . 6 |- ((et /\ y e. RR) -> (E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> th) -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> ta)))
37	36	adantrl 430	. . . . 5 |- ((et /\ (A.x e. RR (0 < x -> E.j e. Z A.k e. W (jRk -> ph)) /\ y e. RR)) -> (E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> th) -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> ta)))
38	29, 37	syld 30	. . . 4 |- ((et /\ (A.x e. RR (0 < x -> E.j e. Z A.k e. W (jRk -> ph)) /\ y e. RR)) -> (0 < y -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> ta)))
39	38	exp32 408	. . 3 |- (et -> (A.x e. RR (0 < x -> E.j e. Z A.k e. W (jRk -> ph)) -> (y e. RR -> (0 < y -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> ta)))))
40	39	r19.21adv 2181	. 2 |- (et -> (A.x e. RR (0 < x -> E.j e. Z A.k e. W (jRk -> ph)) -> A.y e. RR (0 < y -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> ta))))
41		breq2 3342	. . . . 5 |- (x = y -> (0 < x <-> 0 < y))
42		cau3ir.4	. . . . . . . 8 |- (x = y -> (ch <-> ta))
43	42	imbi2d 674	. . . . . . 7 |- (x = y -> (((jRk /\ jRm) -> ch) <-> ((jRk /\ jRm) -> ta)))
44	43	2ralbidv 2140	. . . . . 6 |- (x = y -> (A.k e. W A.m e. W ((jRk /\ jRm) -> ch) <-> A.k e. W A.m e. W ((jRk /\ jRm) -> ta)))
45	44	rexbidv 2124	. . . . 5 |- (x = y -> (E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> ch) <-> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> ta)))
46	41, 45	imbi12d 688	. . . 4 |- (x = y -> ((0 < x -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> ch)) <-> (0 < y -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> ta))))
47	46	cbvralv 2280	. . 3 |- (A.x e. RR (0 < x -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> ch)) <-> A.y e. RR (0 < y -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> ta)))
48		breq1 3341	. . . . . . . . . 10 |- (j = n -> (jRk <-> nRk))
49		breq1 3341	. . . . . . . . . 10 |- (j = n -> (jRm <-> nRm))
50	48, 49	anbi12d 690	. . . . . . . . 9 |- (j = n -> ((jRk /\ jRm) <-> (nRk /\ nRm)))
51	50	imbi1d 675	. . . . . . . 8 |- (j = n -> (((jRk /\ jRm) -> ch) <-> ((nRk /\ nRm) -> ch)))
52	51	2ralbidv 2140	. . . . . . 7 |- (j = n -> (A.k e. W A.m e. W ((jRk /\ jRm) -> ch) <-> A.k e. W A.m e. W ((nRk /\ nRm) -> ch)))
53	52	cbvrexv 2281	. . . . . 6 |- (E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> ch) <-> E.n e. Z A.k e. W A.m e. W ((nRk /\ nRm) -> ch))
54		breq2 3342	. . . . . . . . . . . 12 |- (k = m -> (nRk <-> nRm))
55	54	anbi2d 678	. . . . . . . . . . 11 |- (k = m -> ((nRj /\ nRk) <-> (nRj /\ nRm)))
56	55, 15	imbi12d 688	. . . . . . . . . 10 |- (k = m -> (((nRj /\ nRk) -> ph) <-> ((nRj /\ nRm) -> ps)))
57	56	cbvralv 2280	. . . . . . . . 9 |- (A.k e. W ((nRj /\ nRk) -> ph) <-> A.m e. W ((nRj /\ nRm) -> ps))
58	57	ralbii 2127	. . . . . . . 8 |- (A.j e. W A.k e. W ((nRj /\ nRk) -> ph) <-> A.j e. W A.m e. W ((nRj /\ nRm) -> ps))
59		breq2 3342	. . . . . . . . . . . 12 |- (j = k -> (nRj <-> nRk))
60	59	anbi1d 679	. . . . . . . . . . 11 |- (j = k -> ((nRj /\ nRm) <-> (nRk /\ nRm)))
61		cau3ir.2	. . . . . . . . . . 11 |- (j = k -> (ps <-> ch))
62	60, 61	imbi12d 688	. . . . . . . . . 10 |- (j = k -> (((nRj /\ nRm) -> ps) <-> ((nRk /\ nRm) -> ch)))
63	62	ralbidv 2123	. . . . . . . . 9 |- (j = k -> (A.m e. W ((nRj /\ nRm) -> ps) <-> A.m e. W ((nRk /\ nRm) -> ch)))
64	63	cbvralv 2280	. . . . . . . 8 |- (A.j e. W A.m e. W ((nRj /\ nRm) -> ps) <-> A.k e. W A.m e. W ((nRk /\ nRm) -> ch))
65	58, 64	bitr2i 191	. . . . . . 7 |- (A.k e. W A.m e. W ((nRk /\ nRm) -> ch) <-> A.j e. W A.k e. W ((nRj /\ nRk) -> ph))
66	65	rexbii 2128	. . . . . 6 |- (E.n e. Z A.k e. W A.m e. W ((nRk /\ nRm) -> ch) <-> E.n e. Z A.j e. W A.k e. W ((nRj /\ nRk) -> ph))
67		breq1 3341	. . . . . . . . . 10 |- (n = m -> (nRj <-> mRj))
68		breq1 3341	. . . . . . . . . 10 |- (n = m -> (nRk <-> mRk))
69	67, 68	anbi12d 690	. . . . . . . . 9 |- (n = m -> ((nRj /\ nRk) <-> (mRj /\ mRk)))
70	69	imbi1d 675	. . . . . . . 8 |- (n = m -> (((nRj /\ nRk) -> ph) <-> ((mRj /\ mRk) -> ph)))
71	70	2ralbidv 2140	. . . . . . 7 |- (n = m -> (A.j e. W A.k e. W ((nRj /\ nRk) -> ph) <-> A.j e. W A.k e. W ((mRj /\ mRk) -> ph)))
72	71	cbvrexv 2281	. . . . . 6 |- (E.n e. Z A.j e. W A.k e. W ((nRj /\ nRk) -> ph) <-> E.m e. Z A.j e. W A.k e. W ((mRj /\ mRk) -> ph))
73	53, 66, 72	3bitri 194	. . . . 5 |- (E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> ch) <-> E.m e. Z A.j e. W A.k e. W ((mRj /\ mRk) -> ph))
74	73	imbi2i 202	. . . 4 |- ((0 < x -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> ch)) <-> (0 < x -> E.m e. Z A.j e. W A.k e. W ((mRj /\ mRk) -> ph)))
75	74	ralbii 2127	. . 3 |- (A.x e. RR (0 < x -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> ch)) <-> A.x e. RR (0 < x -> E.m e. Z A.j e. W A.k e. W ((mRj /\ mRk) -> ph)))
76	47, 75	bitr3i 192	. 2 |- (A.y e. RR (0 < y -> E.j e. Z A.k e. W A.m e. W ((jRk /\ jRm) -> ta)) <-> A.x e. RR (0 < x -> E.m e. Z A.j e. W A.k e. W ((mRj /\ mRk) -> ph)))
77	40, 76	syl6ib 229	1 |- (et -> (A.x e. RR (0 < x -> E.j e. Z A.k e. W (jRk -> ph)) -> A.x e. RR (0 < x -> E.m e. Z A.j e. W A.k e. W ((mRj /\ mRk) -> ph))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   class class class wbr 3338  (class class class)co 4884  RRcr 6385  0cc0 6386   / cdiv 6447   < clt 6653  2c2 7145

This theorem is referenced by:  cau3i 8168

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-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-2 7154

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