Theorem cau3iri 7118

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   j,et,k,m,y   ph,m,y   ps,k   th,x   ta,x

Proof of Theorem cau3iri

Step	Hyp	Ref	Expression
1		halfpos2 6183	. . . . . . . . 9 |- (y e. RR -> (0 < y <-> 0 < (y / 2)))
2	1	adantl 388	. . . . . . . 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 2696	. . . . . . . . . . 11 |- (x = (y / 2) -> (0 < x <-> 0 < (y / 2)))
4		cau3ir.3	. . . . . . . . . . . . . 14 |- (x = (y / 2) -> ((ph /\ ps) <-> th))
5	4	imbi2d 615	. . . . . . . . . . . . 13 |- (x = (y / 2) -> (((jRk /\ jRm) -> (ph /\ ps)) <-> ((jRk /\ jRm) -> th)))
6	5	2ralbidv 1726	. . . . . . . . . . . 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 1710	. . . . . . . . . . 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 629	. . . . . . . . . 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 1922	. . . . . . . . 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 6180	. . . . . . . . 9 |- (y e. RR -> (y / 2) e. RR)
11	9, 10	sylan2 453	. . . . . . . 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 201	. . . . . . 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		raaan 2414	. . . . . . . . . . . 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 2696	. . . . . . . . . . . . . . 15 |- (k = m -> (jRk <-> jRm))
15		cau3ir.1	. . . . . . . . . . . . . . 15 |- (k = m -> (ph <-> ps))
16	14, 15	imbi12d 629	. . . . . . . . . . . . . 14 |- (k = m -> ((jRk -> ph) <-> (jRm -> ps)))
17	16	cbvralv 1846	. . . . . . . . . . . . 13 |- (A.k e. W (jRk -> ph) <-> A.m e. W (jRm -> ps))
18	17	anbi2i 483	. . . . . . . . . . . 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 433	. . . . . . . . . . . 12 |- ((A.k e. W (jRk -> ph) /\ A.k e. W (jRk -> ph)) <-> A.k e. W (jRk -> ph))
20	13, 18, 19	3bitr2ri 178	. . . . . . . . . . 11 |- (A.k e. W (jRk -> ph) <-> A.k e. W A.m e. W ((jRk -> ph) /\ (jRm -> ps)))
21		prth 559	. . . . . . . . . . . . 13 |- (((jRk -> ph) /\ (jRm -> ps)) -> ((jRk /\ jRm) -> (ph /\ ps)))
22	21	r19.20si 1752	. . . . . . . . . . . 12 |- (A.m e. W ((jRk -> ph) /\ (jRm -> ps)) -> A.m e. W ((jRk /\ jRm) -> (ph /\ ps)))
23	22	r19.20si 1752	. . . . . . . . . . 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 197	. . . . . . . . . 10 |- (A.k e. W (jRk -> ph) -> A.k e. W A.m e. W ((jRk /\ jRm) -> (ph /\ ps)))
25	24	r19.22si 1780	. . . . . . . . 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 17	. . . . . . . 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	r19.20si 1752	. . . . . . 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 450	. . . . . 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 388	. . . . 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 857	. . . . . . . . . . 11 |- ((et /\ y e. RR) -> (j e. Z -> (k e. W -> (m e. W -> (th -> ta)))))
32	31	imp41 366	. . . . . . . . . 10 |- (((((et /\ y e. RR) /\ j e. Z) /\ k e. W) /\ m e. W) -> (th -> ta))
33	32	imim2d 25	. . . . . . . . 9 |- (((((et /\ y e. RR) /\ j e. Z) /\ k e. W) /\ m e. W) -> (((jRk /\ jRm) -> th) -> ((jRk /\ jRm) -> ta)))
34	33	r19.20dva 1755	. . . . . . . 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	r19.20dva 1755	. . . . . . 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	r19.22dva 1785	. . . . . 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 394	. . . . 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 27	. . . 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 377	. . 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 1764	. 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 2696	. . . . 5 |- (x = y -> (0 < x <-> 0 < y))
42		cau3ir.4	. . . . . . . 8 |- (x = y -> (ch <-> ta))
43	42	imbi2d 615	. . . . . . 7 |- (x = y -> (((jRk /\ jRm) -> ch) <-> ((jRk /\ jRm) -> ta)))
44	43	2ralbidv 1726	. . . . . 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 1710	. . . . 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 629	. . . 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 1846	. . 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 2695	. . . . . . . . . 10 |- (j = n -> (jRk <-> nRk))
49		breq1 2695	. . . . . . . . . 10 |- (j = n -> (jRm <-> nRm))
50	48, 49	anbi12d 631	. . . . . . . . 9 |- (j = n -> ((jRk /\ jRm) <-> (nRk /\ nRm)))
51	50	imbi1d 616	. . . . . . . 8 |- (j = n -> (((jRk /\ jRm) -> ch) <-> ((nRk /\ nRm) -> ch)))
52	51	2ralbidv 1726	. . . . . . 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 1847	. . . . . 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 2696	. . . . . . . . . . . 12 |- (k = m -> (nRk <-> nRm))
55	54	anbi2d 619	. . . . . . . . . . 11 |- (k = m -> ((nRj /\ nRk) <-> (nRj /\ nRm)))
56	55, 15	imbi12d 629	. . . . . . . . . 10 |- (k = m -> (((nRj /\ nRk) -> ph) <-> ((nRj /\ nRm) -> ps)))
57	56	cbvralv 1846	. . . . . . . . 9 |- (A.k e. W ((nRj /\ nRk) -> ph) <-> A.m e. W ((nRj /\ nRm) -> ps))
58	57	ralbii 1713	. . . . . . . 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 2696	. . . . . . . . . . . 12 |- (j = k -> (nRj <-> nRk))
60	59	anbi1d 620	. . . . . . . . . . 11 |- (j = k -> ((nRj /\ nRm) <-> (nRk /\ nRm)))
61		cau3ir.2	. . . . . . . . . . 11 |- (j = k -> (ps <-> ch))
62	60, 61	imbi12d 629	. . . . . . . . . 10 |- (j = k -> (((nRj /\ nRm) -> ps) <-> ((nRk /\ nRm) -> ch)))
63	62	ralbidv 1709	. . . . . . . . 9 |- (j = k -> (A.m e. W ((nRj /\ nRm) -> ps) <-> A.m e. W ((nRk /\ nRm) -> ch)))
64	63	cbvralv 1846	. . . . . . . 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 172	. . . . . . 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 1714	. . . . . 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 2695	. . . . . . . . . 10 |- (n = m -> (nRj <-> mRj))
68		breq1 2695	. . . . . . . . . 10 |- (n = m -> (nRk <-> mRk))
69	67, 68	anbi12d 631	. . . . . . . . 9 |- (n = m -> ((nRj /\ nRk) <-> (mRj /\ mRk)))
70	69	imbi1d 616	. . . . . . . 8 |- (n = m -> (((nRj /\ nRk) -> ph) <-> ((mRj /\ mRk) -> ph)))
71	70	2ralbidv 1726	. . . . . . 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 1847	. . . . . 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 175	. . . . 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 183	. . . 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 1713	. . 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 173	. 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 210	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 144   /\ wa 221   /\ w3a 781   = wceq 992   e. wcel 994  A.wral 1691  E.wrex 1692   class class class wbr 2692  (class class class)co 4021  RRcr 5387  0cc0 5388   / cdiv 5448   < clt 5640  2c2 6107

This theorem is referenced by:  cau3i 7119

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-nel 1631  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-en 4509  df-dom 4510  df-sdom 4511  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-ltr 5324  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400  df-lt 5401  df-sub 5510  df-neg 5512  df-pnf 5641  df-mnf 5642  df-xr 5643  df-ltxr 5644  df-le 5645  df-div 5855  df-2 6116

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