Theorem cau3 11693

Description: Convert between three-quantifier and four-quantifier versions of the Cauchy criterion. (In particular, the four-quantifier version has no occurrence of j in the assertion, so it can be used with rexanuz 11566 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.)

Hypothesis

Ref	Expression
cau3.1	|- Z = ( ZZ>= ` M )

Assertion

Ref	Expression
cau3	|- ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x ) <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ A. m e. ( ZZ>= ` k ) ( abs ` ( ( F ` k ) - ( F ` m ) ) ) < x ) )

Distinct variable groups:   j, k, m, x, F   j, M, k, x   j, Z, k, x

Allowed substitution hints:   M( m)   Z( m)

Proof of Theorem cau3

Step	Hyp	Ref	Expression
1		cau3.1	. . . 4 |- Z = ( ZZ>= ` M )
2		uzssz 9776	. . . 4 |- ( ZZ>= ` M ) C_ ZZ
3	1, 2	eqsstri 3259	. . 3 |- Z C_ ZZ
4		id 19	. . 3 |- ( ( F ` k ) e. CC -> ( F ` k ) e. CC )
5		eleq1 2294	. . 3 |- ( ( F ` k ) = ( F ` j ) -> ( ( F ` k ) e. CC <-> ( F ` j ) e. CC ) )
6		eleq1 2294	. . 3 |- ( ( F ` k ) = ( F ` m ) -> ( ( F ` k ) e. CC <-> ( F ` m ) e. CC ) )
7		abssub 11679	. . . 4 |- ( ( ( F ` j ) e. CC /\ ( F ` k ) e. CC ) -> ( abs ` ( ( F ` j ) - ( F ` k ) ) ) = ( abs ` ( ( F ` k ) - ( F ` j ) ) ) )
8	7	3adant1 1041	. . 3 |- ( ( T. /\ ( F ` j ) e. CC /\ ( F ` k ) e. CC ) -> ( abs ` ( ( F ` j ) - ( F ` k ) ) ) = ( abs ` ( ( F ` k ) - ( F ` j ) ) ) )
9		abssub 11679	. . . 4 |- ( ( ( F ` m ) e. CC /\ ( F ` j ) e. CC ) -> ( abs ` ( ( F ` m ) - ( F ` j ) ) ) = ( abs ` ( ( F ` j ) - ( F ` m ) ) ) )
10	9	3adant1 1041	. . 3 |- ( ( T. /\ ( F ` m ) e. CC /\ ( F ` j ) e. CC ) -> ( abs ` ( ( F ` m ) - ( F ` j ) ) ) = ( abs ` ( ( F ` j ) - ( F ` m ) ) ) )
11		abs3lem 11689	. . . 4 |- ( ( ( ( F ` k ) e. CC /\ ( F ` m ) e. CC ) /\ ( ( F ` j ) e. CC /\ x e. RR ) ) -> ( ( ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < ( x / 2 ) /\ ( abs ` ( ( F ` j ) - ( F ` m ) ) ) < ( x / 2 ) ) -> ( abs ` ( ( F ` k ) - ( F ` m ) ) ) < x ) )
12	11	3adant1 1041	. . 3 |- ( ( T. /\ ( ( F ` k ) e. CC /\ ( F ` m ) e. CC ) /\ ( ( F ` j ) e. CC /\ x e. RR ) ) -> ( ( ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < ( x / 2 ) /\ ( abs ` ( ( F ` j ) - ( F ` m ) ) ) < ( x / 2 ) ) -> ( abs ` ( ( F ` k ) - ( F ` m ) ) ) < x ) )
13	3, 4, 5, 6, 8, 10, 12	cau3lem 11692	. 2 |- ( T. -> ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x ) <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ A. m e. ( ZZ>= ` k ) ( abs ` ( ( F ` k ) - ( F ` m ) ) ) < x ) ) )
14	13	mptru 1406	1 |- ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x ) <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ A. m e. ( ZZ>= ` k ) ( abs ` ( ( F ` k ) - ( F ` m ) ) ) < x ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   T. wtru 1398   e. wcel 2202   A.wral 2510   E.wrex 2511   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   CCcc 8030   RRcr 8031   < clt 8214   - cmin 8350   / cdiv 8852   2c2 9194   ZZcz 9479   ZZ>=cuz 9755   RR+crp 9888   abscabs 11575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-rp 9889  df-seqfrec 10711  df-exp 10802  df-cj 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577

This theorem is referenced by:  cau4  11694  serf0  11930