Theorem cau3 11345

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 11218 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 9650	. . . 4 |- ( ZZ>= ` M ) C_ ZZ
3	1, 2	eqsstri 3224	. . 3 |- Z C_ ZZ
4		id 19	. . 3 |- ( ( F ` k ) e. CC -> ( F ` k ) e. CC )
5		eleq1 2267	. . 3 |- ( ( F ` k ) = ( F ` j ) -> ( ( F ` k ) e. CC <-> ( F ` j ) e. CC ) )
6		eleq1 2267	. . 3 |- ( ( F ` k ) = ( F ` m ) -> ( ( F ` k ) e. CC <-> ( F ` m ) e. CC ) )
7		abssub 11331	. . . 4 |- ( ( ( F ` j ) e. CC /\ ( F ` k ) e. CC ) -> ( abs ` ( ( F ` j ) - ( F ` k ) ) ) = ( abs ` ( ( F ` k ) - ( F ` j ) ) ) )
8	7	3adant1 1017	. . 3 |- ( ( T. /\ ( F ` j ) e. CC /\ ( F ` k ) e. CC ) -> ( abs ` ( ( F ` j ) - ( F ` k ) ) ) = ( abs ` ( ( F ` k ) - ( F ` j ) ) ) )
9		abssub 11331	. . . 4 |- ( ( ( F ` m ) e. CC /\ ( F ` j ) e. CC ) -> ( abs ` ( ( F ` m ) - ( F ` j ) ) ) = ( abs ` ( ( F ` j ) - ( F ` m ) ) ) )
10	9	3adant1 1017	. . 3 |- ( ( T. /\ ( F ` m ) e. CC /\ ( F ` j ) e. CC ) -> ( abs ` ( ( F ` m ) - ( F ` j ) ) ) = ( abs ` ( ( F ` j ) - ( F ` m ) ) ) )
11		abs3lem 11341	. . . 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 1017	. . 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 11344	. 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 1381	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 1372   T. wtru 1373   e. wcel 2175   A.wral 2483   E.wrex 2484   class class class wbr 4043   ` cfv 5268  (class class class)co 5934   CCcc 7905   RRcr 7906   < clt 8089   - cmin 8225   / cdiv 8727   2c2 9069   ZZcz 9354   ZZ>=cuz 9630   RR+crp 9757   abscabs 11227

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-mulrcl 8006  ax-addcom 8007  ax-mulcom 8008  ax-addass 8009  ax-mulass 8010  ax-distr 8011  ax-i2m1 8012  ax-0lt1 8013  ax-1rid 8014  ax-0id 8015  ax-rnegex 8016  ax-precex 8017  ax-cnre 8018  ax-pre-ltirr 8019  ax-pre-ltwlin 8020  ax-pre-lttrn 8021  ax-pre-apti 8022  ax-pre-ltadd 8023  ax-pre-mulgt0 8024  ax-pre-mulext 8025  ax-arch 8026  ax-caucvg 8027

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4338  df-po 4341  df-iso 4342  df-iord 4411  df-on 4413  df-ilim 4414  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-recs 6381  df-frec 6467  df-pnf 8091  df-mnf 8092  df-xr 8093  df-ltxr 8094  df-le 8095  df-sub 8227  df-neg 8228  df-reap 8630  df-ap 8637  df-div 8728  df-inn 9019  df-2 9077  df-3 9078  df-4 9079  df-n0 9278  df-z 9355  df-uz 9631  df-rp 9758  df-seqfrec 10574  df-exp 10665  df-cj 11072  df-re 11073  df-im 11074  df-rsqrt 11228  df-abs 11229

This theorem is referenced by:  cau4  11346  serf0  11582