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Theorem elcncf2 15161
Description: Version of elcncf 15160 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)
Assertion
Ref Expression
elcncf2 |- ( ( A C_ CC /\ B C_ CC ) -> ( F e. ( A -cn-> B ) <-> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) ) ) )
Distinct variable groups:   x, w, y, z, A   w, F, x, y, z   w, B, x, y, z
Proof of Theorem elcncf2
StepHypRef Expression
1 elcncf 15160 . 2 |- ( ( A C_ CC /\ B C_ CC ) -> ( F e. ( A -cn-> B ) <-> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) ) )
2 simplll 533 . . . . . . . . . . . 12 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ ( x e. A /\ w e. A ) ) -> A C_ CC )
3 simprl 529 . . . . . . . . . . . 12 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ ( x e. A /\ w e. A ) ) -> x e. A )
42, 3sseldd 3202 . . . . . . . . . . 11 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ ( x e. A /\ w e. A ) ) -> x e. CC )
5 simprr 531 . . . . . . . . . . . 12 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ ( x e. A /\ w e. A ) ) -> w e. A )
62, 5sseldd 3202 . . . . . . . . . . 11 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ ( x e. A /\ w e. A ) ) -> w e. CC )
74, 6abssubd 11619 . . . . . . . . . 10 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ ( x e. A /\ w e. A ) ) -> ( abs ` ( x - w ) ) = ( abs ` ( w - x ) ) )
87breq1d 4069 . . . . . . . . 9 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ ( x e. A /\ w e. A ) ) -> ( ( abs ` ( x - w ) ) < z <-> ( abs ` ( w - x ) ) < z ) )
9 simpllr 534 . . . . . . . . . . . 12 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ ( x e. A /\ w e. A ) ) -> B C_ CC )
10 simplr 528 . . . . . . . . . . . . 13 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ ( x e. A /\ w e. A ) ) -> F : A --> B )
1110, 3ffvelcdmd 5739 . . . . . . . . . . . 12 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ ( x e. A /\ w e. A ) ) -> ( F ` x ) e. B )
129, 11sseldd 3202 . . . . . . . . . . 11 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ ( x e. A /\ w e. A ) ) -> ( F ` x ) e. CC )
1310, 5ffvelcdmd 5739 . . . . . . . . . . . 12 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ ( x e. A /\ w e. A ) ) -> ( F ` w ) e. B )
149, 13sseldd 3202 . . . . . . . . . . 11 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ ( x e. A /\ w e. A ) ) -> ( F ` w ) e. CC )
1512, 14abssubd 11619 . . . . . . . . . 10 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ ( x e. A /\ w e. A ) ) -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) = ( abs ` ( ( F ` w ) - ( F ` x ) ) ) )
1615breq1d 4069 . . . . . . . . 9 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ ( x e. A /\ w e. A ) ) -> ( ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y <-> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) )
178, 16imbi12d 234 . . . . . . . 8 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ ( x e. A /\ w e. A ) ) -> ( ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) <-> ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) ) )
1817anassrs 400 . . . . . . 7 |- ( ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ x e. A ) /\ w e. A ) -> ( ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) <-> ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) ) )
1918ralbidva 2504 . . . . . 6 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ x e. A ) -> ( A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) <-> A. w e. A ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) ) )
2019rexbidv 2509 . . . . 5 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ x e. A ) -> ( E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) <-> E. z e. RR+ A. w e. A ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) ) )
2120ralbidv 2508 . . . 4 |- ( ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) /\ x e. A ) -> ( A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) <-> A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) ) )
2221ralbidva 2504 . . 3 |- ( ( ( A C_ CC /\ B C_ CC ) /\ F : A --> B ) -> ( A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) <-> A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) ) )
2322pm5.32da 452 . 2 |- ( ( A C_ CC /\ B C_ CC ) -> ( ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) <-> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) ) ) )
241, 23bitrd 188 1 |- ( ( A C_ CC /\ B C_ CC ) -> ( F e. ( A -cn-> B ) <-> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2178   A.wral 2486   E.wrex 2487   C_ wss 3174   class class class wbr 4059   -->wf 5286   ` cfv 5290  (class class class)co 5967   CCcc 7958   < clt 8142   - cmin 8278   RR+crp 9810   abscabs 11423   -cn->ccncf 15157
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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-po 4361  df-iso 4362  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-map 6760  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-2 9130  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-cncf 15158
This theorem is referenced by:  cncfi  15165  cncfcdm  15169  abscncf  15172  recncf  15173  imcncf  15174  cjcncf  15175  mulc1cncf  15176  cncfco  15178  cdivcncfap  15191  mulcncf  15195
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