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Theorem ismrer1 33637
Description: An isometry between RR and RR ^ 1. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
ismrer1.1 |- R = ( ( abs o. - ) |` ( RR X. RR ) )
ismrer1.2 |- F = ( x e. RR |-> ( { A } X. { x } ) )
Assertion
Ref Expression
ismrer1 |- ( A e. V -> F e. ( R Ismty ( Rn ` { A } ) ) )
Distinct variable group:   x, A
Allowed substitution hints:   R( x)   F( x)   V( x)
Proof of Theorem ismrer1
Dummy variables k y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sneq 4187 . . . . . . . 8 |- ( y = A -> { y } = { A } )
21xpeq1d 5138 . . . . . . 7 |- ( y = A -> ( { y } X. { x } ) = ( { A } X. { x } ) )
32mpteq2dv 4745 . . . . . 6 |- ( y = A -> ( x e. RR |-> ( { y } X. { x } ) ) = ( x e. RR |-> ( { A } X. { x } ) ) )
4 ismrer1.2 . . . . . 6 |- F = ( x e. RR |-> ( { A } X. { x } ) )
53, 4syl6eqr 2674 . . . . 5 |- ( y = A -> ( x e. RR |-> ( { y } X. { x } ) ) = F )
6 f1oeq1 6127 . . . . 5 |- ( ( x e. RR |-> ( { y } X. { x } ) ) = F -> ( ( x e. RR |-> ( { y } X. { x } ) ) : RR -1-1-onto-> ( RR ^m { y } ) <-> F : RR -1-1-onto-> ( RR ^m { y } ) ) )
75, 6syl 17 . . . 4 |- ( y = A -> ( ( x e. RR |-> ( { y } X. { x } ) ) : RR -1-1-onto-> ( RR ^m { y } ) <-> F : RR -1-1-onto-> ( RR ^m { y } ) ) )
81oveq2d 6666 . . . . 5 |- ( y = A -> ( RR ^m { y } ) = ( RR ^m { A } ) )
9 f1oeq3 6129 . . . . 5 |- ( ( RR ^m { y } ) = ( RR ^m { A } ) -> ( F : RR -1-1-onto-> ( RR ^m { y } ) <-> F : RR -1-1-onto-> ( RR ^m { A } ) ) )
108, 9syl 17 . . . 4 |- ( y = A -> ( F : RR -1-1-onto-> ( RR ^m { y } ) <-> F : RR -1-1-onto-> ( RR ^m { A } ) ) )
117, 10bitrd 268 . . 3 |- ( y = A -> ( ( x e. RR |-> ( { y } X. { x } ) ) : RR -1-1-onto-> ( RR ^m { y } ) <-> F : RR -1-1-onto-> ( RR ^m { A } ) ) )
12 eqid 2622 . . . 4 |- { y } = { y }
13 reex 10027 . . . 4 |- RR e. _V
14 vex 3203 . . . 4 |- y e. _V
15 eqid 2622 . . . 4 |- ( x e. RR |-> ( { y } X. { x } ) ) = ( x e. RR |-> ( { y } X. { x } ) )
1612, 13, 14, 15mapsnf1o3 7906 . . 3 |- ( x e. RR |-> ( { y } X. { x } ) ) : RR -1-1-onto-> ( RR ^m { y } )
1711, 16vtoclg 3266 . 2 |- ( A e. V -> F : RR -1-1-onto-> ( RR ^m { A } ) )
18 sneq 4187 . . . . . . . . . . . . . . . . 17 |- ( x = y -> { x } = { y } )
1918xpeq2d 5139 . . . . . . . . . . . . . . . 16 |- ( x = y -> ( { A } X. { x } ) = ( { A } X. { y } ) )
20 snex 4908 . . . . . . . . . . . . . . . . 17 |- { A } e. _V
21 snex 4908 . . . . . . . . . . . . . . . . 17 |- { x } e. _V
2220, 21xpex 6962 . . . . . . . . . . . . . . . 16 |- ( { A } X. { x } ) e. _V
2319, 4, 22fvmpt3i 6287 . . . . . . . . . . . . . . 15 |- ( y e. RR -> ( F ` y ) = ( { A } X. { y } ) )
2423fveq1d 6193 . . . . . . . . . . . . . 14 |- ( y e. RR -> ( ( F ` y ) ` A ) = ( ( { A } X. { y } ) ` A ) )
2524adantr 481 . . . . . . . . . . . . 13 |- ( ( y e. RR /\ z e. RR ) -> ( ( F ` y ) ` A ) = ( ( { A } X. { y } ) ` A ) )
26 snidg 4206 . . . . . . . . . . . . . 14 |- ( A e. V -> A e. { A } )
27 fvconst2g 6467 . . . . . . . . . . . . . 14 |- ( ( y e. _V /\ A e. { A } ) -> ( ( { A } X. { y } ) ` A ) = y )
2814, 26, 27sylancr 695 . . . . . . . . . . . . 13 |- ( A e. V -> ( ( { A } X. { y } ) ` A ) = y )
2925, 28sylan9eqr 2678 . . . . . . . . . . . 12 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( ( F ` y ) ` A ) = y )
30 sneq 4187 . . . . . . . . . . . . . . . . 17 |- ( x = z -> { x } = { z } )
3130xpeq2d 5139 . . . . . . . . . . . . . . . 16 |- ( x = z -> ( { A } X. { x } ) = ( { A } X. { z } ) )
3231, 4, 22fvmpt3i 6287 . . . . . . . . . . . . . . 15 |- ( z e. RR -> ( F ` z ) = ( { A } X. { z } ) )
3332fveq1d 6193 . . . . . . . . . . . . . 14 |- ( z e. RR -> ( ( F ` z ) ` A ) = ( ( { A } X. { z } ) ` A ) )
3433adantl 482 . . . . . . . . . . . . 13 |- ( ( y e. RR /\ z e. RR ) -> ( ( F ` z ) ` A ) = ( ( { A } X. { z } ) ` A ) )
35 vex 3203 . . . . . . . . . . . . . 14 |- z e. _V
36 fvconst2g 6467 . . . . . . . . . . . . . 14 |- ( ( z e. _V /\ A e. { A } ) -> ( ( { A } X. { z } ) ` A ) = z )
3735, 26, 36sylancr 695 . . . . . . . . . . . . 13 |- ( A e. V -> ( ( { A } X. { z } ) ` A ) = z )
3834, 37sylan9eqr 2678 . . . . . . . . . . . 12 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( ( F ` z ) ` A ) = z )
3929, 38oveq12d 6668 . . . . . . . . . . 11 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( ( ( F ` y ) ` A ) - ( ( F ` z ) ` A ) ) = ( y - z ) )
4039oveq1d 6665 . . . . . . . . . 10 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( ( ( ( F ` y ) ` A ) - ( ( F ` z ) ` A ) ) ^ 2 ) = ( ( y - z ) ^ 2 ) )
41 resubcl 10345 . . . . . . . . . . . 12 |- ( ( y e. RR /\ z e. RR ) -> ( y - z ) e. RR )
4241adantl 482 . . . . . . . . . . 11 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( y - z ) e. RR )
43 absresq 14042 . . . . . . . . . . 11 |- ( ( y - z ) e. RR -> ( ( abs ` ( y - z ) ) ^ 2 ) = ( ( y - z ) ^ 2 ) )
4442, 43syl 17 . . . . . . . . . 10 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( ( abs ` ( y - z ) ) ^ 2 ) = ( ( y - z ) ^ 2 ) )
4540, 44eqtr4d 2659 . . . . . . . . 9 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( ( ( ( F ` y ) ` A ) - ( ( F ` z ) ` A ) ) ^ 2 ) = ( ( abs ` ( y - z ) ) ^ 2 ) )
4642recnd 10068 . . . . . . . . . . . 12 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( y - z ) e. CC )
4746abscld 14175 . . . . . . . . . . 11 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( abs ` ( y - z ) ) e. RR )
4847recnd 10068 . . . . . . . . . 10 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( abs ` ( y - z ) ) e. CC )
4948sqcld 13006 . . . . . . . . 9 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( ( abs ` ( y - z ) ) ^ 2 ) e. CC )
5045, 49eqeltrd 2701 . . . . . . . 8 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( ( ( ( F ` y ) ` A ) - ( ( F ` z ) ` A ) ) ^ 2 ) e. CC )
51 fveq2 6191 . . . . . . . . . . 11 |- ( k = A -> ( ( F ` y ) ` k ) = ( ( F ` y ) ` A ) )
52 fveq2 6191 . . . . . . . . . . 11 |- ( k = A -> ( ( F ` z ) ` k ) = ( ( F ` z ) ` A ) )
5351, 52oveq12d 6668 . . . . . . . . . 10 |- ( k = A -> ( ( ( F ` y ) ` k ) - ( ( F ` z ) ` k ) ) = ( ( ( F ` y ) ` A ) - ( ( F ` z ) ` A ) ) )
5453oveq1d 6665 . . . . . . . . 9 |- ( k = A -> ( ( ( ( F ` y ) ` k ) - ( ( F ` z ) ` k ) ) ^ 2 ) = ( ( ( ( F ` y ) ` A ) - ( ( F ` z ) ` A ) ) ^ 2 ) )
5554sumsn 14475 . . . . . . . 8 |- ( ( A e. V /\ ( ( ( ( F ` y ) ` A ) - ( ( F ` z ) ` A ) ) ^ 2 ) e. CC ) -> sum_ k e. { A } ( ( ( ( F ` y ) ` k ) - ( ( F ` z ) ` k ) ) ^ 2 ) = ( ( ( ( F ` y ) ` A ) - ( ( F ` z ) ` A ) ) ^ 2 ) )
5650, 55syldan 487 . . . . . . 7 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> sum_ k e. { A } ( ( ( ( F ` y ) ` k ) - ( ( F ` z ) ` k ) ) ^ 2 ) = ( ( ( ( F ` y ) ` A ) - ( ( F ` z ) ` A ) ) ^ 2 ) )
5756, 45eqtrd 2656 . . . . . 6 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> sum_ k e. { A } ( ( ( ( F ` y ) ` k ) - ( ( F ` z ) ` k ) ) ^ 2 ) = ( ( abs ` ( y - z ) ) ^ 2 ) )
5857fveq2d 6195 . . . . 5 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( sqr ` sum_ k e. { A } ( ( ( ( F ` y ) ` k ) - ( ( F ` z ) ` k ) ) ^ 2 ) ) = ( sqr ` ( ( abs ` ( y - z ) ) ^ 2 ) ) )
5946absge0d 14183 . . . . . 6 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> 0 <_ ( abs ` ( y - z ) ) )
6047, 59sqrtsqd 14158 . . . . 5 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( sqr ` ( ( abs ` ( y - z ) ) ^ 2 ) ) = ( abs ` ( y - z ) ) )
6158, 60eqtrd 2656 . . . 4 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( sqr ` sum_ k e. { A } ( ( ( ( F ` y ) ` k ) - ( ( F ` z ) ` k ) ) ^ 2 ) ) = ( abs ` ( y - z ) ) )
62 f1of 6137 . . . . . . . 8 |- ( F : RR -1-1-onto-> ( RR ^m { A } ) -> F : RR --> ( RR ^m { A } ) )
6317, 62syl 17 . . . . . . 7 |- ( A e. V -> F : RR --> ( RR ^m { A } ) )
6463ffvelrnda 6359 . . . . . 6 |- ( ( A e. V /\ y e. RR ) -> ( F ` y ) e. ( RR ^m { A } ) )
6563ffvelrnda 6359 . . . . . 6 |- ( ( A e. V /\ z e. RR ) -> ( F ` z ) e. ( RR ^m { A } ) )
6664, 65anim12dan 882 . . . . 5 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( ( F ` y ) e. ( RR ^m { A } ) /\ ( F ` z ) e. ( RR ^m { A } ) ) )
67 snfi 8038 . . . . . 6 |- { A } e. Fin
68 eqid 2622 . . . . . . 7 |- ( RR ^m { A } ) = ( RR ^m { A } )
6968rrnmval 33627 . . . . . 6 |- ( ( { A } e. Fin /\ ( F ` y ) e. ( RR ^m { A } ) /\ ( F ` z ) e. ( RR ^m { A } ) ) -> ( ( F ` y ) ( Rn ` { A } ) ( F ` z ) ) = ( sqr ` sum_ k e. { A } ( ( ( ( F ` y ) ` k ) - ( ( F ` z ) ` k ) ) ^ 2 ) ) )
7067, 69mp3an1 1411 . . . . 5 |- ( ( ( F ` y ) e. ( RR ^m { A } ) /\ ( F ` z ) e. ( RR ^m { A } ) ) -> ( ( F ` y ) ( Rn ` { A } ) ( F ` z ) ) = ( sqr ` sum_ k e. { A } ( ( ( ( F ` y ) ` k ) - ( ( F ` z ) ` k ) ) ^ 2 ) ) )
7166, 70syl 17 . . . 4 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( ( F ` y ) ( Rn ` { A } ) ( F ` z ) ) = ( sqr ` sum_ k e. { A } ( ( ( ( F ` y ) ` k ) - ( ( F ` z ) ` k ) ) ^ 2 ) ) )
72 ismrer1.1 . . . . . 6 |- R = ( ( abs o. - ) |` ( RR X. RR ) )
7372remetdval 22592 . . . . 5 |- ( ( y e. RR /\ z e. RR ) -> ( y R z ) = ( abs ` ( y - z ) ) )
7473adantl 482 . . . 4 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( y R z ) = ( abs ` ( y - z ) ) )
7561, 71, 743eqtr4rd 2667 . . 3 |- ( ( A e. V /\ ( y e. RR /\ z e. RR ) ) -> ( y R z ) = ( ( F ` y ) ( Rn ` { A } ) ( F ` z ) ) )
7675ralrimivva 2971 . 2 |- ( A e. V -> A. y e. RR A. z e. RR ( y R z ) = ( ( F ` y ) ( Rn ` { A } ) ( F ` z ) ) )
7772rexmet 22594 . . 3 |- R e. ( *Met ` RR )
7868rrnmet 33628 . . . 4 |- ( { A } e. Fin -> ( Rn ` { A } ) e. ( Met ` ( RR ^m { A } ) ) )
79 metxmet 22139 . . . 4 |- ( ( Rn ` { A } ) e. ( Met ` ( RR ^m { A } ) ) -> ( Rn ` { A } ) e. ( *Met ` ( RR ^m { A } ) ) )
8067, 78, 79mp2b 10 . . 3 |- ( Rn ` { A } ) e. ( *Met ` ( RR ^m { A } ) )
81 isismty 33600 . . 3 |- ( ( R e. ( *Met ` RR ) /\ ( Rn ` { A } ) e. ( *Met ` ( RR ^m { A } ) ) ) -> ( F e. ( R Ismty ( Rn ` { A } ) ) <-> ( F : RR -1-1-onto-> ( RR ^m { A } ) /\ A. y e. RR A. z e. RR ( y R z ) = ( ( F ` y ) ( Rn ` { A } ) ( F ` z ) ) ) ) )
8277, 80, 81mp2an 708 . 2 |- ( F e. ( R Ismty ( Rn ` { A } ) ) <-> ( F : RR -1-1-onto-> ( RR ^m { A } ) /\ A. y e. RR A. z e. RR ( y R z ) = ( ( F ` y ) ( Rn ` { A } ) ( F ` z ) ) ) )
8317, 76, 82sylanbrc 698 1 |- ( A e. V -> F e. ( R Ismty ( Rn ` { A } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   {csn 4177   |-> cmpt 4729   X. cxp 5112   |` cres 5116   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   CCcc 9934   RRcr 9935   - cmin 10266   2c2 11070   ^cexp 12860   sqrcsqrt 13973   abscabs 13974   sum_csu 14416   *Metcxmt 19731   Metcme 19732   Ismty cismty 33597   Rncrrn 33624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-ico 12181  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-xmet 19739  df-met 19740  df-ismty 33598  df-rrn 33625
This theorem is referenced by:  reheibor  33638
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