Theorem ballotfilemrinv 13279

Description: R is its own inverse : it is an involution. (Contributed by Thierry Arnoux, 10-Apr-2017.)

Hypotheses

Ref	Expression
ballotth.m	|- M e. NN
ballotth.n	|- N e. NN
ballotfilem.o	|- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
ballotfilem.p	|- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
ballotth.f	|- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
ballotth.e	|- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
ballotth.mgtn	|- N < M
ballotth.i	|- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
ballotth.s	|- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
ballotth.r	|- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )

Assertion

Ref	Expression
ballotfilemrinv	|- `' R = R

Distinct variable groups:   M, c   N, c   O, c   i, M   i, N   i, O   k, M   k, N   k, O   i, c, F, k   i, E, k   k, I, c   E, c   i, I, c   S, k, i, c   R, i, k   x, c, F   x, M   x, N, i, k

Allowed substitution hints:   P( x, i, k, c)   R( x, c)   S( x)   E( x)   I( x)   O( x)

Proof of Theorem ballotfilemrinv

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotth.m	. . . . . . . 8 |- M e. NN
2		ballotth.n	. . . . . . . 8 |- N e. NN
3		ballotfilem.o	. . . . . . . 8 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
4		ballotfilem.p	. . . . . . . 8 |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
5		ballotth.f	. . . . . . . 8 |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
6		ballotth.e	. . . . . . . 8 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
7		ballotth.mgtn	. . . . . . . 8 |- N < M
8		ballotth.i	. . . . . . . 8 |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
9		ballotth.s	. . . . . . . 8 |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
10		ballotth.r	. . . . . . . 8 |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotfilemrinv0 13278	. . . . . . 7 |- ( ( c e. ( O \ E ) /\ d = ( ( S ` c ) " c ) ) -> ( d e. ( O \ E ) /\ c = ( ( S ` d ) " d ) ) )
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotfilemrinv0 13278	. . . . . . 7 |- ( ( d e. ( O \ E ) /\ c = ( ( S ` d ) " d ) ) -> ( c e. ( O \ E ) /\ d = ( ( S ` c ) " c ) ) )
13	11, 12	impbii 126	. . . . . 6 |- ( ( c e. ( O \ E ) /\ d = ( ( S ` c ) " c ) ) <-> ( d e. ( O \ E ) /\ c = ( ( S ` d ) " d ) ) )
14	13	a1i 9	. . . . 5 |- ( T. -> ( ( c e. ( O \ E ) /\ d = ( ( S ` c ) " c ) ) <-> ( d e. ( O \ E ) /\ c = ( ( S ` d ) " d ) ) ) )
15	14	mptcnv 5190	. . . 4 |- ( T. -> `' ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) ) = ( d e. ( O \ E ) |-> ( ( S ` d ) " d ) ) )
16	15	mptru 1411	. . 3 |- `' ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) ) = ( d e. ( O \ E ) |-> ( ( S ` d ) " d ) )
17		fveq2 5695	. . . . 5 |- ( d = c -> ( S ` d ) = ( S ` c ) )
18		id 19	. . . . 5 |- ( d = c -> d = c )
19	17, 18	imaeq12d 5127	. . . 4 |- ( d = c -> ( ( S ` d ) " d ) = ( ( S ` c ) " c ) )
20	19	cbvmptv 4227	. . 3 |- ( d e. ( O \ E ) |-> ( ( S ` d ) " d ) ) = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
21	16, 20	eqtri 2259	. 2 |- `' ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) ) = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
22	10	cnveqi 4955	. 2 |- `' R = `' ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
23	21, 22, 10	3eqtr4i 2269	1 |- `' R = R

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   T. wtru 1403   e. wcel 2209   A.wral 2528   {crab 2532   \ cdif 3217   i^i cin 3219   ifcif 3638   ~Pcpw 3688   class class class wbr 4130   |-> cmpt 4192   `'ccnv 4773   "cima 4777   ` cfv 5377  (class class class)co 6085   Fincfn 7022  infcinf 7323   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   < clt 8360   <_ cle 8361   - cmin 8497   / cdiv 9003   NNcn 9305   ZZcz 9646   ...cfz 10413  ♯chash 11216

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-n0 9566  df-z 9647  df-uz 9924  df-q 10022  df-rp 10057  df-fz 10414  df-fzo 10552  df-ihash 11217

This theorem is used by:  ballotfilem7  13281