Theorem ballotfilemrinv0 13259

Description: Lemma for ballotfilemrinv 13260. (Contributed by Thierry Arnoux, 18-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
ballotfilemrinv0	|- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> ( D e. ( O \ E ) /\ C = ( ( S ` D ) " D ) ) )

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   C, i, k   i, E, k   C, k   k, I, c   E, c   i, I, c   S, k   D, i, k   S, i, c   R, i, k   x, c   x, C   x, F   x, M   x, N, i, k

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

Proof of Theorem ballotfilemrinv0

Step	Hyp	Ref	Expression
1		ballotth.m	. . . . . 6 |- M e. NN
2		ballotth.n	. . . . . 6 |- N e. NN
3		ballotfilem.o	. . . . . 6 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
4		ballotfilem.p	. . . . . 6 |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
5		ballotth.f	. . . . . 6 |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
6		ballotth.e	. . . . . 6 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
7		ballotth.mgtn	. . . . . 6 |- N < M
8		ballotth.i	. . . . . 6 |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
9		ballotth.s	. . . . . 6 |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
10		ballotth.r	. . . . . 6 |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotfilemrval 13244	. . . . 5 |- ( C e. ( O \ E ) -> ( R ` C ) = ( ( S ` C ) " C ) )
12	11	adantr 276	. . . 4 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> ( R ` C ) = ( ( S ` C ) " C ) )
13		simpr 110	. . . 4 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> D = ( ( S ` C ) " C ) )
14	12, 13	eqtr4d 2274	. . 3 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> ( R ` C ) = D )
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotfilemrc 13257	. . . 4 |- ( C e. ( O \ E ) -> ( R ` C ) e. ( O \ E ) )
16	15	adantr 276	. . 3 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> ( R ` C ) e. ( O \ E ) )
17	14, 16	eqeltrrd 2316	. 2 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> D e. ( O \ E ) )
18	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotfilemsf1o 13240	. . . . . . 7 |- ( C e. ( O \ E ) -> ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-onto-> ( 1 ... ( M + N ) ) /\ `' ( S ` C ) = ( S ` C ) ) )
19	18	simprd 114	. . . . . 6 |- ( C e. ( O \ E ) -> `' ( S ` C ) = ( S ` C ) )
20	19	adantr 276	. . . . 5 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> `' ( S ` C ) = ( S ` C ) )
21	20	eqcomd 2244	. . . 4 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> ( S ` C ) = `' ( S ` C ) )
22	21, 13	imaeq12d 5125	. . 3 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> ( ( S ` C ) " D ) = ( `' ( S ` C ) " ( ( S ` C ) " C ) ) )
23		simpl 109	. . . . 5 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> C e. ( O \ E ) )
24	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotfilemirc 13258	. . . . . . 7 |- ( C e. ( O \ E ) -> ( I ` ( R ` C ) ) = ( I ` C ) )
25	24	adantr 276	. . . . . 6 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> ( I ` ( R ` C ) ) = ( I ` C ) )
26	14	fveq2d 5697	. . . . . 6 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> ( I ` ( R ` C ) ) = ( I ` D ) )
27	25, 26	eqtr3d 2273	. . . . 5 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> ( I ` C ) = ( I ` D ) )
28	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotfilemieq 13243	. . . . 5 |- ( ( C e. ( O \ E ) /\ D e. ( O \ E ) /\ ( I ` C ) = ( I ` D ) ) -> ( S ` C ) = ( S ` D ) )
29	23, 17, 27, 28	syl3anc 1278	. . . 4 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> ( S ` C ) = ( S ` D ) )
30	29	imaeq1d 5123	. . 3 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> ( ( S ` C ) " D ) = ( ( S ` D ) " D ) )
31	18	simpld 112	. . . . 5 |- ( C e. ( O \ E ) -> ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-onto-> ( 1 ... ( M + N ) ) )
32		f1of1 5636	. . . . 5 |- ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-onto-> ( 1 ... ( M + N ) ) -> ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-> ( 1 ... ( M + N ) ) )
33	23, 31, 32	3syl 17	. . . 4 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-> ( 1 ... ( M + N ) ) )
34		eldifi 3351	. . . . 5 |- ( C e. ( O \ E ) -> C e. O )
35	1, 2, 3	ballotfilemelo 13205	. . . . . 6 |- ( C e. O <-> ( C C_ ( 1 ... ( M + N ) ) /\ C e. Fin /\ (♯ ` C ) = M ) )
36	35	simp1bi 1043	. . . . 5 |- ( C e. O -> C C_ ( 1 ... ( M + N ) ) )
37	23, 34, 36	3syl 17	. . . 4 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> C C_ ( 1 ... ( M + N ) ) )
38		f1imacnv 5654	. . . 4 |- ( ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-> ( 1 ... ( M + N ) ) /\ C C_ ( 1 ... ( M + N ) ) ) -> ( `' ( S ` C ) " ( ( S ` C ) " C ) ) = C )
39	33, 37, 38	syl2anc 415	. . 3 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> ( `' ( S ` C ) " ( ( S ` C ) " C ) ) = C )
40	22, 30, 39	3eqtr3rd 2280	. 2 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> C = ( ( S ` D ) " D ) )
41	17, 40	jca 306	1 |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> ( D e. ( O \ E ) /\ C = ( ( S ` D ) " D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   \ cdif 3217   i^i cin 3219   C_ wss 3220   ifcif 3638   ~Pcpw 3688   class class class wbr 4128   |-> cmpt 4190   `'ccnv 4771   "cima 4775   -1-1->wf1 5372   -1-1-onto->wf1o 5374   ` cfv 5375  (class class class)co 6079   Fincfn 7016  infcinf 7317   RRcr 8172   0cc0 8173   1c1 8174   + caddc 8176   < clt 8354   <_ cle 8355   - cmin 8491   / cdiv 8996   NNcn 9287   ZZcz 9627   ...cfz 10394  ♯chash 11197

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 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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-oadd 6685  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-sup 7318  df-inf 7319  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-fzo 10533  df-ihash 11198

This theorem is referenced by:  ballotfilemrinv  13260