Theorem ballotfilemieq 13318

Description: If two countings share the same first tie, they also have the same swap function. (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 ) ) )

Assertion

Ref	Expression
ballotfilemieq	|- ( ( C e. ( O \ E ) /\ D e. ( O \ E ) /\ ( I ` C ) = ( I ` D ) ) -> ( S ` C ) = ( S ` 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

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

Proof of Theorem ballotfilemieq

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . 6 |- ( ( ( I ` C ) = ( I ` D ) /\ i e. ( 1 ... ( M + N ) ) ) -> ( I ` C ) = ( I ` D ) )
2	1	breq2d 4143	. . . . 5 |- ( ( ( I ` C ) = ( I ` D ) /\ i e. ( 1 ... ( M + N ) ) ) -> ( i <_ ( I ` C ) <-> i <_ ( I ` D ) ) )
3	1	oveq1d 6103	. . . . . 6 |- ( ( ( I ` C ) = ( I ` D ) /\ i e. ( 1 ... ( M + N ) ) ) -> ( ( I ` C ) + 1 ) = ( ( I ` D ) + 1 ) )
4	3	oveq1d 6103	. . . . 5 |- ( ( ( I ` C ) = ( I ` D ) /\ i e. ( 1 ... ( M + N ) ) ) -> ( ( ( I ` C ) + 1 ) - i ) = ( ( ( I ` D ) + 1 ) - i ) )
5	2, 4	ifbieq1d 3663	. . . 4 |- ( ( ( I ` C ) = ( I ` D ) /\ i e. ( 1 ... ( M + N ) ) ) -> if ( i <_ ( I ` C ) , ( ( ( I ` C ) + 1 ) - i ) , i ) = if ( i <_ ( I ` D ) , ( ( ( I ` D ) + 1 ) - i ) , i ) )
6	5	mpteq2dva 4223	. . 3 |- ( ( I ` C ) = ( I ` D ) -> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` C ) , ( ( ( I ` C ) + 1 ) - i ) , i ) ) = ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` D ) , ( ( ( I ` D ) + 1 ) - i ) , i ) ) )
7	6	3ad2ant3 1051	. 2 |- ( ( C e. ( O \ E ) /\ D e. ( O \ E ) /\ ( I ` C ) = ( I ` D ) ) -> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` C ) , ( ( ( I ` C ) + 1 ) - i ) , i ) ) = ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` D ) , ( ( ( I ` D ) + 1 ) - i ) , i ) ) )
8		ballotth.m	. . . 4 |- M e. NN
9		ballotth.n	. . . 4 |- N e. NN
10		ballotfilem.o	. . . 4 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
11		ballotfilem.p	. . . 4 |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
12		ballotth.f	. . . 4 |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
13		ballotth.e	. . . 4 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
14		ballotth.mgtn	. . . 4 |- N < M
15		ballotth.i	. . . 4 |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
16		ballotth.s	. . . 4 |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
17	8, 9, 10, 11, 12, 13, 14, 15, 16	ballotfilemsval 13310	. . 3 |- ( C e. ( O \ E ) -> ( S ` C ) = ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` C ) , ( ( ( I ` C ) + 1 ) - i ) , i ) ) )
18	17	3ad2ant1 1049	. 2 |- ( ( C e. ( O \ E ) /\ D e. ( O \ E ) /\ ( I ` C ) = ( I ` D ) ) -> ( S ` C ) = ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` C ) , ( ( ( I ` C ) + 1 ) - i ) , i ) ) )
19	8, 9, 10, 11, 12, 13, 14, 15, 16	ballotfilemsval 13310	. . 3 |- ( D e. ( O \ E ) -> ( S ` D ) = ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` D ) , ( ( ( I ` D ) + 1 ) - i ) , i ) ) )
20	19	3ad2ant2 1050	. 2 |- ( ( C e. ( O \ E ) /\ D e. ( O \ E ) /\ ( I ` C ) = ( I ` D ) ) -> ( S ` D ) = ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` D ) , ( ( ( I ` D ) + 1 ) - i ) , i ) ) )
21	7, 18, 20	3eqtr4d 2281	1 |- ( ( C e. ( O \ E ) /\ D e. ( O \ E ) /\ ( I ` C ) = ( I ` D ) ) -> ( S ` C ) = ( S ` D ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   \ cdif 3217   i^i cin 3219   ifcif 3638   ~Pcpw 3688   class class class wbr 4131   |-> cmpt 4193   ` cfv 5380  (class class class)co 6088   Fincfn 7026  infcinf 7329   RRcr 8184   0cc0 8185   1c1 8186   + caddc 8188   < clt 8366   <_ cle 8367   - cmin 8504   / cdiv 9010   NNcn 9313   ZZcz 9655   ...cfz 10428  ♯chash 11236

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301

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-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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-en 7027  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429

This theorem is used by:  ballotfilemrinv0  13334