Theorem ballotlemieq 30578

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
ballotth.o	|- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
ballotth.p	|- P = ( x e. ~P O |-> ( ( # ` 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
ballotlemieq	|- ( ( 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 ballotlemieq

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . 6 |- ( ( ( I ` C ) = ( I ` D ) /\ i e. ( 1 ... ( M + N ) ) ) -> ( I ` C ) = ( I ` D ) )
2	1	breq2d 4665	. . . . 5 |- ( ( ( I ` C ) = ( I ` D ) /\ i e. ( 1 ... ( M + N ) ) ) -> ( i <_ ( I ` C ) <-> i <_ ( I ` D ) ) )
3	1	oveq1d 6665	. . . . . 6 |- ( ( ( I ` C ) = ( I ` D ) /\ i e. ( 1 ... ( M + N ) ) ) -> ( ( I ` C ) + 1 ) = ( ( I ` D ) + 1 ) )
4	3	oveq1d 6665	. . . . 5 |- ( ( ( I ` C ) = ( I ` D ) /\ i e. ( 1 ... ( M + N ) ) ) -> ( ( ( I ` C ) + 1 ) - i ) = ( ( ( I ` D ) + 1 ) - i ) )
5	2, 4	ifbieq1d 4109	. . . 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 4744	. . 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 1084	. 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		ballotth.o	. . . 4 |- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
11		ballotth.p	. . . 4 |- P = ( x e. ~P O |-> ( ( # ` 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	ballotlemsval 30570	. . 3 |- ( C e. ( O \ E ) -> ( S ` C ) = ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` C ) , ( ( ( I ` C ) + 1 ) - i ) , i ) ) )
18	17	3ad2ant1 1082	. 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	ballotlemsval 30570	. . 3 |- ( D e. ( O \ E ) -> ( S ` D ) = ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` D ) , ( ( ( I ` D ) + 1 ) - i ) , i ) ) )
20	19	3ad2ant2 1083	. 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 2666	1 |- ( ( C e. ( O \ E ) /\ D e. ( O \ E ) /\ ( I ` C ) = ( I ` D ) ) -> ( S ` C ) = ( S ` D ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ cdif 3571   i^i cin 3573   ifcif 4086   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650  infcinf 8347   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   ZZcz 11377   ...cfz 12326   #chash 13117

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-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-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653

This theorem is referenced by:  ballotlemrinv0  30594