Theorem summolem2 12189

Description: Lemma for summo 12190. (Contributed by Mario Carneiro, 3-Apr-2014.)

Hypotheses

Ref	Expression
summo.1	|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
summo.2	|- ( ( ph /\ k e. A ) -> B e. CC )
summo.3	|- G = ( n e. NN |-> [_ ( f ` n ) / k ]_ B )

Assertion

Ref	Expression
summolem2	|- ( ( ph /\ E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , F ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G ) ` m ) ) -> x = y ) )

Distinct variable groups:   f, k, m, n, x, y, A   f, F, k, m, n, x, y   k, G, m, n, x, y   ph, k, m, n, y   B, f, m, n, x, y   ph, x, f

Allowed substitution hints:   B( k)   G( f)

Proof of Theorem summolem2

Dummy variables g j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5525	. . . . 5 |- ( m = j -> ( ZZ>= ` m ) = ( ZZ>= ` j ) )
2	1	sseq2d 3206	. . . 4 |- ( m = j -> ( A C_ ( ZZ>= ` m ) <-> A C_ ( ZZ>= ` j ) ) )
3		seqeq1 11049	. . . . 5 |- ( m = j -> seq m ( + , F ) = seq j ( + , F ) )
4	3	breq1d 4033	. . . 4 |- ( m = j -> ( seq m ( + , F ) ~~> x <-> seq j ( + , F ) ~~> x ) )
5	2, 4	anbi12d 691	. . 3 |- ( m = j -> ( ( A C_ ( ZZ>= ` m ) /\ seq m ( + , F ) ~~> x ) <-> ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) )
6	5	cbvrexv 2765	. 2 |- ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , F ) ~~> x ) <-> E. j e. ZZ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) )
7		simplrr 737	. . . . . . . . . . 11 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> seq j ( + , F ) ~~> x )
8		simplrl 736	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> A C_ ( ZZ>= ` j ) )
9		uzssz 10247	. . . . . . . . . . . . . . . 16 |- ( ZZ>= ` j ) C_ ZZ
10		zssre 10031	. . . . . . . . . . . . . . . 16 |- ZZ C_ RR
11	9, 10	sstri 3188	. . . . . . . . . . . . . . 15 |- ( ZZ>= ` j ) C_ RR
12	8, 11	syl6ss 3191	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> A C_ RR )
13		ltso 8903	. . . . . . . . . . . . . 14 |- < Or RR
14		soss 4332	. . . . . . . . . . . . . 14 |- ( A C_ RR -> ( < Or RR -> < Or A ) )
15	12, 13, 14	ee10 1366	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> < Or A )
16		fzfi 11034	. . . . . . . . . . . . . 14 |- ( 1 ... m ) e. Fin
17		ovex 5883	. . . . . . . . . . . . . . . . 17 |- ( 1 ... m ) e. _V
18	17	f1oen 6882	. . . . . . . . . . . . . . . 16 |- ( f : ( 1 ... m ) -1-1-onto-> A -> ( 1 ... m ) ~~ A )
19	18	ad2antll 709	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> ( 1 ... m ) ~~ A )
20		ensym 6910	. . . . . . . . . . . . . . 15 |- ( ( 1 ... m ) ~~ A -> A ~~ ( 1 ... m ) )
21	19, 20	syl 15	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> A ~~ ( 1 ... m ) )
22		enfii 7080	. . . . . . . . . . . . . 14 |- ( ( ( 1 ... m ) e. Fin /\ A ~~ ( 1 ... m ) ) -> A e. Fin )
23	16, 21, 22	sylancr 644	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> A e. Fin )
24		fz1iso 11400	. . . . . . . . . . . . 13 |- ( ( < Or A /\ A e. Fin ) -> E. g g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) )
25	15, 23, 24	syl2anc 642	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> E. g g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) )
26		summo.1	. . . . . . . . . . . . . . 15 |- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
27		simplll 734	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) ) -> ph )
28		summo.2	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ k e. A ) -> B e. CC )
29	27, 28	sylan 457	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) ) /\ k e. A ) -> B e. CC )
30		summo.3	. . . . . . . . . . . . . . 15 |- G = ( n e. NN |-> [_ ( f ` n ) / k ]_ B )
31		eqid 2283	. . . . . . . . . . . . . . 15 |- ( n e. NN |-> [_ ( g ` n ) / k ]_ B ) = ( n e. NN |-> [_ ( g ` n ) / k ]_ B )
32		simprll 738	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) ) -> m e. NN )
33		simpllr 735	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) ) -> j e. ZZ )
34		simplrl 736	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) ) -> A C_ ( ZZ>= ` j ) )
35		simprlr 739	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) ) -> f : ( 1 ... m ) -1-1-onto-> A )
36		simprr 733	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) ) -> g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) )
37	26, 29, 30, 31, 32, 33, 34, 35, 36	summolem2a 12188	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) ) -> seq j ( + , F ) ~~> ( seq 1 ( + , G ) ` m ) )
38	37	expr 598	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> ( g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) -> seq j ( + , F ) ~~> ( seq 1 ( + , G ) ` m ) ) )
39	38	exlimdv 1664	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> ( E. g g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) -> seq j ( + , F ) ~~> ( seq 1 ( + , G ) ` m ) ) )
40	25, 39	mpd 14	. . . . . . . . . . 11 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> seq j ( + , F ) ~~> ( seq 1 ( + , G ) ` m ) )
41		climuni 12026	. . . . . . . . . . 11 |- ( ( seq j ( + , F ) ~~> x /\ seq j ( + , F ) ~~> ( seq 1 ( + , G ) ` m ) ) -> x = ( seq 1 ( + , G ) ` m ) )
42	7, 40, 41	syl2anc 642	. . . . . . . . . 10 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> x = ( seq 1 ( + , G ) ` m ) )
43	42	anassrs 629	. . . . . . . . 9 |- ( ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ m e. NN ) /\ f : ( 1 ... m ) -1-1-onto-> A ) -> x = ( seq 1 ( + , G ) ` m ) )
44		eqeq2 2292	. . . . . . . . 9 |- ( y = ( seq 1 ( + , G ) ` m ) -> ( x = y <-> x = ( seq 1 ( + , G ) ` m ) ) )
45	43, 44	syl5ibrcom 213	. . . . . . . 8 |- ( ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ m e. NN ) /\ f : ( 1 ... m ) -1-1-onto-> A ) -> ( y = ( seq 1 ( + , G ) ` m ) -> x = y ) )
46	45	expimpd 586	. . . . . . 7 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ m e. NN ) -> ( ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G ) ` m ) ) -> x = y ) )
47	46	exlimdv 1664	. . . . . 6 |- ( ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) /\ m e. NN ) -> ( E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G ) ` m ) ) -> x = y ) )
48	47	rexlimdva 2667	. . . . 5 |- ( ( ( ph /\ j e. ZZ ) /\ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G ) ` m ) ) -> x = y ) )
49	48	ex 423	. . . 4 |- ( ( ph /\ j e. ZZ ) -> ( ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G ) ` m ) ) -> x = y ) ) )
50	49	rexlimdva 2667	. . 3 |- ( ph -> ( E. j e. ZZ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G ) ` m ) ) -> x = y ) ) )
51	50	imp 418	. 2 |- ( ( ph /\ E. j e. ZZ ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G ) ` m ) ) -> x = y ) )
52	6, 51	sylan2b 461	1 |- ( ( ph /\ E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , F ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G ) ` m ) ) -> x = y ) )

Syntax hints:   -> wi 4   /\ wa 358   E.wex 1528   = wceq 1623   e. wcel 1684   E.wrex 2544   [_csb 3081   C_ wss 3152   ifcif 3565   class class class wbr 4023   e. cmpt 4077   Or wor 4313   -1-1-onto->wf1o 5254   ` cfv 5255   Isom wiso 5256  (class class class)co 5858   ~~ cen 6860   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738   + caddc 8740   < clt 8867   NNcn 9746   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   seq cseq 11046   #chash 11337   ~~> cli 11958

This theorem is referenced by:  summo  12190

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962