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Theorem summolem2 14447
Description: Lemma for summo 14448. (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.
StepHypRef Expression
1 fveq2 6191 . . . . 5 |- ( m = j -> ( ZZ>= ` m ) = ( ZZ>= ` j ) )
21sseq2d 3633 . . . 4 |- ( m = j -> ( A C_ ( ZZ>= ` m ) <-> A C_ ( ZZ>= ` j ) ) )
3 seqeq1 12804 . . . . 5 |- ( m = j -> seq m ( + , F ) = seq j ( + , F ) )
43breq1d 4663 . . . 4 |- ( m = j -> ( seq m ( + , F ) ~~> x <-> seq j ( + , F ) ~~> x ) )
52, 4anbi12d 747 . . 3 |- ( m = j -> ( ( A C_ ( ZZ>= ` m ) /\ seq m ( + , F ) ~~> x ) <-> ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) )
65cbvrexv 3172 . 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 801 . . . . . . . . 9 |- ( ( ( ( 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 800 . . . . . . . . . . . . 13 |- ( ( ( ( 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 11707 . . . . . . . . . . . . . 14 |- ( ZZ>= ` j ) C_ ZZ
10 zssre 11384 . . . . . . . . . . . . . 14 |- ZZ C_ RR
119, 10sstri 3612 . . . . . . . . . . . . 13 |- ( ZZ>= ` j ) C_ RR
128, 11syl6ss 3615 . . . . . . . . . . . 12 |- ( ( ( ( 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 10118 . . . . . . . . . . . 12 |- < Or RR
14 soss 5053 . . . . . . . . . . . 12 |- ( A C_ RR -> ( < Or RR -> < Or A ) )
1512, 13, 14mpisyl 21 . . . . . . . . . . 11 |- ( ( ( ( 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 12771 . . . . . . . . . . . 12 |- ( 1 ... m ) e. Fin
17 ovex 6678 . . . . . . . . . . . . . . 15 |- ( 1 ... m ) e. _V
1817f1oen 7976 . . . . . . . . . . . . . 14 |- ( f : ( 1 ... m ) -1-1-onto-> A -> ( 1 ... m ) ~~ A )
1918ad2antll 765 . . . . . . . . . . . . 13 |- ( ( ( ( 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 )
2019ensymd 8007 . . . . . . . . . . . 12 |- ( ( ( ( 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 ) )
21 enfii 8177 . . . . . . . . . . . 12 |- ( ( ( 1 ... m ) e. Fin /\ A ~~ ( 1 ... m ) ) -> A e. Fin )
2216, 20, 21sylancr 695 . . . . . . . . . . 11 |- ( ( ( ( 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 )
23 fz1iso 13246 . . . . . . . . . . 11 |- ( ( < Or A /\ A e. Fin ) -> E. g g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) )
2415, 22, 23syl2anc 693 . . . . . . . . . 10 |- ( ( ( ( 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 ) )
25 summo.1 . . . . . . . . . . . . 13 |- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
26 simplll 798 . . . . . . . . . . . . . 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 ) ) ) -> ph )
27 summo.2 . . . . . . . . . . . . . 14 |- ( ( ph /\ k e. A ) -> B e. CC )
2826, 27sylan 488 . . . . . . . . . . . . 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 ) ) ) /\ k e. A ) -> B e. CC )
29 summo.3 . . . . . . . . . . . . 13 |- G = ( n e. NN |-> [_ ( f ` n ) / k ]_ B )
30 eqid 2622 . . . . . . . . . . . . 13 |- ( n e. NN |-> [_ ( g ` n ) / k ]_ B ) = ( n e. NN |-> [_ ( g ` n ) / k ]_ B )
31 simprll 802 . . . . . . . . . . . . 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 ) ) ) -> m e. NN )
32 simpllr 799 . . . . . . . . . . . . 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 ) ) ) -> j e. ZZ )
33 simplrl 800 . . . . . . . . . . . . 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 ) ) ) -> A C_ ( ZZ>= ` j ) )
34 simprlr 803 . . . . . . . . . . . . 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 ) ) ) -> f : ( 1 ... m ) -1-1-onto-> A )
35 simprr 796 . . . . . . . . . . . . 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 ) ) ) -> g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) )
3625, 28, 29, 30, 31, 32, 33, 34, 35summolem2a 14446 . . . . . . . . . . . 12 |- ( ( ( ( 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 ) )
3736expr 643 . . . . . . . . . . 11 |- ( ( ( ( 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 ) ) )
3837exlimdv 1861 . . . . . . . . . 10 |- ( ( ( ( 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 ) ) )
3924, 38mpd 15 . . . . . . . . 9 |- ( ( ( ( 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 ) )
40 climuni 14283 . . . . . . . . 9 |- ( ( seq j ( + , F ) ~~> x /\ seq j ( + , F ) ~~> ( seq 1 ( + , G ) ` m ) ) -> x = ( seq 1 ( + , G ) ` m ) )
417, 39, 40syl2anc 693 . . . . . . . 8 |- ( ( ( ( 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 ) )
4241anassrs 680 . . . . . . 7 |- ( ( ( ( ( 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 eqeq2 2633 . . . . . . 7 |- ( y = ( seq 1 ( + , G ) ` m ) -> ( x = y <-> x = ( seq 1 ( + , G ) ` m ) ) )
4442, 43syl5ibrcom 237 . . . . . 6 |- ( ( ( ( ( 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 ) )
4544expimpd 629 . . . . 5 |- ( ( ( ( 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 ) )
4645exlimdv 1861 . . . 4 |- ( ( ( ( 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 ) )
4746rexlimdva 3031 . . 3 |- ( ( ( 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 ) )
4847r19.29an 3077 . 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 ) )
496, 48sylan2b 492 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 ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   [_csb 3533   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   Or wor 5034   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   ~~ cen 7952   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   #chash 13117   ~~> cli 14215
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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219
This theorem is referenced by:  summo  14448
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