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Theorem prodmolem2 14665
Description: Lemma for prodmo 14666. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypotheses
Ref Expression
prodmo.1 |- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )
prodmo.2 |- ( ( ph /\ k e. A ) -> B e. CC )
prodmo.3 |- G = ( j e. NN |-> [_ ( f ` j ) / k ]_ B )
Assertion
Ref Expression
prodmolem2 |- ( ( ph /\ E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ E. n e. ( ZZ>= ` m ) E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ seq m ( x. , F ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( x. , G ) ` m ) ) -> x = z ) )
Distinct variable groups:   A, k, n   k, F, n   ph, k, n   A, f, j, m   B, j   f, F, j, k, m   ph, f   x, f   z, f   j, G   j, k, m, ph   x, j   k, m, x   ph, m   x, m   z, m
Allowed substitution hints:   ph( x, y, z)   A( x, y, z)   B( x, y, z, f, k, m, n)   F( x, y, z)   G( x, y, z, f, k, m, n)
Proof of Theorem prodmolem2
Dummy variables g w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpb 1059 . . 3 |- ( ( A C_ ( ZZ>= ` m ) /\ E. n e. ( ZZ>= ` m ) E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ seq m ( x. , F ) ~~> x ) -> ( A C_ ( ZZ>= ` m ) /\ seq m ( x. , F ) ~~> x ) )
21reximi 3011 . 2 |- ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ E. n e. ( ZZ>= ` m ) E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ seq m ( x. , F ) ~~> x ) -> E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( x. , F ) ~~> x ) )
3 fveq2 6191 . . . . . 6 |- ( m = w -> ( ZZ>= ` m ) = ( ZZ>= ` w ) )
43sseq2d 3633 . . . . 5 |- ( m = w -> ( A C_ ( ZZ>= ` m ) <-> A C_ ( ZZ>= ` w ) ) )
5 seqeq1 12804 . . . . . 6 |- ( m = w -> seq m ( x. , F ) = seq w ( x. , F ) )
65breq1d 4663 . . . . 5 |- ( m = w -> ( seq m ( x. , F ) ~~> x <-> seq w ( x. , F ) ~~> x ) )
74, 6anbi12d 747 . . . 4 |- ( m = w -> ( ( A C_ ( ZZ>= ` m ) /\ seq m ( x. , F ) ~~> x ) <-> ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) ) )
87cbvrexv 3172 . . 3 |- ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( x. , F ) ~~> x ) <-> E. w e. ZZ ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) )
9 reeanv 3107 . . . . 5 |- ( E. w e. ZZ E. m e. NN ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( x. , G ) ` m ) ) ) <-> ( E. w e. ZZ ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( x. , G ) ` m ) ) ) )
10 simprlr 803 . . . . . . . . . . . 12 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> seq w ( x. , F ) ~~> x )
11 simprll 802 . . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> A C_ ( ZZ>= ` w ) )
12 uzssz 11707 . . . . . . . . . . . . . . . . 17 |- ( ZZ>= ` w ) C_ ZZ
13 zssre 11384 . . . . . . . . . . . . . . . . 17 |- ZZ C_ RR
1412, 13sstri 3612 . . . . . . . . . . . . . . . 16 |- ( ZZ>= ` w ) C_ RR
1511, 14syl6ss 3615 . . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> A C_ RR )
16 ltso 10118 . . . . . . . . . . . . . . 15 |- < Or RR
17 soss 5053 . . . . . . . . . . . . . . 15 |- ( A C_ RR -> ( < Or RR -> < Or A ) )
1815, 16, 17mpisyl 21 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> < Or A )
19 fzfi 12771 . . . . . . . . . . . . . . 15 |- ( 1 ... m ) e. Fin
20 ovex 6678 . . . . . . . . . . . . . . . . . 18 |- ( 1 ... m ) e. _V
2120f1oen 7976 . . . . . . . . . . . . . . . . 17 |- ( f : ( 1 ... m ) -1-1-onto-> A -> ( 1 ... m ) ~~ A )
2221ad2antll 765 . . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> ( 1 ... m ) ~~ A )
2322ensymd 8007 . . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> A ~~ ( 1 ... m ) )
24 enfii 8177 . . . . . . . . . . . . . . 15 |- ( ( ( 1 ... m ) e. Fin /\ A ~~ ( 1 ... m ) ) -> A e. Fin )
2519, 23, 24sylancr 695 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> A e. Fin )
26 fz1iso 13246 . . . . . . . . . . . . . 14 |- ( ( < Or A /\ A e. Fin ) -> E. g g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) )
2718, 25, 26syl2anc 693 . . . . . . . . . . . . 13 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> E. g g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) )
28 prodmo.1 . . . . . . . . . . . . . . . 16 |- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )
29 simpll 790 . . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) ) -> ph )
30 prodmo.2 . . . . . . . . . . . . . . . . 17 |- ( ( ph /\ k e. A ) -> B e. CC )
3129, 30sylan 488 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) ) /\ k e. A ) -> B e. CC )
32 prodmo.3 . . . . . . . . . . . . . . . 16 |- G = ( j e. NN |-> [_ ( f ` j ) / k ]_ B )
33 eqid 2622 . . . . . . . . . . . . . . . 16 |- ( j e. NN |-> [_ ( g ` j ) / k ]_ B ) = ( j e. NN |-> [_ ( g ` j ) / k ]_ B )
34 simplrr 801 . . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) ) -> m e. NN )
35 simplrl 800 . . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) ) -> w e. ZZ )
36 simplll 798 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) -> A C_ ( ZZ>= ` w ) )
3736adantl 482 . . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) ) -> A C_ ( ZZ>= ` w ) )
38 simprlr 803 . . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) ) -> f : ( 1 ... m ) -1-1-onto-> A )
39 simprr 796 . . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) ) -> g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) )
4028, 31, 32, 33, 34, 35, 37, 38, 39prodmolem2a 14664 . . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) ) ) -> seq w ( x. , F ) ~~> ( seq 1 ( x. , G ) ` m ) )
4140expr 643 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> ( g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) -> seq w ( x. , F ) ~~> ( seq 1 ( x. , G ) ` m ) ) )
4241exlimdv 1861 . . . . . . . . . . . . 13 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> ( E. g g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) -> seq w ( x. , F ) ~~> ( seq 1 ( x. , G ) ` m ) ) )
4327, 42mpd 15 . . . . . . . . . . . 12 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> seq w ( x. , F ) ~~> ( seq 1 ( x. , G ) ` m ) )
44 climuni 14283 . . . . . . . . . . . 12 |- ( ( seq w ( x. , F ) ~~> x /\ seq w ( x. , F ) ~~> ( seq 1 ( x. , G ) ` m ) ) -> x = ( seq 1 ( x. , G ) ` m ) )
4510, 43, 44syl2anc 693 . . . . . . . . . . 11 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> x = ( seq 1 ( x. , G ) ` m ) )
46 eqeq2 2633 . . . . . . . . . . 11 |- ( z = ( seq 1 ( x. , G ) ` m ) -> ( x = z <-> x = ( seq 1 ( x. , G ) ` m ) ) )
4745, 46syl5ibrcom 237 . . . . . . . . . 10 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> ( z = ( seq 1 ( x. , G ) ` m ) -> x = z ) )
4847expr 643 . . . . . . . . 9 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) ) -> ( f : ( 1 ... m ) -1-1-onto-> A -> ( z = ( seq 1 ( x. , G ) ` m ) -> x = z ) ) )
4948impd 447 . . . . . . . 8 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) ) -> ( ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( x. , G ) ` m ) ) -> x = z ) )
5049exlimdv 1861 . . . . . . 7 |- ( ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) /\ ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) ) -> ( E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( x. , G ) ` m ) ) -> x = z ) )
5150expimpd 629 . . . . . 6 |- ( ( ph /\ ( w e. ZZ /\ m e. NN ) ) -> ( ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( x. , G ) ` m ) ) ) -> x = z ) )
5251rexlimdvva 3038 . . . . 5 |- ( ph -> ( E. w e. ZZ E. m e. NN ( ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( x. , G ) ` m ) ) ) -> x = z ) )
539, 52syl5bir 233 . . . 4 |- ( ph -> ( ( E. w e. ZZ ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) /\ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( x. , G ) ` m ) ) ) -> x = z ) )
5453expdimp 453 . . 3 |- ( ( ph /\ E. w e. ZZ ( A C_ ( ZZ>= ` w ) /\ seq w ( x. , F ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( x. , G ) ` m ) ) -> x = z ) )
558, 54sylan2b 492 . 2 |- ( ( ph /\ E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( x. , F ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( x. , G ) ` m ) ) -> x = z ) )
562, 55sylan2 491 1 |- ( ( ph /\ E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ E. n e. ( ZZ>= ` m ) E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ seq m ( x. , F ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( x. , G ) ` m ) ) -> x = z ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   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   x. cmul 9941   < 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:  prodmo  14666
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