Theorem summolem2 12515

Description: Lemma for summo 12516. (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 5731	. . . . 5 |- ( m = j -> ( ZZ>= ` m ) = ( ZZ>= ` j ) )
2	1	sseq2d 3378	. . . 4 |- ( m = j -> ( A C_ ( ZZ>= ` m ) <-> A C_ ( ZZ>= ` j ) ) )
3		seqeq1 11331	. . . . 5 |- ( m = j -> seq m ( + , F ) = seq j ( + , F ) )
4	3	breq1d 4225	. . . 4 |- ( m = j -> ( seq m ( + , F ) ~~> x <-> seq j ( + , F ) ~~> x ) )
5	2, 4	anbi12d 693	. . 3 |- ( m = j -> ( ( A C_ ( ZZ>= ` m ) /\ seq m ( + , F ) ~~> x ) <-> ( A C_ ( ZZ>= ` j ) /\ seq j ( + , F ) ~~> x ) ) )
6	5	cbvrexv 2935	. 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 739	. . . . . . . . . . 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 738	. . . . . . . . . . . . . . 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 10510	. . . . . . . . . . . . . . . 16 |- ( ZZ>= ` j ) C_ ZZ
10		zssre 10294	. . . . . . . . . . . . . . . 16 |- ZZ C_ RR
11	9, 10	sstri 3359	. . . . . . . . . . . . . . 15 |- ( ZZ>= ` j ) C_ RR
12	8, 11	syl6ss 3362	. . . . . . . . . . . . . 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 9161	. . . . . . . . . . . . . 14 |- < Or RR
14		soss 4524	. . . . . . . . . . . . . 14 |- ( A C_ RR -> ( < Or RR -> < Or A ) )
15	12, 13, 14	ee10 1386	. . . . . . . . . . . . 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 11316	. . . . . . . . . . . . . 14 |- ( 1 ... m ) e. Fin
17		ovex 6109	. . . . . . . . . . . . . . . . 17 |- ( 1 ... m ) e. _V
18	17	f1oen 7131	. . . . . . . . . . . . . . . 16 |- ( f : ( 1 ... m ) -1-1-onto-> A -> ( 1 ... m ) ~~ A )
19	18	ad2antll 711	. . . . . . . . . . . . . . 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	19	ensymd 7161	. . . . . . . . . . . . . 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 ) )
21		enfii 7329	. . . . . . . . . . . . . 14 |- ( ( ( 1 ... m ) e. Fin /\ A ~~ ( 1 ... m ) ) -> A e. Fin )
22	16, 20, 21	sylancr 646	. . . . . . . . . . . . 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 )
23		fz1iso 11716	. . . . . . . . . . . . 13 |- ( ( < Or A /\ A e. Fin ) -> E. g g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) )
24	15, 22, 23	syl2anc 644	. . . . . . . . . . . 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 ) )
25		summo.1	. . . . . . . . . . . . . . 15 |- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
26		simplll 736	. . . . . . . . . . . . . . . 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 )
27		summo.2	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ k e. A ) -> B e. CC )
28	26, 27	sylan 459	. . . . . . . . . . . . . . 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 )
29		summo.3	. . . . . . . . . . . . . . 15 |- G = ( n e. NN |-> [_ ( f ` n ) / k ]_ B )
30		eqid 2438	. . . . . . . . . . . . . . 15 |- ( n e. NN |-> [_ ( g ` n ) / k ]_ B ) = ( n e. NN |-> [_ ( g ` n ) / k ]_ B )
31		simprll 740	. . . . . . . . . . . . . . 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 )
32		simpllr 737	. . . . . . . . . . . . . . 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 )
33		simplrl 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 ) ) ) -> A C_ ( ZZ>= ` j ) )
34		simprlr 741	. . . . . . . . . . . . . . 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 )
35		simprr 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 ) ) ) -> g Isom < , < ( ( 1 ... ( # ` A ) ) , A ) )
36	25, 28, 29, 30, 31, 32, 33, 34, 35	summolem2a 12514	. . . . . . . . . . . . . 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 ) )
37	36	expr 600	. . . . . . . . . . . . 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 ) ) )
38	37	exlimdv 1647	. . . . . . . . . . . 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 ) ) )
39	24, 38	mpd 15	. . . . . . . . . . 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 ) )
40		climuni 12351	. . . . . . . . . . 11 |- ( ( seq j ( + , F ) ~~> x /\ seq j ( + , F ) ~~> ( seq 1 ( + , G ) ` m ) ) -> x = ( seq 1 ( + , G ) ` m ) )
41	7, 39, 40	syl2anc 644	. . . . . . . . . 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 ) )
42	41	anassrs 631	. . . . . . . . 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 ) )
43		eqeq2 2447	. . . . . . . . 9 |- ( y = ( seq 1 ( + , G ) ` m ) -> ( x = y <-> x = ( seq 1 ( + , G ) ` m ) ) )
44	42, 43	syl5ibrcom 215	. . . . . . . 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 ) )
45	44	expimpd 588	. . . . . . 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 ) )
46	45	exlimdv 1647	. . . . . 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 ) )
47	46	rexlimdva 2832	. . . . 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 ) )
48	47	ex 425	. . . 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 ) ) )
49	48	rexlimdva 2832	. . 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 ) ) )
50	49	imp 420	. 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 ) )
51	6, 50	sylan2b 463	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 360   E.wex 1551   = wceq 1653   e. wcel 1726   E.wrex 2708   [_csb 3253   C_ wss 3322   ifcif 3741   class class class wbr 4215   e. cmpt 4269   Or wor 4505   -1-1-onto->wf1o 5456   ` cfv 5457   Isom wiso 5458  (class class class)co 6084   ~~ cen 7109   Fincfn 7112   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996   + caddc 8998   < clt 9125   NNcn 10005   ZZcz 10287   ZZ>=cuz 10493   ...cfz 11048   seq cseq 11328   #chash 11623   ~~> cli 12283

This theorem is referenced by:  summo  12516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287