Theorem gsumval3lem1 18306

Description: Lemma 1 for gsumval3 18308. (Contributed by AV, 31-May-2019.)

Hypotheses

Ref	Expression
gsumval3.b	|- B = ( Base ` G )
gsumval3.0	|- .0. = ( 0g ` G )
gsumval3.p	|- .+ = ( +g ` G )
gsumval3.z	|- Z = (Cntz ` G )
gsumval3.g	|- ( ph -> G e. Mnd )
gsumval3.a	|- ( ph -> A e. V )
gsumval3.f	|- ( ph -> F : A --> B )
gsumval3.c	|- ( ph -> ran F C_ ( Z ` ran F ) )
gsumval3.m	|- ( ph -> M e. NN )
gsumval3.h	|- ( ph -> H : ( 1 ... M ) -1-1-> A )
gsumval3.n	|- ( ph -> ( F supp .0. ) C_ ran H )
gsumval3.w	|- W = ( ( F o. H ) supp .0. )

Assertion

Ref	Expression
gsumval3lem1	|- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H o. f ) : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) )

Distinct variable groups:   .+ , f   A, f   ph, f   f, G   f, M   B, f   f, F   f, H   f, W

Allowed substitution hints:   V( f)   .0. ( f)   Z( f)

Proof of Theorem gsumval3lem1

Step	Hyp	Ref	Expression
1		gsumval3.h	. . . . . . 7 |- ( ph -> H : ( 1 ... M ) -1-1-> A )
2	1	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> H : ( 1 ... M ) -1-1-> A )
3		gsumval3.w	. . . . . . . . 9 |- W = ( ( F o. H ) supp .0. )
4		suppssdm 7308	. . . . . . . . 9 |- ( ( F o. H ) supp .0. ) C_ dom ( F o. H )
5	3, 4	eqsstri 3635	. . . . . . . 8 |- W C_ dom ( F o. H )
6		gsumval3.f	. . . . . . . . . 10 |- ( ph -> F : A --> B )
7		f1f 6101	. . . . . . . . . . 11 |- ( H : ( 1 ... M ) -1-1-> A -> H : ( 1 ... M ) --> A )
8	1, 7	syl 17	. . . . . . . . . 10 |- ( ph -> H : ( 1 ... M ) --> A )
9		fco 6058	. . . . . . . . . 10 |- ( ( F : A --> B /\ H : ( 1 ... M ) --> A ) -> ( F o. H ) : ( 1 ... M ) --> B )
10	6, 8, 9	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( F o. H ) : ( 1 ... M ) --> B )
11		fdm 6051	. . . . . . . . 9 |- ( ( F o. H ) : ( 1 ... M ) --> B -> dom ( F o. H ) = ( 1 ... M ) )
12	10, 11	syl 17	. . . . . . . 8 |- ( ph -> dom ( F o. H ) = ( 1 ... M ) )
13	5, 12	syl5sseq 3653	. . . . . . 7 |- ( ph -> W C_ ( 1 ... M ) )
14	13	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> W C_ ( 1 ... M ) )
15		f1ores 6151	. . . . . 6 |- ( ( H : ( 1 ... M ) -1-1-> A /\ W C_ ( 1 ... M ) ) -> ( H |` W ) : W -1-1-onto-> ( H " W ) )
16	2, 14, 15	syl2anc 693	. . . . 5 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H |` W ) : W -1-1-onto-> ( H " W ) )
17	3	imaeq2i 5464	. . . . . . 7 |- ( H " W ) = ( H " ( ( F o. H ) supp .0. ) )
18		gsumval3.a	. . . . . . . . . . 11 |- ( ph -> A e. V )
19		fex 6490	. . . . . . . . . . 11 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
20	6, 18, 19	syl2anc 693	. . . . . . . . . 10 |- ( ph -> F e. _V )
21		ovex 6678	. . . . . . . . . . . 12 |- ( 1 ... M ) e. _V
22		fex 6490	. . . . . . . . . . . 12 |- ( ( H : ( 1 ... M ) --> A /\ ( 1 ... M ) e. _V ) -> H e. _V )
23	7, 21, 22	sylancl 694	. . . . . . . . . . 11 |- ( H : ( 1 ... M ) -1-1-> A -> H e. _V )
24	1, 23	syl 17	. . . . . . . . . 10 |- ( ph -> H e. _V )
25		f1fun 6103	. . . . . . . . . . . 12 |- ( H : ( 1 ... M ) -1-1-> A -> Fun H )
26	1, 25	syl 17	. . . . . . . . . . 11 |- ( ph -> Fun H )
27		gsumval3.n	. . . . . . . . . . 11 |- ( ph -> ( F supp .0. ) C_ ran H )
28	26, 27	jca 554	. . . . . . . . . 10 |- ( ph -> ( Fun H /\ ( F supp .0. ) C_ ran H ) )
29	20, 24, 28	jca31 557	. . . . . . . . 9 |- ( ph -> ( ( F e. _V /\ H e. _V ) /\ ( Fun H /\ ( F supp .0. ) C_ ran H ) ) )
30	29	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( ( F e. _V /\ H e. _V ) /\ ( Fun H /\ ( F supp .0. ) C_ ran H ) ) )
31		imacosupp 7335	. . . . . . . . 9 |- ( ( F e. _V /\ H e. _V ) -> ( ( Fun H /\ ( F supp .0. ) C_ ran H ) -> ( H " ( ( F o. H ) supp .0. ) ) = ( F supp .0. ) ) )
32	31	imp 445	. . . . . . . 8 |- ( ( ( F e. _V /\ H e. _V ) /\ ( Fun H /\ ( F supp .0. ) C_ ran H ) ) -> ( H " ( ( F o. H ) supp .0. ) ) = ( F supp .0. ) )
33	30, 32	syl 17	. . . . . . 7 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H " ( ( F o. H ) supp .0. ) ) = ( F supp .0. ) )
34	17, 33	syl5eq 2668	. . . . . 6 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H " W ) = ( F supp .0. ) )
35		f1oeq3 6129	. . . . . 6 |- ( ( H " W ) = ( F supp .0. ) -> ( ( H |` W ) : W -1-1-onto-> ( H " W ) <-> ( H |` W ) : W -1-1-onto-> ( F supp .0. ) ) )
36	34, 35	syl 17	. . . . 5 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( ( H |` W ) : W -1-1-onto-> ( H " W ) <-> ( H |` W ) : W -1-1-onto-> ( F supp .0. ) ) )
37	16, 36	mpbid 222	. . . 4 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H |` W ) : W -1-1-onto-> ( F supp .0. ) )
38		isof1o 6573	. . . . 5 |- ( f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) -> f : ( 1 ... ( # ` W ) ) -1-1-onto-> W )
39	38	ad2antll 765	. . . 4 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> f : ( 1 ... ( # ` W ) ) -1-1-onto-> W )
40		f1oco 6159	. . . 4 |- ( ( ( H |` W ) : W -1-1-onto-> ( F supp .0. ) /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) -> ( ( H |` W ) o. f ) : ( 1 ... ( # ` W ) ) -1-1-onto-> ( F supp .0. ) )
41	37, 39, 40	syl2anc 693	. . 3 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( ( H |` W ) o. f ) : ( 1 ... ( # ` W ) ) -1-1-onto-> ( F supp .0. ) )
42		f1of 6137	. . . . 5 |- ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W -> f : ( 1 ... ( # ` W ) ) --> W )
43		frn 6053	. . . . 5 |- ( f : ( 1 ... ( # ` W ) ) --> W -> ran f C_ W )
44	39, 42, 43	3syl 18	. . . 4 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ran f C_ W )
45		cores 5638	. . . 4 |- ( ran f C_ W -> ( ( H |` W ) o. f ) = ( H o. f ) )
46		f1oeq1 6127	. . . 4 |- ( ( ( H |` W ) o. f ) = ( H o. f ) -> ( ( ( H |` W ) o. f ) : ( 1 ... ( # ` W ) ) -1-1-onto-> ( F supp .0. ) <-> ( H o. f ) : ( 1 ... ( # ` W ) ) -1-1-onto-> ( F supp .0. ) ) )
47	44, 45, 46	3syl 18	. . 3 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( ( ( H |` W ) o. f ) : ( 1 ... ( # ` W ) ) -1-1-onto-> ( F supp .0. ) <-> ( H o. f ) : ( 1 ... ( # ` W ) ) -1-1-onto-> ( F supp .0. ) ) )
48	41, 47	mpbid 222	. 2 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H o. f ) : ( 1 ... ( # ` W ) ) -1-1-onto-> ( F supp .0. ) )
49		fzfi 12771	. . . . . . . . . 10 |- ( 1 ... M ) e. Fin
50	49	a1i 11	. . . . . . . . 9 |- ( ph -> ( 1 ... M ) e. Fin )
51		fex2 7121	. . . . . . . . 9 |- ( ( H : ( 1 ... M ) --> A /\ ( 1 ... M ) e. Fin /\ A e. V ) -> H e. _V )
52	8, 50, 18, 51	syl3anc 1326	. . . . . . . 8 |- ( ph -> H e. _V )
53		resexg 5442	. . . . . . . 8 |- ( H e. _V -> ( H |` W ) e. _V )
54	52, 53	syl 17	. . . . . . 7 |- ( ph -> ( H |` W ) e. _V )
55	54	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H |` W ) e. _V )
56	3	a1i 11	. . . . . . . . . 10 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> W = ( ( F o. H ) supp .0. ) )
57	56	imaeq2d 5466	. . . . . . . . 9 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H " W ) = ( H " ( ( F o. H ) supp .0. ) ) )
58	20, 52, 28	jca31 557	. . . . . . . . . . 11 |- ( ph -> ( ( F e. _V /\ H e. _V ) /\ ( Fun H /\ ( F supp .0. ) C_ ran H ) ) )
59	58	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( ( F e. _V /\ H e. _V ) /\ ( Fun H /\ ( F supp .0. ) C_ ran H ) ) )
60	59, 32	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H " ( ( F o. H ) supp .0. ) ) = ( F supp .0. ) )
61	57, 60	eqtrd 2656	. . . . . . . 8 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H " W ) = ( F supp .0. ) )
62	61, 35	syl 17	. . . . . . 7 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( ( H |` W ) : W -1-1-onto-> ( H " W ) <-> ( H |` W ) : W -1-1-onto-> ( F supp .0. ) ) )
63	16, 62	mpbid 222	. . . . . 6 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H |` W ) : W -1-1-onto-> ( F supp .0. ) )
64		f1oen3g 7971	. . . . . 6 |- ( ( ( H |` W ) e. _V /\ ( H |` W ) : W -1-1-onto-> ( F supp .0. ) ) -> W ~~ ( F supp .0. ) )
65	55, 63, 64	syl2anc 693	. . . . 5 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> W ~~ ( F supp .0. ) )
66		ssfi 8180	. . . . . . . 8 |- ( ( ( 1 ... M ) e. Fin /\ W C_ ( 1 ... M ) ) -> W e. Fin )
67	49, 13, 66	sylancr 695	. . . . . . 7 |- ( ph -> W e. Fin )
68	67	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> W e. Fin )
69		f1f1orn 6148	. . . . . . . . . . . 12 |- ( H : ( 1 ... M ) -1-1-> A -> H : ( 1 ... M ) -1-1-onto-> ran H )
70	1, 69	syl 17	. . . . . . . . . . 11 |- ( ph -> H : ( 1 ... M ) -1-1-onto-> ran H )
71		f1oen3g 7971	. . . . . . . . . . 11 |- ( ( H e. _V /\ H : ( 1 ... M ) -1-1-onto-> ran H ) -> ( 1 ... M ) ~~ ran H )
72	52, 70, 71	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( 1 ... M ) ~~ ran H )
73		enfi 8176	. . . . . . . . . 10 |- ( ( 1 ... M ) ~~ ran H -> ( ( 1 ... M ) e. Fin <-> ran H e. Fin ) )
74	72, 73	syl 17	. . . . . . . . 9 |- ( ph -> ( ( 1 ... M ) e. Fin <-> ran H e. Fin ) )
75	49, 74	mpbii 223	. . . . . . . 8 |- ( ph -> ran H e. Fin )
76		ssfi 8180	. . . . . . . 8 |- ( ( ran H e. Fin /\ ( F supp .0. ) C_ ran H ) -> ( F supp .0. ) e. Fin )
77	75, 27, 76	syl2anc 693	. . . . . . 7 |- ( ph -> ( F supp .0. ) e. Fin )
78	77	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( F supp .0. ) e. Fin )
79		hashen 13135	. . . . . 6 |- ( ( W e. Fin /\ ( F supp .0. ) e. Fin ) -> ( ( # ` W ) = ( # ` ( F supp .0. ) ) <-> W ~~ ( F supp .0. ) ) )
80	68, 78, 79	syl2anc 693	. . . . 5 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( ( # ` W ) = ( # ` ( F supp .0. ) ) <-> W ~~ ( F supp .0. ) ) )
81	65, 80	mpbird 247	. . . 4 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( # ` W ) = ( # ` ( F supp .0. ) ) )
82	81	oveq2d 6666	. . 3 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( 1 ... ( # ` W ) ) = ( 1 ... ( # ` ( F supp .0. ) ) ) )
83		f1oeq2 6128	. . 3 |- ( ( 1 ... ( # ` W ) ) = ( 1 ... ( # ` ( F supp .0. ) ) ) -> ( ( H o. f ) : ( 1 ... ( # ` W ) ) -1-1-onto-> ( F supp .0. ) <-> ( H o. f ) : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) )
84	82, 83	syl 17	. 2 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( ( H o. f ) : ( 1 ... ( # ` W ) ) -1-1-onto-> ( F supp .0. ) <-> ( H o. f ) : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) )
85	48, 84	mpbid 222	1 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H o. f ) : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Fun wfun 5882   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   supp csupp 7295   ~~ cen 7952   Fincfn 7955   1c1 9937   < clt 10074   NNcn 11020   ...cfz 12326   #chash 13117   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Mndcmnd 17294  Cntzccntz 17748

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-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

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-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-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-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118

This theorem is referenced by:  gsumval3lem2  18307