Theorem gsumressfi 14267

Description: The group sum in a substructure is the same as the group sum in the original structure. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
gsumress.b	|- B = ( Base ` G )
gsumress.o	|- .+ = ( +g ` G )
gsumress.h	|- H = ( G ↾s S )
gsumressfi.g	|- ( ph -> G e. CMnd )
gsumressfi.h	|- ( ph -> H e. CMnd )
gsumressfi.a	|- ( ph -> A e. Fin )
gsumress.s	|- ( ph -> S C_ B )
gsumress.f	|- ( ph -> F : A --> S )
gsumress.z	|- ( ph -> .0. e. S )
gsumress.c	|- ( ( ph /\ x e. B ) -> ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )

Assertion

Ref	Expression
gsumressfi	|- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Distinct variable groups:   x, .+   x, .0.   x, A   x, B   x, G   x, H   x, S   ph, x

Allowed substitution hint:   F( x)

Proof of Theorem gsumressfi

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumressfi.a	. . . 4 |- ( ph -> A e. Fin )
2		isfinite4im 11253	. . . 4 |- ( A e. Fin -> ( 1 ... (♯ ` A ) ) ~~ A )
3	1, 2	syl 14	. . 3 |- ( ph -> ( 1 ... (♯ ` A ) ) ~~ A )
4		bren 7034	. . 3 |- ( ( 1 ... (♯ ` A ) ) ~~ A <-> E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
5	3, 4	sylib 122	. 2 |- ( ph -> E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
6		gsumress.b	. . . 4 |- B = ( Base ` G )
7		gsumress.o	. . . 4 |- .+ = ( +g ` G )
8		gsumress.h	. . . 4 |- H = ( G ↾s S )
9		gsumressfi.g	. . . . 5 |- ( ph -> G e. CMnd )
10	9	adantr 276	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> G e. CMnd )
11		1zzd 9682	. . . . 5 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> 1 e. ZZ )
12	1	adantr 276	. . . . . . 7 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> A e. Fin )
13		hashcl 11242	. . . . . . 7 |- ( A e. Fin -> (♯ ` A ) e. NN0 )
14	12, 13	syl 14	. . . . . 6 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> (♯ ` A ) e. NN0 )
15	14	nn0zd 9777	. . . . 5 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> (♯ ` A ) e. ZZ )
16	11, 15	fzfigd 10889	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( 1 ... (♯ ` A ) ) e. Fin )
17		gsumress.s	. . . . 5 |- ( ph -> S C_ B )
18	17	adantr 276	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> S C_ B )
19		gsumress.f	. . . . . 6 |- ( ph -> F : A --> S )
20	19	adantr 276	. . . . 5 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> F : A --> S )
21		f1of 5642	. . . . . 6 |- ( f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A -> f : ( 1 ... (♯ ` A ) ) --> A )
22	21	adantl 277	. . . . 5 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> f : ( 1 ... (♯ ` A ) ) --> A )
23	20, 22	fcod 5556	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( F o. f ) : ( 1 ... (♯ ` A ) ) --> S )
24		gsumress.z	. . . . 5 |- ( ph -> .0. e. S )
25	24	adantr 276	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> .0. e. S )
26		gsumress.c	. . . . 5 |- ( ( ph /\ x e. B ) -> ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
27	26	adantlr 481	. . . 4 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ x e. B ) -> ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
28	6, 7, 8, 10, 16, 18, 23, 25, 27	gzsumress 13771	. . 3 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( G gzsumgz ( F o. f ) ) = ( H gzsumgz ( F o. f ) ) )
29	19, 17	fssd 5550	. . . . 5 |- ( ph -> F : A --> B )
30	29	adantr 276	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> F : A --> B )
31		simpr 110	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
32	6, 10, 30, 12, 31	gsumvalfi 14242	. . 3 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( G gsumg F ) = ( G gzsumgz ( F o. f ) ) )
33		eqid 2238	. . . 4 |- ( Base ` H ) = ( Base ` H )
34		gsumressfi.h	. . . . 5 |- ( ph -> H e. CMnd )
35	34	adantr 276	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> H e. CMnd )
36	8	a1i 9	. . . . . . . 8 |- ( ph -> H = ( G ↾s S ) )
37	6	a1i 9	. . . . . . . 8 |- ( ph -> B = ( Base ` G ) )
38	36, 37, 9, 17	ressbas2d 13481	. . . . . . 7 |- ( ph -> S = ( Base ` H ) )
39	38	feq3d 5525	. . . . . 6 |- ( ph -> ( F : A --> S <-> F : A --> ( Base ` H ) ) )
40	19, 39	mpbid 147	. . . . 5 |- ( ph -> F : A --> ( Base ` H ) )
41	40	adantr 276	. . . 4 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> F : A --> ( Base ` H ) )
42	33, 35, 41, 12, 31	gsumvalfi 14242	. . 3 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( H gsumg F ) = ( H gzsumgz ( F o. f ) ) )
43	28, 32, 42	3eqtr4d 2281	. 2 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( G gsumg F ) = ( H gsumg F ) )
44	5, 43	exlimddv 1954	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   C_ wss 3220   class class class wbr 4131   o. ccom 4780   -->wf 5376   -1-1-onto->wf1o 5379   ` cfv 5380  (class class class)co 6088   ~~ cen 7024   Fincfn 7026   1c1 8186   NN0cn0 9574   ...cfz 10428  ♯chash 11236   Basecbs 13410   ↾_s cress 13411   +g cplusg 13490   gzsum_gz cgzsu 13670  CMndccmn 14177   gsum_g cgsu 14240

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-ihash 11237  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-iress 13418  df-plusg 13503  df-0g 13671  df-gzsum 13672  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-cmn 14179  df-gsumfi 14241

This theorem is used by:  gsumsubmfi  14268