Theorem gsumply1subr 19604

Description: Evaluate a group sum in a polynomial ring over a subring. (Contributed by AV, 22-Sep-2019.) (Proof shortened by AV, 31-Jan-2020.)

Hypotheses

Ref	Expression
subrgply1.s	|- S = (Poly1 ` R )
subrgply1.h	|- H = ( R ↾s T )
subrgply1.u	|- U = (Poly1 ` H )
subrgply1.b	|- B = ( Base ` U )
gsumply1subr.s	|- ( ph -> T e. (SubRing ` R ) )
gsumply1subr.a	|- ( ph -> A e. V )
gsumply1subr.f	|- ( ph -> F : A --> B )

Assertion

Ref	Expression
gsumply1subr	|- ( ph -> ( S gsumg F ) = ( U gsumg F ) )

Proof of Theorem gsumply1subr

Dummy variables s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumply1subr.a	. . 3 |- ( ph -> A e. V )
2		gsumply1subr.s	. . . 4 |- ( ph -> T e. (SubRing ` R ) )
3		subrgply1.s	. . . . 5 |- S = (Poly1 ` R )
4		subrgply1.h	. . . . 5 |- H = ( R ↾s T )
5		subrgply1.u	. . . . 5 |- U = (Poly1 ` H )
6		subrgply1.b	. . . . 5 |- B = ( Base ` U )
7	3, 4, 5, 6	subrgply1 19603	. . . 4 |- ( T e. (SubRing ` R ) -> B e. (SubRing ` S ) )
8		subrgsubg 18786	. . . . 5 |- ( B e. (SubRing ` S ) -> B e. (SubGrp ` S ) )
9		subgsubm 17616	. . . . 5 |- ( B e. (SubGrp ` S ) -> B e. (SubMnd ` S ) )
10	8, 9	syl 17	. . . 4 |- ( B e. (SubRing ` S ) -> B e. (SubMnd ` S ) )
11	2, 7, 10	3syl 18	. . 3 |- ( ph -> B e. (SubMnd ` S ) )
12		gsumply1subr.f	. . 3 |- ( ph -> F : A --> B )
13		eqid 2622	. . 3 |- ( S ↾s B ) = ( S ↾s B )
14	1, 11, 12, 13	gsumsubm 17373	. 2 |- ( ph -> ( S gsumg F ) = ( ( S ↾s B ) gsumg F ) )
15		fex 6490	. . . 4 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
16	12, 1, 15	syl2anc 693	. . 3 |- ( ph -> F e. _V )
17		ovexd 6680	. . 3 |- ( ph -> ( S ↾s B ) e. _V )
18		fvex 6201	. . . . 5 |- (Poly1 ` H ) e. _V
19	5, 18	eqeltri 2697	. . . 4 |- U e. _V
20	19	a1i 11	. . 3 |- ( ph -> U e. _V )
21		eqid 2622	. . . . 5 |- ( Base ` U ) = ( Base ` U )
22	6	oveq2i 6661	. . . . 5 |- ( S ↾s B ) = ( S ↾s ( Base ` U ) )
23	3, 4, 5, 21, 2, 22	ressply1bas 19599	. . . 4 |- ( ph -> ( Base ` U ) = ( Base ` ( S ↾s B ) ) )
24	23	eqcomd 2628	. . 3 |- ( ph -> ( Base ` ( S ↾s B ) ) = ( Base ` U ) )
25	13	subrgring 18783	. . . . 5 |- ( B e. (SubRing ` S ) -> ( S ↾s B ) e. Ring )
26	7, 25	syl 17	. . . 4 |- ( T e. (SubRing ` R ) -> ( S ↾s B ) e. Ring )
27		ringmgm 18557	. . . 4 |- ( ( S ↾s B ) e. Ring -> ( S ↾s B ) e. Mgm )
28	2, 26, 27	3syl 18	. . 3 |- ( ph -> ( S ↾s B ) e. Mgm )
29		simpl 473	. . . . 5 |- ( ( ph /\ ( s e. ( Base ` ( S ↾s B ) ) /\ t e. ( Base ` ( S ↾s B ) ) ) ) -> ph )
30	3, 4, 5, 6, 2, 13	ressply1bas 19599	. . . . . . . . . 10 |- ( ph -> B = ( Base ` ( S ↾s B ) ) )
31	30	eqcomd 2628	. . . . . . . . 9 |- ( ph -> ( Base ` ( S ↾s B ) ) = B )
32	31	eleq2d 2687	. . . . . . . 8 |- ( ph -> ( s e. ( Base ` ( S ↾s B ) ) <-> s e. B ) )
33	32	biimpcd 239	. . . . . . 7 |- ( s e. ( Base ` ( S ↾s B ) ) -> ( ph -> s e. B ) )
34	33	adantr 481	. . . . . 6 |- ( ( s e. ( Base ` ( S ↾s B ) ) /\ t e. ( Base ` ( S ↾s B ) ) ) -> ( ph -> s e. B ) )
35	34	impcom 446	. . . . 5 |- ( ( ph /\ ( s e. ( Base ` ( S ↾s B ) ) /\ t e. ( Base ` ( S ↾s B ) ) ) ) -> s e. B )
36	31	eleq2d 2687	. . . . . . . 8 |- ( ph -> ( t e. ( Base ` ( S ↾s B ) ) <-> t e. B ) )
37	36	biimpcd 239	. . . . . . 7 |- ( t e. ( Base ` ( S ↾s B ) ) -> ( ph -> t e. B ) )
38	37	adantl 482	. . . . . 6 |- ( ( s e. ( Base ` ( S ↾s B ) ) /\ t e. ( Base ` ( S ↾s B ) ) ) -> ( ph -> t e. B ) )
39	38	impcom 446	. . . . 5 |- ( ( ph /\ ( s e. ( Base ` ( S ↾s B ) ) /\ t e. ( Base ` ( S ↾s B ) ) ) ) -> t e. B )
40	3, 4, 5, 6, 2, 13	ressply1add 19600	. . . . 5 |- ( ( ph /\ ( s e. B /\ t e. B ) ) -> ( s ( +g ` U ) t ) = ( s ( +g ` ( S ↾s B ) ) t ) )
41	29, 35, 39, 40	syl12anc 1324	. . . 4 |- ( ( ph /\ ( s e. ( Base ` ( S ↾s B ) ) /\ t e. ( Base ` ( S ↾s B ) ) ) ) -> ( s ( +g ` U ) t ) = ( s ( +g ` ( S ↾s B ) ) t ) )
42	41	eqcomd 2628	. . 3 |- ( ( ph /\ ( s e. ( Base ` ( S ↾s B ) ) /\ t e. ( Base ` ( S ↾s B ) ) ) ) -> ( s ( +g ` ( S ↾s B ) ) t ) = ( s ( +g ` U ) t ) )
43		ffun 6048	. . . 4 |- ( F : A --> B -> Fun F )
44	12, 43	syl 17	. . 3 |- ( ph -> Fun F )
45		frn 6053	. . . . 5 |- ( F : A --> B -> ran F C_ B )
46	12, 45	syl 17	. . . 4 |- ( ph -> ran F C_ B )
47	46, 30	sseqtrd 3641	. . 3 |- ( ph -> ran F C_ ( Base ` ( S ↾s B ) ) )
48	16, 17, 20, 24, 28, 42, 44, 47	gsummgmpropd 17275	. 2 |- ( ph -> ( ( S ↾s B ) gsumg F ) = ( U gsumg F ) )
49	14, 48	eqtrd 2656	1 |- ( ph -> ( S gsumg F ) = ( U gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ran crn 5115   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   gsum_g cgsu 16101  Mgmcmgm 17240  SubMndcsubmnd 17334  SubGrpcsubg 17588   Ringcrg 18547  SubRingcsubrg 18776  Poly₁cpl1 19547

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

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-iin 4523  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-of 6897  df-ofr 6898  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-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  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-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-tset 15960  df-ple 15961  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-psr 19356  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-ply1 19552

This theorem is referenced by: (None)