Theorem coe1fzgsumd 19672

Description: Value of an evaluated coefficient in a finite group sum of polynomials. (Contributed by AV, 8-Oct-2019.)

Hypotheses

Ref	Expression
coe1fzgsumd.p	|- P = (Poly1 ` R )
coe1fzgsumd.b	|- B = ( Base ` P )
coe1fzgsumd.r	|- ( ph -> R e. Ring )
coe1fzgsumd.k	|- ( ph -> K e. NN0 )
coe1fzgsumd.m	|- ( ph -> A. x e. N M e. B )
coe1fzgsumd.n	|- ( ph -> N e. Fin )

Assertion

Ref	Expression
coe1fzgsumd	|- ( ph -> ( (coe1 ` ( P gsumg ( x e. N |-> M ) ) ) ` K ) = ( R gsumg ( x e. N |-> ( (coe1 ` M ) ` K ) ) ) )

Distinct variable groups:   x, B   x, K   x, N

Allowed substitution hints:   ph( x)   P( x)   R( x)   M( x)

Proof of Theorem coe1fzgsumd

Dummy variables a m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		coe1fzgsumd.m	. 2 |- ( ph -> A. x e. N M e. B )
2		coe1fzgsumd.n	. . 3 |- ( ph -> N e. Fin )
3		raleq 3138	. . . . . . 7 |- ( n = (/) -> ( A. x e. n M e. B <-> A. x e. (/) M e. B ) )
4	3	anbi2d 740	. . . . . 6 |- ( n = (/) -> ( ( ph /\ A. x e. n M e. B ) <-> ( ph /\ A. x e. (/) M e. B ) ) )
5		mpteq1 4737	. . . . . . . . . 10 |- ( n = (/) -> ( x e. n |-> M ) = ( x e. (/) |-> M ) )
6	5	oveq2d 6666	. . . . . . . . 9 |- ( n = (/) -> ( P gsumg ( x e. n |-> M ) ) = ( P gsumg ( x e. (/) |-> M ) ) )
7	6	fveq2d 6195	. . . . . . . 8 |- ( n = (/) -> (coe1 ` ( P gsumg ( x e. n |-> M ) ) ) = (coe1 ` ( P gsumg ( x e. (/) |-> M ) ) ) )
8	7	fveq1d 6193	. . . . . . 7 |- ( n = (/) -> ( (coe1 ` ( P gsumg ( x e. n |-> M ) ) ) ` K ) = ( (coe1 ` ( P gsumg ( x e. (/) |-> M ) ) ) ` K ) )
9		mpteq1 4737	. . . . . . . 8 |- ( n = (/) -> ( x e. n |-> ( (coe1 ` M ) ` K ) ) = ( x e. (/) |-> ( (coe1 ` M ) ` K ) ) )
10	9	oveq2d 6666	. . . . . . 7 |- ( n = (/) -> ( R gsumg ( x e. n |-> ( (coe1 ` M ) ` K ) ) ) = ( R gsumg ( x e. (/) |-> ( (coe1 ` M ) ` K ) ) ) )
11	8, 10	eqeq12d 2637	. . . . . 6 |- ( n = (/) -> ( ( (coe1 ` ( P gsumg ( x e. n |-> M ) ) ) ` K ) = ( R gsumg ( x e. n |-> ( (coe1 ` M ) ` K ) ) ) <-> ( (coe1 ` ( P gsumg ( x e. (/) |-> M ) ) ) ` K ) = ( R gsumg ( x e. (/) |-> ( (coe1 ` M ) ` K ) ) ) ) )
12	4, 11	imbi12d 334	. . . . 5 |- ( n = (/) -> ( ( ( ph /\ A. x e. n M e. B ) -> ( (coe1 ` ( P gsumg ( x e. n |-> M ) ) ) ` K ) = ( R gsumg ( x e. n |-> ( (coe1 ` M ) ` K ) ) ) ) <-> ( ( ph /\ A. x e. (/) M e. B ) -> ( (coe1 ` ( P gsumg ( x e. (/) |-> M ) ) ) ` K ) = ( R gsumg ( x e. (/) |-> ( (coe1 ` M ) ` K ) ) ) ) ) )
13		raleq 3138	. . . . . . 7 |- ( n = m -> ( A. x e. n M e. B <-> A. x e. m M e. B ) )
14	13	anbi2d 740	. . . . . 6 |- ( n = m -> ( ( ph /\ A. x e. n M e. B ) <-> ( ph /\ A. x e. m M e. B ) ) )
15		mpteq1 4737	. . . . . . . . . 10 |- ( n = m -> ( x e. n |-> M ) = ( x e. m |-> M ) )
16	15	oveq2d 6666	. . . . . . . . 9 |- ( n = m -> ( P gsumg ( x e. n |-> M ) ) = ( P gsumg ( x e. m |-> M ) ) )
17	16	fveq2d 6195	. . . . . . . 8 |- ( n = m -> (coe1 ` ( P gsumg ( x e. n |-> M ) ) ) = (coe1 ` ( P gsumg ( x e. m |-> M ) ) ) )
18	17	fveq1d 6193	. . . . . . 7 |- ( n = m -> ( (coe1 ` ( P gsumg ( x e. n |-> M ) ) ) ` K ) = ( (coe1 ` ( P gsumg ( x e. m |-> M ) ) ) ` K ) )
19		mpteq1 4737	. . . . . . . 8 |- ( n = m -> ( x e. n |-> ( (coe1 ` M ) ` K ) ) = ( x e. m |-> ( (coe1 ` M ) ` K ) ) )
20	19	oveq2d 6666	. . . . . . 7 |- ( n = m -> ( R gsumg ( x e. n |-> ( (coe1 ` M ) ` K ) ) ) = ( R gsumg ( x e. m |-> ( (coe1 ` M ) ` K ) ) ) )
21	18, 20	eqeq12d 2637	. . . . . 6 |- ( n = m -> ( ( (coe1 ` ( P gsumg ( x e. n |-> M ) ) ) ` K ) = ( R gsumg ( x e. n |-> ( (coe1 ` M ) ` K ) ) ) <-> ( (coe1 ` ( P gsumg ( x e. m |-> M ) ) ) ` K ) = ( R gsumg ( x e. m |-> ( (coe1 ` M ) ` K ) ) ) ) )
22	14, 21	imbi12d 334	. . . . 5 |- ( n = m -> ( ( ( ph /\ A. x e. n M e. B ) -> ( (coe1 ` ( P gsumg ( x e. n |-> M ) ) ) ` K ) = ( R gsumg ( x e. n |-> ( (coe1 ` M ) ` K ) ) ) ) <-> ( ( ph /\ A. x e. m M e. B ) -> ( (coe1 ` ( P gsumg ( x e. m |-> M ) ) ) ` K ) = ( R gsumg ( x e. m |-> ( (coe1 ` M ) ` K ) ) ) ) ) )
23		raleq 3138	. . . . . . 7 |- ( n = ( m u. { a } ) -> ( A. x e. n M e. B <-> A. x e. ( m u. { a } ) M e. B ) )
24	23	anbi2d 740	. . . . . 6 |- ( n = ( m u. { a } ) -> ( ( ph /\ A. x e. n M e. B ) <-> ( ph /\ A. x e. ( m u. { a } ) M e. B ) ) )
25		mpteq1 4737	. . . . . . . . . 10 |- ( n = ( m u. { a } ) -> ( x e. n |-> M ) = ( x e. ( m u. { a } ) |-> M ) )
26	25	oveq2d 6666	. . . . . . . . 9 |- ( n = ( m u. { a } ) -> ( P gsumg ( x e. n |-> M ) ) = ( P gsumg ( x e. ( m u. { a } ) |-> M ) ) )
27	26	fveq2d 6195	. . . . . . . 8 |- ( n = ( m u. { a } ) -> (coe1 ` ( P gsumg ( x e. n |-> M ) ) ) = (coe1 ` ( P gsumg ( x e. ( m u. { a } ) |-> M ) ) ) )
28	27	fveq1d 6193	. . . . . . 7 |- ( n = ( m u. { a } ) -> ( (coe1 ` ( P gsumg ( x e. n |-> M ) ) ) ` K ) = ( (coe1 ` ( P gsumg ( x e. ( m u. { a } ) |-> M ) ) ) ` K ) )
29		mpteq1 4737	. . . . . . . 8 |- ( n = ( m u. { a } ) -> ( x e. n |-> ( (coe1 ` M ) ` K ) ) = ( x e. ( m u. { a } ) |-> ( (coe1 ` M ) ` K ) ) )
30	29	oveq2d 6666	. . . . . . 7 |- ( n = ( m u. { a } ) -> ( R gsumg ( x e. n |-> ( (coe1 ` M ) ` K ) ) ) = ( R gsumg ( x e. ( m u. { a } ) |-> ( (coe1 ` M ) ` K ) ) ) )
31	28, 30	eqeq12d 2637	. . . . . 6 |- ( n = ( m u. { a } ) -> ( ( (coe1 ` ( P gsumg ( x e. n |-> M ) ) ) ` K ) = ( R gsumg ( x e. n |-> ( (coe1 ` M ) ` K ) ) ) <-> ( (coe1 ` ( P gsumg ( x e. ( m u. { a } ) |-> M ) ) ) ` K ) = ( R gsumg ( x e. ( m u. { a } ) |-> ( (coe1 ` M ) ` K ) ) ) ) )
32	24, 31	imbi12d 334	. . . . 5 |- ( n = ( m u. { a } ) -> ( ( ( ph /\ A. x e. n M e. B ) -> ( (coe1 ` ( P gsumg ( x e. n |-> M ) ) ) ` K ) = ( R gsumg ( x e. n |-> ( (coe1 ` M ) ` K ) ) ) ) <-> ( ( ph /\ A. x e. ( m u. { a } ) M e. B ) -> ( (coe1 ` ( P gsumg ( x e. ( m u. { a } ) |-> M ) ) ) ` K ) = ( R gsumg ( x e. ( m u. { a } ) |-> ( (coe1 ` M ) ` K ) ) ) ) ) )
33		raleq 3138	. . . . . . 7 |- ( n = N -> ( A. x e. n M e. B <-> A. x e. N M e. B ) )
34	33	anbi2d 740	. . . . . 6 |- ( n = N -> ( ( ph /\ A. x e. n M e. B ) <-> ( ph /\ A. x e. N M e. B ) ) )
35		mpteq1 4737	. . . . . . . . . 10 |- ( n = N -> ( x e. n |-> M ) = ( x e. N |-> M ) )
36	35	oveq2d 6666	. . . . . . . . 9 |- ( n = N -> ( P gsumg ( x e. n |-> M ) ) = ( P gsumg ( x e. N |-> M ) ) )
37	36	fveq2d 6195	. . . . . . . 8 |- ( n = N -> (coe1 ` ( P gsumg ( x e. n |-> M ) ) ) = (coe1 ` ( P gsumg ( x e. N |-> M ) ) ) )
38	37	fveq1d 6193	. . . . . . 7 |- ( n = N -> ( (coe1 ` ( P gsumg ( x e. n |-> M ) ) ) ` K ) = ( (coe1 ` ( P gsumg ( x e. N |-> M ) ) ) ` K ) )
39		mpteq1 4737	. . . . . . . 8 |- ( n = N -> ( x e. n |-> ( (coe1 ` M ) ` K ) ) = ( x e. N |-> ( (coe1 ` M ) ` K ) ) )
40	39	oveq2d 6666	. . . . . . 7 |- ( n = N -> ( R gsumg ( x e. n |-> ( (coe1 ` M ) ` K ) ) ) = ( R gsumg ( x e. N |-> ( (coe1 ` M ) ` K ) ) ) )
41	38, 40	eqeq12d 2637	. . . . . 6 |- ( n = N -> ( ( (coe1 ` ( P gsumg ( x e. n |-> M ) ) ) ` K ) = ( R gsumg ( x e. n |-> ( (coe1 ` M ) ` K ) ) ) <-> ( (coe1 ` ( P gsumg ( x e. N |-> M ) ) ) ` K ) = ( R gsumg ( x e. N |-> ( (coe1 ` M ) ` K ) ) ) ) )
42	34, 41	imbi12d 334	. . . . 5 |- ( n = N -> ( ( ( ph /\ A. x e. n M e. B ) -> ( (coe1 ` ( P gsumg ( x e. n |-> M ) ) ) ` K ) = ( R gsumg ( x e. n |-> ( (coe1 ` M ) ` K ) ) ) ) <-> ( ( ph /\ A. x e. N M e. B ) -> ( (coe1 ` ( P gsumg ( x e. N |-> M ) ) ) ` K ) = ( R gsumg ( x e. N |-> ( (coe1 ` M ) ` K ) ) ) ) ) )
43		mpt0 6021	. . . . . . . . . . . . 13 |- ( x e. (/) |-> M ) = (/)
44	43	oveq2i 6661	. . . . . . . . . . . 12 |- ( P gsumg ( x e. (/) |-> M ) ) = ( P gsumg (/) )
45		eqid 2622	. . . . . . . . . . . . 13 |- ( 0g ` P ) = ( 0g ` P )
46	45	gsum0 17278	. . . . . . . . . . . 12 |- ( P gsumg (/) ) = ( 0g ` P )
47	44, 46	eqtri 2644	. . . . . . . . . . 11 |- ( P gsumg ( x e. (/) |-> M ) ) = ( 0g ` P )
48	47	fveq2i 6194	. . . . . . . . . 10 |- (coe1 ` ( P gsumg ( x e. (/) |-> M ) ) ) = (coe1 ` ( 0g ` P ) )
49	48	a1i 11	. . . . . . . . 9 |- ( ph -> (coe1 ` ( P gsumg ( x e. (/) |-> M ) ) ) = (coe1 ` ( 0g ` P ) ) )
50	49	fveq1d 6193	. . . . . . . 8 |- ( ph -> ( (coe1 ` ( P gsumg ( x e. (/) |-> M ) ) ) ` K ) = ( (coe1 ` ( 0g ` P ) ) ` K ) )
51		coe1fzgsumd.r	. . . . . . . . . 10 |- ( ph -> R e. Ring )
52		coe1fzgsumd.p	. . . . . . . . . . 11 |- P = (Poly1 ` R )
53		eqid 2622	. . . . . . . . . . 11 |- ( 0g ` R ) = ( 0g ` R )
54	52, 45, 53	coe1z 19633	. . . . . . . . . 10 |- ( R e. Ring -> (coe1 ` ( 0g ` P ) ) = ( NN0 X. { ( 0g ` R ) } ) )
55	51, 54	syl 17	. . . . . . . . 9 |- ( ph -> (coe1 ` ( 0g ` P ) ) = ( NN0 X. { ( 0g ` R ) } ) )
56	55	fveq1d 6193	. . . . . . . 8 |- ( ph -> ( (coe1 ` ( 0g ` P ) ) ` K ) = ( ( NN0 X. { ( 0g ` R ) } ) ` K ) )
57		fvex 6201	. . . . . . . . 9 |- ( 0g ` R ) e. _V
58		coe1fzgsumd.k	. . . . . . . . 9 |- ( ph -> K e. NN0 )
59		fvconst2g 6467	. . . . . . . . 9 |- ( ( ( 0g ` R ) e. _V /\ K e. NN0 ) -> ( ( NN0 X. { ( 0g ` R ) } ) ` K ) = ( 0g ` R ) )
60	57, 58, 59	sylancr 695	. . . . . . . 8 |- ( ph -> ( ( NN0 X. { ( 0g ` R ) } ) ` K ) = ( 0g ` R ) )
61	50, 56, 60	3eqtrd 2660	. . . . . . 7 |- ( ph -> ( (coe1 ` ( P gsumg ( x e. (/) |-> M ) ) ) ` K ) = ( 0g ` R ) )
62		mpt0 6021	. . . . . . . . 9 |- ( x e. (/) |-> ( (coe1 ` M ) ` K ) ) = (/)
63	62	oveq2i 6661	. . . . . . . 8 |- ( R gsumg ( x e. (/) |-> ( (coe1 ` M ) ` K ) ) ) = ( R gsumg (/) )
64	53	gsum0 17278	. . . . . . . 8 |- ( R gsumg (/) ) = ( 0g ` R )
65	63, 64	eqtri 2644	. . . . . . 7 |- ( R gsumg ( x e. (/) |-> ( (coe1 ` M ) ` K ) ) ) = ( 0g ` R )
66	61, 65	syl6eqr 2674	. . . . . 6 |- ( ph -> ( (coe1 ` ( P gsumg ( x e. (/) |-> M ) ) ) ` K ) = ( R gsumg ( x e. (/) |-> ( (coe1 ` M ) ` K ) ) ) )
67	66	adantr 481	. . . . 5 |- ( ( ph /\ A. x e. (/) M e. B ) -> ( (coe1 ` ( P gsumg ( x e. (/) |-> M ) ) ) ` K ) = ( R gsumg ( x e. (/) |-> ( (coe1 ` M ) ` K ) ) ) )
68		coe1fzgsumd.b	. . . . . . . . 9 |- B = ( Base ` P )
69	52, 68, 51, 58	coe1fzgsumdlem 19671	. . . . . . . 8 |- ( ( m e. Fin /\ -. a e. m /\ ph ) -> ( ( A. x e. m M e. B -> ( (coe1 ` ( P gsumg ( x e. m |-> M ) ) ) ` K ) = ( R gsumg ( x e. m |-> ( (coe1 ` M ) ` K ) ) ) ) -> ( A. x e. ( m u. { a } ) M e. B -> ( (coe1 ` ( P gsumg ( x e. ( m u. { a } ) |-> M ) ) ) ` K ) = ( R gsumg ( x e. ( m u. { a } ) |-> ( (coe1 ` M ) ` K ) ) ) ) ) )
70	69	3expia 1267	. . . . . . 7 |- ( ( m e. Fin /\ -. a e. m ) -> ( ph -> ( ( A. x e. m M e. B -> ( (coe1 ` ( P gsumg ( x e. m |-> M ) ) ) ` K ) = ( R gsumg ( x e. m |-> ( (coe1 ` M ) ` K ) ) ) ) -> ( A. x e. ( m u. { a } ) M e. B -> ( (coe1 ` ( P gsumg ( x e. ( m u. { a } ) |-> M ) ) ) ` K ) = ( R gsumg ( x e. ( m u. { a } ) |-> ( (coe1 ` M ) ` K ) ) ) ) ) ) )
71	70	a2d 29	. . . . . 6 |- ( ( m e. Fin /\ -. a e. m ) -> ( ( ph -> ( A. x e. m M e. B -> ( (coe1 ` ( P gsumg ( x e. m |-> M ) ) ) ` K ) = ( R gsumg ( x e. m |-> ( (coe1 ` M ) ` K ) ) ) ) ) -> ( ph -> ( A. x e. ( m u. { a } ) M e. B -> ( (coe1 ` ( P gsumg ( x e. ( m u. { a } ) |-> M ) ) ) ` K ) = ( R gsumg ( x e. ( m u. { a } ) |-> ( (coe1 ` M ) ` K ) ) ) ) ) ) )
72		impexp 462	. . . . . 6 |- ( ( ( ph /\ A. x e. m M e. B ) -> ( (coe1 ` ( P gsumg ( x e. m |-> M ) ) ) ` K ) = ( R gsumg ( x e. m |-> ( (coe1 ` M ) ` K ) ) ) ) <-> ( ph -> ( A. x e. m M e. B -> ( (coe1 ` ( P gsumg ( x e. m |-> M ) ) ) ` K ) = ( R gsumg ( x e. m |-> ( (coe1 ` M ) ` K ) ) ) ) ) )
73		impexp 462	. . . . . 6 |- ( ( ( ph /\ A. x e. ( m u. { a } ) M e. B ) -> ( (coe1 ` ( P gsumg ( x e. ( m u. { a } ) |-> M ) ) ) ` K ) = ( R gsumg ( x e. ( m u. { a } ) |-> ( (coe1 ` M ) ` K ) ) ) ) <-> ( ph -> ( A. x e. ( m u. { a } ) M e. B -> ( (coe1 ` ( P gsumg ( x e. ( m u. { a } ) |-> M ) ) ) ` K ) = ( R gsumg ( x e. ( m u. { a } ) |-> ( (coe1 ` M ) ` K ) ) ) ) ) )
74	71, 72, 73	3imtr4g 285	. . . . 5 |- ( ( m e. Fin /\ -. a e. m ) -> ( ( ( ph /\ A. x e. m M e. B ) -> ( (coe1 ` ( P gsumg ( x e. m |-> M ) ) ) ` K ) = ( R gsumg ( x e. m |-> ( (coe1 ` M ) ` K ) ) ) ) -> ( ( ph /\ A. x e. ( m u. { a } ) M e. B ) -> ( (coe1 ` ( P gsumg ( x e. ( m u. { a } ) |-> M ) ) ) ` K ) = ( R gsumg ( x e. ( m u. { a } ) |-> ( (coe1 ` M ) ` K ) ) ) ) ) )
75	12, 22, 32, 42, 67, 74	findcard2s 8201	. . . 4 |- ( N e. Fin -> ( ( ph /\ A. x e. N M e. B ) -> ( (coe1 ` ( P gsumg ( x e. N |-> M ) ) ) ` K ) = ( R gsumg ( x e. N |-> ( (coe1 ` M ) ` K ) ) ) ) )
76	75	expd 452	. . 3 |- ( N e. Fin -> ( ph -> ( A. x e. N M e. B -> ( (coe1 ` ( P gsumg ( x e. N |-> M ) ) ) ` K ) = ( R gsumg ( x e. N |-> ( (coe1 ` M ) ` K ) ) ) ) ) )
77	2, 76	mpcom 38	. 2 |- ( ph -> ( A. x e. N M e. B -> ( (coe1 ` ( P gsumg ( x e. N |-> M ) ) ) ` K ) = ( R gsumg ( x e. N |-> ( (coe1 ` M ) ` K ) ) ) ) )
78	1, 77	mpd 15	1 |- ( ph -> ( (coe1 ` ( P gsumg ( x e. N |-> M ) ) ) ` K ) = ( R gsumg ( x e. N |-> ( (coe1 ` M ) ` K ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   u. cun 3572   (/)c0 3915   {csn 4177   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   Fincfn 7955   NN0cn0 11292   Basecbs 15857   0gc0g 16100   gsum_g cgsu 16101   Ringcrg 18547  Poly₁cpl1 19547  coe₁cco1 19548

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-fal 1489  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  df-coe1 19553

This theorem is referenced by:  gsummoncoe1  19674  cpmatmcllem  20523  decpmatmullem  20576  mp2pm2mplem4  20614