Theorem lply1binomsc 19677

Description: The binomial theorem for linear polynomials (monic polynomials of degree 1) over commutative rings, expressed by an element of this ring: ( X + A ) ^ N is the sum from k = 0 to N of ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( X ^ k ) ). (Contributed by AV, 25-Aug-2019.)

Hypotheses

Ref	Expression
cply1binom.p	|- P = (Poly1 ` R )
cply1binom.x	|- X = (var1 ` R )
cply1binom.a	|- .+ = ( +g ` P )
cply1binom.m	|- .X. = ( .r ` P )
cply1binom.t	|- .x. = (.g ` P )
cply1binom.g	|- G = (mulGrp ` P )
cply1binom.e	|- .^ = (.g ` G )
lply1binomsc.k	|- K = ( Base ` R )
lply1binomsc.s	|- S = (algSc ` P )
lply1binomsc.h	|- H = (mulGrp ` R )
lply1binomsc.e	|- E = (.g ` H )

Assertion

Ref	Expression
lply1binomsc	|- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> ( N .^ ( X .+ ( S ` A ) ) ) = ( P gsumg ( k e. ( 0 ... N ) |-> ( ( N _C k ) .x. ( ( S ` ( ( N - k ) E A ) ) .X. ( k .^ X ) ) ) ) ) )

Distinct variable groups:   A, k   k, K   k, N   P, k   R, k   k, X   .X. , k   .x. , k   .^ , k   .+ , k   S, k

Allowed substitution hints:   E( k)   G( k)   H( k)

Proof of Theorem lply1binomsc

Step	Hyp	Ref	Expression
1		lply1binomsc.s	. . . . . 6 |- S = (algSc ` P )
2		eqid 2622	. . . . . 6 |- (Scalar ` P ) = (Scalar ` P )
3		crngring 18558	. . . . . . . 8 |- ( R e. CRing -> R e. Ring )
4		cply1binom.p	. . . . . . . . 9 |- P = (Poly1 ` R )
5	4	ply1ring 19618	. . . . . . . 8 |- ( R e. Ring -> P e. Ring )
6	3, 5	syl 17	. . . . . . 7 |- ( R e. CRing -> P e. Ring )
7	6	3ad2ant1 1082	. . . . . 6 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> P e. Ring )
8	4	ply1lmod 19622	. . . . . . . 8 |- ( R e. Ring -> P e. LMod )
9	3, 8	syl 17	. . . . . . 7 |- ( R e. CRing -> P e. LMod )
10	9	3ad2ant1 1082	. . . . . 6 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> P e. LMod )
11		eqid 2622	. . . . . 6 |- ( Base ` (Scalar ` P ) ) = ( Base ` (Scalar ` P ) )
12		eqid 2622	. . . . . 6 |- ( Base ` P ) = ( Base ` P )
13	1, 2, 7, 10, 11, 12	asclf 19337	. . . . 5 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> S : ( Base ` (Scalar ` P ) ) --> ( Base ` P ) )
14		lply1binomsc.k	. . . . . . 7 |- K = ( Base ` R )
15	4	ply1sca 19623	. . . . . . . . 9 |- ( R e. CRing -> R = (Scalar ` P ) )
16	15	3ad2ant1 1082	. . . . . . . 8 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> R = (Scalar ` P ) )
17	16	fveq2d 6195	. . . . . . 7 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> ( Base ` R ) = ( Base ` (Scalar ` P ) ) )
18	14, 17	syl5eq 2668	. . . . . 6 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> K = ( Base ` (Scalar ` P ) ) )
19	18	feq2d 6031	. . . . 5 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> ( S : K --> ( Base ` P ) <-> S : ( Base ` (Scalar ` P ) ) --> ( Base ` P ) ) )
20	13, 19	mpbird 247	. . . 4 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> S : K --> ( Base ` P ) )
21		simp3 1063	. . . 4 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> A e. K )
22	20, 21	ffvelrnd 6360	. . 3 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> ( S ` A ) e. ( Base ` P ) )
23		cply1binom.x	. . . 4 |- X = (var1 ` R )
24		cply1binom.a	. . . 4 |- .+ = ( +g ` P )
25		cply1binom.m	. . . 4 |- .X. = ( .r ` P )
26		cply1binom.t	. . . 4 |- .x. = (.g ` P )
27		cply1binom.g	. . . 4 |- G = (mulGrp ` P )
28		cply1binom.e	. . . 4 |- .^ = (.g ` G )
29	4, 23, 24, 25, 26, 27, 28, 12	lply1binom 19676	. . 3 |- ( ( R e. CRing /\ N e. NN0 /\ ( S ` A ) e. ( Base ` P ) ) -> ( N .^ ( X .+ ( S ` A ) ) ) = ( P gsumg ( k e. ( 0 ... N ) |-> ( ( N _C k ) .x. ( ( ( N - k ) .^ ( S ` A ) ) .X. ( k .^ X ) ) ) ) ) )
30	22, 29	syld3an3 1371	. 2 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> ( N .^ ( X .+ ( S ` A ) ) ) = ( P gsumg ( k e. ( 0 ... N ) |-> ( ( N _C k ) .x. ( ( ( N - k ) .^ ( S ` A ) ) .X. ( k .^ X ) ) ) ) ) )
31	4	ply1assa 19569	. . . . . . . . . . 11 |- ( R e. CRing -> P e. AssAlg )
32	31	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> P e. AssAlg )
33	32	adantr 481	. . . . . . . . 9 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> P e. AssAlg )
34		fznn0sub 12373	. . . . . . . . . 10 |- ( k e. ( 0 ... N ) -> ( N - k ) e. NN0 )
35	34	adantl 482	. . . . . . . . 9 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( N - k ) e. NN0 )
36	15	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( R e. CRing -> ( Base ` R ) = ( Base ` (Scalar ` P ) ) )
37	14, 36	syl5eq 2668	. . . . . . . . . . . . 13 |- ( R e. CRing -> K = ( Base ` (Scalar ` P ) ) )
38	37	eleq2d 2687	. . . . . . . . . . . 12 |- ( R e. CRing -> ( A e. K <-> A e. ( Base ` (Scalar ` P ) ) ) )
39	38	biimpa 501	. . . . . . . . . . 11 |- ( ( R e. CRing /\ A e. K ) -> A e. ( Base ` (Scalar ` P ) ) )
40	39	3adant2 1080	. . . . . . . . . 10 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> A e. ( Base ` (Scalar ` P ) ) )
41	40	adantr 481	. . . . . . . . 9 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> A e. ( Base ` (Scalar ` P ) ) )
42		eqid 2622	. . . . . . . . . . . . 13 |- ( 1r ` P ) = ( 1r ` P )
43	12, 42	ringidcl 18568	. . . . . . . . . . . 12 |- ( P e. Ring -> ( 1r ` P ) e. ( Base ` P ) )
44	6, 43	syl 17	. . . . . . . . . . 11 |- ( R e. CRing -> ( 1r ` P ) e. ( Base ` P ) )
45	44	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> ( 1r ` P ) e. ( Base ` P ) )
46	45	adantr 481	. . . . . . . . 9 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( 1r ` P ) e. ( Base ` P ) )
47		eqid 2622	. . . . . . . . . 10 |- ( .s ` P ) = ( .s ` P )
48		eqid 2622	. . . . . . . . . 10 |- (mulGrp ` (Scalar ` P ) ) = (mulGrp ` (Scalar ` P ) )
49		eqid 2622	. . . . . . . . . 10 |- (.g ` (mulGrp ` (Scalar ` P ) ) ) = (.g ` (mulGrp ` (Scalar ` P ) ) )
50	12, 2, 11, 47, 48, 49, 27, 28	assamulgscm 19350	. . . . . . . . 9 |- ( ( P e. AssAlg /\ ( ( N - k ) e. NN0 /\ A e. ( Base ` (Scalar ` P ) ) /\ ( 1r ` P ) e. ( Base ` P ) ) ) -> ( ( N - k ) .^ ( A ( .s ` P ) ( 1r ` P ) ) ) = ( ( ( N - k ) (.g ` (mulGrp ` (Scalar ` P ) ) ) A ) ( .s ` P ) ( ( N - k ) .^ ( 1r ` P ) ) ) )
51	33, 35, 41, 46, 50	syl13anc 1328	. . . . . . . 8 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( ( N - k ) .^ ( A ( .s ` P ) ( 1r ` P ) ) ) = ( ( ( N - k ) (.g ` (mulGrp ` (Scalar ` P ) ) ) A ) ( .s ` P ) ( ( N - k ) .^ ( 1r ` P ) ) ) )
52		lply1binomsc.e	. . . . . . . . . . . . . 14 |- E = (.g ` H )
53		lply1binomsc.h	. . . . . . . . . . . . . . . 16 |- H = (mulGrp ` R )
54	15	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( R e. CRing -> (mulGrp ` R ) = (mulGrp ` (Scalar ` P ) ) )
55	53, 54	syl5eq 2668	. . . . . . . . . . . . . . 15 |- ( R e. CRing -> H = (mulGrp ` (Scalar ` P ) ) )
56	55	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( R e. CRing -> (.g ` H ) = (.g ` (mulGrp ` (Scalar ` P ) ) ) )
57	52, 56	syl5eq 2668	. . . . . . . . . . . . 13 |- ( R e. CRing -> E = (.g ` (mulGrp ` (Scalar ` P ) ) ) )
58	57	3ad2ant1 1082	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> E = (.g ` (mulGrp ` (Scalar ` P ) ) ) )
59	58	adantr 481	. . . . . . . . . . 11 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> E = (.g ` (mulGrp ` (Scalar ` P ) ) ) )
60	59	eqcomd 2628	. . . . . . . . . 10 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> (.g ` (mulGrp ` (Scalar ` P ) ) ) = E )
61	60	oveqd 6667	. . . . . . . . 9 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( ( N - k ) (.g ` (mulGrp ` (Scalar ` P ) ) ) A ) = ( ( N - k ) E A ) )
62	27	ringmgp 18553	. . . . . . . . . . . 12 |- ( P e. Ring -> G e. Mnd )
63	6, 62	syl 17	. . . . . . . . . . 11 |- ( R e. CRing -> G e. Mnd )
64	63	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> G e. Mnd )
65	27, 12	mgpbas 18495	. . . . . . . . . . 11 |- ( Base ` P ) = ( Base ` G )
66	27, 42	ringidval 18503	. . . . . . . . . . 11 |- ( 1r ` P ) = ( 0g ` G )
67	65, 28, 66	mulgnn0z 17567	. . . . . . . . . 10 |- ( ( G e. Mnd /\ ( N - k ) e. NN0 ) -> ( ( N - k ) .^ ( 1r ` P ) ) = ( 1r ` P ) )
68	64, 34, 67	syl2an 494	. . . . . . . . 9 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( ( N - k ) .^ ( 1r ` P ) ) = ( 1r ` P ) )
69	61, 68	oveq12d 6668	. . . . . . . 8 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( ( ( N - k ) (.g ` (mulGrp ` (Scalar ` P ) ) ) A ) ( .s ` P ) ( ( N - k ) .^ ( 1r ` P ) ) ) = ( ( ( N - k ) E A ) ( .s ` P ) ( 1r ` P ) ) )
70	51, 69	eqtrd 2656	. . . . . . 7 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( ( N - k ) .^ ( A ( .s ` P ) ( 1r ` P ) ) ) = ( ( ( N - k ) E A ) ( .s ` P ) ( 1r ` P ) ) )
71	1, 2, 11, 47, 42	asclval 19335	. . . . . . . . 9 |- ( A e. ( Base ` (Scalar ` P ) ) -> ( S ` A ) = ( A ( .s ` P ) ( 1r ` P ) ) )
72	41, 71	syl 17	. . . . . . . 8 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( S ` A ) = ( A ( .s ` P ) ( 1r ` P ) ) )
73	72	oveq2d 6666	. . . . . . 7 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( ( N - k ) .^ ( S ` A ) ) = ( ( N - k ) .^ ( A ( .s ` P ) ( 1r ` P ) ) ) )
74	53	ringmgp 18553	. . . . . . . . . . . . 13 |- ( R e. Ring -> H e. Mnd )
75	3, 74	syl 17	. . . . . . . . . . . 12 |- ( R e. CRing -> H e. Mnd )
76	75	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> H e. Mnd )
77	76	adantr 481	. . . . . . . . . 10 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> H e. Mnd )
78		simpr 477	. . . . . . . . . . . . 13 |- ( ( R e. CRing /\ A e. K ) -> A e. K )
79	53, 14	mgpbas 18495	. . . . . . . . . . . . 13 |- K = ( Base ` H )
80	78, 79	syl6eleq 2711	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ A e. K ) -> A e. ( Base ` H ) )
81	80	3adant2 1080	. . . . . . . . . . 11 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> A e. ( Base ` H ) )
82	81	adantr 481	. . . . . . . . . 10 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> A e. ( Base ` H ) )
83		eqid 2622	. . . . . . . . . . 11 |- ( Base ` H ) = ( Base ` H )
84	83, 52	mulgnn0cl 17558	. . . . . . . . . 10 |- ( ( H e. Mnd /\ ( N - k ) e. NN0 /\ A e. ( Base ` H ) ) -> ( ( N - k ) E A ) e. ( Base ` H ) )
85	77, 35, 82, 84	syl3anc 1326	. . . . . . . . 9 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( ( N - k ) E A ) e. ( Base ` H ) )
86	16	adantr 481	. . . . . . . . . . . 12 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> R = (Scalar ` P ) )
87	86	eqcomd 2628	. . . . . . . . . . 11 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> (Scalar ` P ) = R )
88	87	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( Base ` (Scalar ` P ) ) = ( Base ` R ) )
89		eqid 2622	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
90	53, 89	mgpbas 18495	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` H )
91	88, 90	syl6eq 2672	. . . . . . . . 9 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( Base ` (Scalar ` P ) ) = ( Base ` H ) )
92	85, 91	eleqtrrd 2704	. . . . . . . 8 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( ( N - k ) E A ) e. ( Base ` (Scalar ` P ) ) )
93	1, 2, 11, 47, 42	asclval 19335	. . . . . . . 8 |- ( ( ( N - k ) E A ) e. ( Base ` (Scalar ` P ) ) -> ( S ` ( ( N - k ) E A ) ) = ( ( ( N - k ) E A ) ( .s ` P ) ( 1r ` P ) ) )
94	92, 93	syl 17	. . . . . . 7 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( S ` ( ( N - k ) E A ) ) = ( ( ( N - k ) E A ) ( .s ` P ) ( 1r ` P ) ) )
95	70, 73, 94	3eqtr4d 2666	. . . . . 6 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( ( N - k ) .^ ( S ` A ) ) = ( S ` ( ( N - k ) E A ) ) )
96	95	oveq1d 6665	. . . . 5 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( ( ( N - k ) .^ ( S ` A ) ) .X. ( k .^ X ) ) = ( ( S ` ( ( N - k ) E A ) ) .X. ( k .^ X ) ) )
97	96	oveq2d 6666	. . . 4 |- ( ( ( R e. CRing /\ N e. NN0 /\ A e. K ) /\ k e. ( 0 ... N ) ) -> ( ( N _C k ) .x. ( ( ( N - k ) .^ ( S ` A ) ) .X. ( k .^ X ) ) ) = ( ( N _C k ) .x. ( ( S ` ( ( N - k ) E A ) ) .X. ( k .^ X ) ) ) )
98	97	mpteq2dva 4744	. . 3 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> ( k e. ( 0 ... N ) |-> ( ( N _C k ) .x. ( ( ( N - k ) .^ ( S ` A ) ) .X. ( k .^ X ) ) ) ) = ( k e. ( 0 ... N ) |-> ( ( N _C k ) .x. ( ( S ` ( ( N - k ) E A ) ) .X. ( k .^ X ) ) ) ) )
99	98	oveq2d 6666	. 2 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> ( P gsumg ( k e. ( 0 ... N ) |-> ( ( N _C k ) .x. ( ( ( N - k ) .^ ( S ` A ) ) .X. ( k .^ X ) ) ) ) ) = ( P gsumg ( k e. ( 0 ... N ) |-> ( ( N _C k ) .x. ( ( S ` ( ( N - k ) E A ) ) .X. ( k .^ X ) ) ) ) ) )
100	30, 99	eqtrd 2656	1 |- ( ( R e. CRing /\ N e. NN0 /\ A e. K ) -> ( N .^ ( X .+ ( S ` A ) ) ) = ( P gsumg ( k e. ( 0 ... N ) |-> ( ( N _C k ) .x. ( ( S ` ( ( N - k ) E A ) ) .X. ( k .^ X ) ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   - cmin 10266   NN0cn0 11292   ...cfz 12326   _C cbc 13089   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   gsum_g cgsu 16101   Mndcmnd 17294  ._gcmg 17540  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547   CRingccrg 18548   LModclmod 18863  AssAlgcasa 19309  algSccascl 19311  var₁cv1 19546  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-div 10685  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-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-fac 13061  df-bc 13090  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-sbg 17427  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-srg 18506  df-ring 18549  df-cring 18550  df-subrg 18778  df-lmod 18865  df-lss 18933  df-assa 19312  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-vr1 19551  df-ply1 19552

This theorem is referenced by:  chpscmatgsumbin  20649