Theorem cply1coe0bi 19670

Description: A polynomial is constant (i.e. a "lifted scalar") iff all but the first coefficient are zero. (Contributed by AV, 16-Nov-2019.)

Hypotheses

Ref	Expression
cply1coe0.k	|- K = ( Base ` R )
cply1coe0.0	|- .0. = ( 0g ` R )
cply1coe0.p	|- P = (Poly1 ` R )
cply1coe0.b	|- B = ( Base ` P )
cply1coe0.a	|- A = (algSc ` P )

Assertion

Ref	Expression
cply1coe0bi	|- ( ( R e. Ring /\ M e. B ) -> ( E. s e. K M = ( A ` s ) <-> A. n e. NN ( (coe1 ` M ) ` n ) = .0. ) )

Distinct variable groups:   n, K   R, n   A, n, s   B, n, s   K, s   n, M, s   R, s   .0. , s

Allowed substitution hints:   P( n, s)   .0. ( n)

Proof of Theorem cply1coe0bi

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . . . 8 |- ( ( R e. Ring /\ M e. B ) -> R e. Ring )
2	1	anim1i 592	. . . . . . 7 |- ( ( ( R e. Ring /\ M e. B ) /\ s e. K ) -> ( R e. Ring /\ s e. K ) )
3	2	adantr 481	. . . . . 6 |- ( ( ( ( R e. Ring /\ M e. B ) /\ s e. K ) /\ M = ( A ` s ) ) -> ( R e. Ring /\ s e. K ) )
4		cply1coe0.k	. . . . . . 7 |- K = ( Base ` R )
5		cply1coe0.0	. . . . . . 7 |- .0. = ( 0g ` R )
6		cply1coe0.p	. . . . . . 7 |- P = (Poly1 ` R )
7		cply1coe0.b	. . . . . . 7 |- B = ( Base ` P )
8		cply1coe0.a	. . . . . . 7 |- A = (algSc ` P )
9	4, 5, 6, 7, 8	cply1coe0 19669	. . . . . 6 |- ( ( R e. Ring /\ s e. K ) -> A. n e. NN ( (coe1 ` ( A ` s ) ) ` n ) = .0. )
10	3, 9	syl 17	. . . . 5 |- ( ( ( ( R e. Ring /\ M e. B ) /\ s e. K ) /\ M = ( A ` s ) ) -> A. n e. NN ( (coe1 ` ( A ` s ) ) ` n ) = .0. )
11		fveq2 6191	. . . . . . . . 9 |- ( M = ( A ` s ) -> (coe1 ` M ) = (coe1 ` ( A ` s ) ) )
12	11	fveq1d 6193	. . . . . . . 8 |- ( M = ( A ` s ) -> ( (coe1 ` M ) ` n ) = ( (coe1 ` ( A ` s ) ) ` n ) )
13	12	eqeq1d 2624	. . . . . . 7 |- ( M = ( A ` s ) -> ( ( (coe1 ` M ) ` n ) = .0. <-> ( (coe1 ` ( A ` s ) ) ` n ) = .0. ) )
14	13	ralbidv 2986	. . . . . 6 |- ( M = ( A ` s ) -> ( A. n e. NN ( (coe1 ` M ) ` n ) = .0. <-> A. n e. NN ( (coe1 ` ( A ` s ) ) ` n ) = .0. ) )
15	14	adantl 482	. . . . 5 |- ( ( ( ( R e. Ring /\ M e. B ) /\ s e. K ) /\ M = ( A ` s ) ) -> ( A. n e. NN ( (coe1 ` M ) ` n ) = .0. <-> A. n e. NN ( (coe1 ` ( A ` s ) ) ` n ) = .0. ) )
16	10, 15	mpbird 247	. . . 4 |- ( ( ( ( R e. Ring /\ M e. B ) /\ s e. K ) /\ M = ( A ` s ) ) -> A. n e. NN ( (coe1 ` M ) ` n ) = .0. )
17	16	ex 450	. . 3 |- ( ( ( R e. Ring /\ M e. B ) /\ s e. K ) -> ( M = ( A ` s ) -> A. n e. NN ( (coe1 ` M ) ` n ) = .0. ) )
18	17	rexlimdva 3031	. 2 |- ( ( R e. Ring /\ M e. B ) -> ( E. s e. K M = ( A ` s ) -> A. n e. NN ( (coe1 ` M ) ` n ) = .0. ) )
19		simpr 477	. . . . . 6 |- ( ( R e. Ring /\ M e. B ) -> M e. B )
20		0nn0 11307	. . . . . 6 |- 0 e. NN0
21		eqid 2622	. . . . . . 7 |- (coe1 ` M ) = (coe1 ` M )
22	21, 7, 6, 4	coe1fvalcl 19582	. . . . . 6 |- ( ( M e. B /\ 0 e. NN0 ) -> ( (coe1 ` M ) ` 0 ) e. K )
23	19, 20, 22	sylancl 694	. . . . 5 |- ( ( R e. Ring /\ M e. B ) -> ( (coe1 ` M ) ` 0 ) e. K )
24	23	adantr 481	. . . 4 |- ( ( ( R e. Ring /\ M e. B ) /\ A. n e. NN ( (coe1 ` M ) ` n ) = .0. ) -> ( (coe1 ` M ) ` 0 ) e. K )
25		fveq2 6191	. . . . . 6 |- ( s = ( (coe1 ` M ) ` 0 ) -> ( A ` s ) = ( A ` ( (coe1 ` M ) ` 0 ) ) )
26	25	eqeq2d 2632	. . . . 5 |- ( s = ( (coe1 ` M ) ` 0 ) -> ( M = ( A ` s ) <-> M = ( A ` ( (coe1 ` M ) ` 0 ) ) ) )
27	26	adantl 482	. . . 4 |- ( ( ( ( R e. Ring /\ M e. B ) /\ A. n e. NN ( (coe1 ` M ) ` n ) = .0. ) /\ s = ( (coe1 ` M ) ` 0 ) ) -> ( M = ( A ` s ) <-> M = ( A ` ( (coe1 ` M ) ` 0 ) ) ) )
28		eqid 2622	. . . . . . . . . 10 |- (Scalar ` P ) = (Scalar ` P )
29	6	ply1ring 19618	. . . . . . . . . 10 |- ( R e. Ring -> P e. Ring )
30	6	ply1lmod 19622	. . . . . . . . . 10 |- ( R e. Ring -> P e. LMod )
31		eqid 2622	. . . . . . . . . 10 |- ( Base ` (Scalar ` P ) ) = ( Base ` (Scalar ` P ) )
32	8, 28, 29, 30, 31, 7	asclf 19337	. . . . . . . . 9 |- ( R e. Ring -> A : ( Base ` (Scalar ` P ) ) --> B )
33	32	adantr 481	. . . . . . . 8 |- ( ( R e. Ring /\ M e. B ) -> A : ( Base ` (Scalar ` P ) ) --> B )
34		eqid 2622	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
35	21, 7, 6, 34	coe1fvalcl 19582	. . . . . . . . . 10 |- ( ( M e. B /\ 0 e. NN0 ) -> ( (coe1 ` M ) ` 0 ) e. ( Base ` R ) )
36	19, 20, 35	sylancl 694	. . . . . . . . 9 |- ( ( R e. Ring /\ M e. B ) -> ( (coe1 ` M ) ` 0 ) e. ( Base ` R ) )
37	6	ply1sca 19623	. . . . . . . . . . . 12 |- ( R e. Ring -> R = (Scalar ` P ) )
38	37	eqcomd 2628	. . . . . . . . . . 11 |- ( R e. Ring -> (Scalar ` P ) = R )
39	38	fveq2d 6195	. . . . . . . . . 10 |- ( R e. Ring -> ( Base ` (Scalar ` P ) ) = ( Base ` R ) )
40	39	adantr 481	. . . . . . . . 9 |- ( ( R e. Ring /\ M e. B ) -> ( Base ` (Scalar ` P ) ) = ( Base ` R ) )
41	36, 40	eleqtrrd 2704	. . . . . . . 8 |- ( ( R e. Ring /\ M e. B ) -> ( (coe1 ` M ) ` 0 ) e. ( Base ` (Scalar ` P ) ) )
42	33, 41	ffvelrnd 6360	. . . . . . 7 |- ( ( R e. Ring /\ M e. B ) -> ( A ` ( (coe1 ` M ) ` 0 ) ) e. B )
43	1, 19, 42	3jca 1242	. . . . . 6 |- ( ( R e. Ring /\ M e. B ) -> ( R e. Ring /\ M e. B /\ ( A ` ( (coe1 ` M ) ` 0 ) ) e. B ) )
44	43	adantr 481	. . . . 5 |- ( ( ( R e. Ring /\ M e. B ) /\ A. n e. NN ( (coe1 ` M ) ` n ) = .0. ) -> ( R e. Ring /\ M e. B /\ ( A ` ( (coe1 ` M ) ` 0 ) ) e. B ) )
45		simpr 477	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ M e. B ) /\ n e. NN ) /\ ( (coe1 ` M ) ` n ) = .0. ) -> ( (coe1 ` M ) ` n ) = .0. )
46	6, 8, 4, 5	coe1scl 19657	. . . . . . . . . . . . . . 15 |- ( ( R e. Ring /\ ( (coe1 ` M ) ` 0 ) e. K ) -> (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) = ( k e. NN0 |-> if ( k = 0 , ( (coe1 ` M ) ` 0 ) , .0. ) ) )
47	23, 46	syldan 487	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ M e. B ) -> (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) = ( k e. NN0 |-> if ( k = 0 , ( (coe1 ` M ) ` 0 ) , .0. ) ) )
48	47	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( R e. Ring /\ M e. B ) /\ n e. NN ) -> (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) = ( k e. NN0 |-> if ( k = 0 , ( (coe1 ` M ) ` 0 ) , .0. ) ) )
49		nnne0 11053	. . . . . . . . . . . . . . . . . 18 |- ( n e. NN -> n =/= 0 )
50	49	neneqd 2799	. . . . . . . . . . . . . . . . 17 |- ( n e. NN -> -. n = 0 )
51	50	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( R e. Ring /\ M e. B ) /\ n e. NN ) -> -. n = 0 )
52	51	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( R e. Ring /\ M e. B ) /\ n e. NN ) /\ k = n ) -> -. n = 0 )
53		eqeq1 2626	. . . . . . . . . . . . . . . . 17 |- ( k = n -> ( k = 0 <-> n = 0 ) )
54	53	notbid 308	. . . . . . . . . . . . . . . 16 |- ( k = n -> ( -. k = 0 <-> -. n = 0 ) )
55	54	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ( ( R e. Ring /\ M e. B ) /\ n e. NN ) /\ k = n ) -> ( -. k = 0 <-> -. n = 0 ) )
56	52, 55	mpbird 247	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. Ring /\ M e. B ) /\ n e. NN ) /\ k = n ) -> -. k = 0 )
57	56	iffalsed 4097	. . . . . . . . . . . . 13 |- ( ( ( ( R e. Ring /\ M e. B ) /\ n e. NN ) /\ k = n ) -> if ( k = 0 , ( (coe1 ` M ) ` 0 ) , .0. ) = .0. )
58		nnnn0 11299	. . . . . . . . . . . . . 14 |- ( n e. NN -> n e. NN0 )
59	58	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( R e. Ring /\ M e. B ) /\ n e. NN ) -> n e. NN0 )
60		fvex 6201	. . . . . . . . . . . . . . 15 |- ( 0g ` R ) e. _V
61	5, 60	eqeltri 2697	. . . . . . . . . . . . . 14 |- .0. e. _V
62	61	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( R e. Ring /\ M e. B ) /\ n e. NN ) -> .0. e. _V )
63	48, 57, 59, 62	fvmptd 6288	. . . . . . . . . . . 12 |- ( ( ( R e. Ring /\ M e. B ) /\ n e. NN ) -> ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) = .0. )
64	63	eqcomd 2628	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ M e. B ) /\ n e. NN ) -> .0. = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) )
65	64	adantr 481	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ M e. B ) /\ n e. NN ) /\ ( (coe1 ` M ) ` n ) = .0. ) -> .0. = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) )
66	45, 65	eqtrd 2656	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ M e. B ) /\ n e. NN ) /\ ( (coe1 ` M ) ` n ) = .0. ) -> ( (coe1 ` M ) ` n ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) )
67	66	ex 450	. . . . . . . 8 |- ( ( ( R e. Ring /\ M e. B ) /\ n e. NN ) -> ( ( (coe1 ` M ) ` n ) = .0. -> ( (coe1 ` M ) ` n ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) ) )
68	67	ralimdva 2962	. . . . . . 7 |- ( ( R e. Ring /\ M e. B ) -> ( A. n e. NN ( (coe1 ` M ) ` n ) = .0. -> A. n e. NN ( (coe1 ` M ) ` n ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) ) )
69	68	imp 445	. . . . . 6 |- ( ( ( R e. Ring /\ M e. B ) /\ A. n e. NN ( (coe1 ` M ) ` n ) = .0. ) -> A. n e. NN ( (coe1 ` M ) ` n ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) )
70	6, 8, 4	ply1sclid 19658	. . . . . . . 8 |- ( ( R e. Ring /\ ( (coe1 ` M ) ` 0 ) e. K ) -> ( (coe1 ` M ) ` 0 ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` 0 ) )
71	23, 70	syldan 487	. . . . . . 7 |- ( ( R e. Ring /\ M e. B ) -> ( (coe1 ` M ) ` 0 ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` 0 ) )
72	71	adantr 481	. . . . . 6 |- ( ( ( R e. Ring /\ M e. B ) /\ A. n e. NN ( (coe1 ` M ) ` n ) = .0. ) -> ( (coe1 ` M ) ` 0 ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` 0 ) )
73		df-n0 11293	. . . . . . . 8 |- NN0 = ( NN u. { 0 } )
74	73	raleqi 3142	. . . . . . 7 |- ( A. n e. NN0 ( (coe1 ` M ) ` n ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) <-> A. n e. ( NN u. { 0 } ) ( (coe1 ` M ) ` n ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) )
75		c0ex 10034	. . . . . . . 8 |- 0 e. _V
76		fveq2 6191	. . . . . . . . . 10 |- ( n = 0 -> ( (coe1 ` M ) ` n ) = ( (coe1 ` M ) ` 0 ) )
77		fveq2 6191	. . . . . . . . . 10 |- ( n = 0 -> ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` 0 ) )
78	76, 77	eqeq12d 2637	. . . . . . . . 9 |- ( n = 0 -> ( ( (coe1 ` M ) ` n ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) <-> ( (coe1 ` M ) ` 0 ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` 0 ) ) )
79	78	ralunsn 4422	. . . . . . . 8 |- ( 0 e. _V -> ( A. n e. ( NN u. { 0 } ) ( (coe1 ` M ) ` n ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) <-> ( A. n e. NN ( (coe1 ` M ) ` n ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) /\ ( (coe1 ` M ) ` 0 ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` 0 ) ) ) )
80	75, 79	mp1i 13	. . . . . . 7 |- ( ( ( R e. Ring /\ M e. B ) /\ A. n e. NN ( (coe1 ` M ) ` n ) = .0. ) -> ( A. n e. ( NN u. { 0 } ) ( (coe1 ` M ) ` n ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) <-> ( A. n e. NN ( (coe1 ` M ) ` n ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) /\ ( (coe1 ` M ) ` 0 ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` 0 ) ) ) )
81	74, 80	syl5bb 272	. . . . . 6 |- ( ( ( R e. Ring /\ M e. B ) /\ A. n e. NN ( (coe1 ` M ) ` n ) = .0. ) -> ( A. n e. NN0 ( (coe1 ` M ) ` n ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) <-> ( A. n e. NN ( (coe1 ` M ) ` n ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) /\ ( (coe1 ` M ) ` 0 ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` 0 ) ) ) )
82	69, 72, 81	mpbir2and 957	. . . . 5 |- ( ( ( R e. Ring /\ M e. B ) /\ A. n e. NN ( (coe1 ` M ) ` n ) = .0. ) -> A. n e. NN0 ( (coe1 ` M ) ` n ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) )
83		eqid 2622	. . . . . 6 |- (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) = (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) )
84	6, 7, 21, 83	eqcoe1ply1eq 19667	. . . . 5 |- ( ( R e. Ring /\ M e. B /\ ( A ` ( (coe1 ` M ) ` 0 ) ) e. B ) -> ( A. n e. NN0 ( (coe1 ` M ) ` n ) = ( (coe1 ` ( A ` ( (coe1 ` M ) ` 0 ) ) ) ` n ) -> M = ( A ` ( (coe1 ` M ) ` 0 ) ) ) )
85	44, 82, 84	sylc 65	. . . 4 |- ( ( ( R e. Ring /\ M e. B ) /\ A. n e. NN ( (coe1 ` M ) ` n ) = .0. ) -> M = ( A ` ( (coe1 ` M ) ` 0 ) ) )
86	24, 27, 85	rspcedvd 3317	. . 3 |- ( ( ( R e. Ring /\ M e. B ) /\ A. n e. NN ( (coe1 ` M ) ` n ) = .0. ) -> E. s e. K M = ( A ` s ) )
87	86	ex 450	. 2 |- ( ( R e. Ring /\ M e. B ) -> ( A. n e. NN ( (coe1 ` M ) ` n ) = .0. -> E. s e. K M = ( A ` s ) ) )
88	18, 87	impbid 202	1 |- ( ( R e. Ring /\ M e. B ) -> ( E. s e. K M = ( A ` s ) <-> A. n e. NN ( (coe1 ` M ) ` n ) = .0. ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   u. cun 3572   ifcif 4086   {csn 4177   |-> cmpt 4729   -->wf 5884   ` cfv 5888   0cc0 9936   NNcn 11020   NN0cn0 11292   Basecbs 15857  Scalarcsca 15944   0gc0g 16100   Ringcrg 18547  algSccascl 19311  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-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-subrg 18778  df-lmod 18865  df-lss 18933  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553

This theorem is referenced by:  cpmatel2  20518