Theorem rrxcph 22338

Description: Generalized Euclidean real spaces are pre-Hilbert spaces. (Contributed by Thierry Arnoux, 23-Jun-2019.) (Proof shortened by AV, 22-Jul-2019.)

Hypotheses

Ref	Expression
rrxval.r	|- H = (ℝ^ ` I )
rrxbase.b	|- B = ( Base ` H )

Assertion

Ref	Expression
rrxcph	|- ( I e. V -> H e. CPreHil )

Proof of Theorem rrxcph

Dummy variables f x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrxval.r	. . 3 |- H = (ℝ^ ` I )
2	1	rrxval 22333	. 2 |- ( I e. V -> H = (toCHil ` (RRfld freeLMod I ) ) )
3		eqid 2422	. . 3 |- (toCHil ` (RRfld freeLMod I ) ) = (toCHil ` (RRfld freeLMod I ) )
4		eqid 2422	. . 3 |- ( Base ` (RRfld freeLMod I ) ) = ( Base ` (RRfld freeLMod I ) )
5		eqid 2422	. . 3 |- (Scalar ` (RRfld freeLMod I ) ) = (Scalar ` (RRfld freeLMod I ) )
6		eqid 2422	. . . 4 |- (RRfld freeLMod I ) = (RRfld freeLMod I )
7		rebase 19161	. . . 4 |- RR = ( Base ` RRfld )
8		remulr 19166	. . . 4 |- x. = ( .r ` RRfld )
9		eqid 2422	. . . 4 |- ( .i ` (RRfld freeLMod I ) ) = ( .i ` (RRfld freeLMod I ) )
10		eqid 2422	. . . 4 |- ( 0g ` (RRfld freeLMod I ) ) = ( 0g ` (RRfld freeLMod I ) )
11		re0g 19167	. . . 4 |- 0 = ( 0g ` RRfld )
12		refldcj 19175	. . . 4 |- * = ( *r ` RRfld )
13		refld 19174	. . . . 5 |- RRfld e. Field
14	13	a1i 11	. . . 4 |- ( I e. V -> RRfld e. Field )
15		fconstmpt 4894	. . . . 5 |- ( I X. { 0 } ) = ( x e. I |-> 0 )
16	6, 7, 4	frlmbasf 19310	. . . . . . . 8 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> f : I --> RR )
17		ffn 5743	. . . . . . . 8 |- ( f : I --> RR -> f Fn I )
18	16, 17	syl 17	. . . . . . 7 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> f Fn I )
19	18	3adant3 1025	. . . . . 6 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> f Fn I )
20		simpl 458	. . . . . . . . . . . . . . . . 17 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> I e. V )
21	13	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> RRfld e. Field )
22		simpr 462	. . . . . . . . . . . . . . . . 17 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> f e. ( Base ` (RRfld freeLMod I ) ) )
23	6, 7, 8, 4, 9	frlmipval 19324	. . . . . . . . . . . . . . . . 17 |- ( ( ( I e. V /\ RRfld e. Field ) /\ ( f e. ( Base ` (RRfld freeLMod I ) ) /\ f e. ( Base ` (RRfld freeLMod I ) ) ) ) -> ( f ( .i ` (RRfld freeLMod I ) ) f ) = (RRfld gsumg ( f oF x. f ) ) )
24	20, 21, 22, 22, 23	syl22anc 1265	. . . . . . . . . . . . . . . 16 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f ( .i ` (RRfld freeLMod I ) ) f ) = (RRfld gsumg ( f oF x. f ) ) )
25		ovex 6330	. . . . . . . . . . . . . . . . . . . 20 |- ( ( f ` x ) x. ( f ` x ) ) e. _V
26	25	a1i 11	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> ( ( f ` x ) x. ( f ` x ) ) e. _V )
27		inidm 3671	. . . . . . . . . . . . . . . . . . . 20 |- ( I i^i I ) = I
28		eqidd 2423	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> ( f ` x ) = ( f ` x ) )
29	18, 18, 20, 20, 27, 28, 28	offval 6549	. . . . . . . . . . . . . . . . . . 19 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f oF x. f ) = ( x e. I |-> ( ( f ` x ) x. ( f ` x ) ) ) )
30	18, 18, 20, 20, 27, 28, 28	ofval 6551	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> ( ( f oF x. f ) ` x ) = ( ( f ` x ) x. ( f ` x ) ) )
31	16	ffvelrnda 6034	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> ( f ` x ) e. RR )
32	31, 31	remulcld 9672	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> ( ( f ` x ) x. ( f ` x ) ) e. RR )
33	30, 32	eqeltrd 2510	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> ( ( f oF x. f ) ` x ) e. RR )
34	26, 29, 33	fmpt2d 6065	. . . . . . . . . . . . . . . . . 18 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f oF x. f ) : I --> RR )
35		ovex 6330	. . . . . . . . . . . . . . . . . . . 20 |- ( f oF x. f ) e. _V
36	35	a1i 11	. . . . . . . . . . . . . . . . . . 19 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f oF x. f ) e. _V )
37		ffun 5745	. . . . . . . . . . . . . . . . . . . 20 |- ( ( f oF x. f ) : I --> RR -> Fun ( f oF x. f ) )
38	34, 37	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> Fun ( f oF x. f ) )
39	6, 11, 4	frlmbasfsupp 19308	. . . . . . . . . . . . . . . . . . 19 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> f finSupp 0 )
40		0red 9645	. . . . . . . . . . . . . . . . . . . 20 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> 0 e. RR )
41		simpr 462	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. RR ) -> x e. RR )
42	41	recnd 9670	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. RR ) -> x e. CC )
43	42	mul02d 9832	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. RR ) -> ( 0 x. x ) = 0 )
44	20, 40, 16, 16, 43	suppofss1d 6960	. . . . . . . . . . . . . . . . . . 19 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( ( f oF x. f ) supp 0 ) C_ ( f supp 0 ) )
45		fsuppsssupp 7902	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( f oF x. f ) e. _V /\ Fun ( f oF x. f ) ) /\ ( f finSupp 0 /\ ( ( f oF x. f ) supp 0 ) C_ ( f supp 0 ) ) ) -> ( f oF x. f ) finSupp 0 )
46	36, 38, 39, 44, 45	syl22anc 1265	. . . . . . . . . . . . . . . . . 18 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f oF x. f ) finSupp 0 )
47		regsumsupp 19177	. . . . . . . . . . . . . . . . . 18 |- ( ( ( f oF x. f ) : I --> RR /\ ( f oF x. f ) finSupp 0 /\ I e. V ) -> (RRfld gsumg ( f oF x. f ) ) = sum_ x e. ( ( f oF x. f ) supp 0 ) ( ( f oF x. f ) ` x ) )
48	34, 46, 20, 47	syl3anc 1264	. . . . . . . . . . . . . . . . 17 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> (RRfld gsumg ( f oF x. f ) ) = sum_ x e. ( ( f oF x. f ) supp 0 ) ( ( f oF x. f ) ` x ) )
49		suppssdm 6935	. . . . . . . . . . . . . . . . . . . . . 22 |- ( f supp 0 ) C_ dom f
50		fdm 5747	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( f : I --> RR -> dom f = I )
51	16, 50	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> dom f = I )
52	49, 51	syl5sseq 3512	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f supp 0 ) C_ I )
53	44, 52	sstrd 3474	. . . . . . . . . . . . . . . . . . . 20 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( ( f oF x. f ) supp 0 ) C_ I )
54	53	sselda 3464	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. ( ( f oF x. f ) supp 0 ) ) -> x e. I )
55	54, 30	syldan 472	. . . . . . . . . . . . . . . . . 18 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. ( ( f oF x. f ) supp 0 ) ) -> ( ( f oF x. f ) ` x ) = ( ( f ` x ) x. ( f ` x ) ) )
56	55	sumeq2dv 13757	. . . . . . . . . . . . . . . . 17 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> sum_ x e. ( ( f oF x. f ) supp 0 ) ( ( f oF x. f ) ` x ) = sum_ x e. ( ( f oF x. f ) supp 0 ) ( ( f ` x ) x. ( f ` x ) ) )
57	48, 56	eqtrd 2463	. . . . . . . . . . . . . . . 16 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> (RRfld gsumg ( f oF x. f ) ) = sum_ x e. ( ( f oF x. f ) supp 0 ) ( ( f ` x ) x. ( f ` x ) ) )
58	24, 57	eqtrd 2463	. . . . . . . . . . . . . . 15 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f ( .i ` (RRfld freeLMod I ) ) f ) = sum_ x e. ( ( f oF x. f ) supp 0 ) ( ( f ` x ) x. ( f ` x ) ) )
59	58	3adant3 1025	. . . . . . . . . . . . . 14 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> ( f ( .i ` (RRfld freeLMod I ) ) f ) = sum_ x e. ( ( f oF x. f ) supp 0 ) ( ( f ` x ) x. ( f ` x ) ) )
60		simp3 1007	. . . . . . . . . . . . . 14 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 )
61	59, 60	eqtr3d 2465	. . . . . . . . . . . . 13 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> sum_ x e. ( ( f oF x. f ) supp 0 ) ( ( f ` x ) x. ( f ` x ) ) = 0 )
62	39	fsuppimpd 7893	. . . . . . . . . . . . . . . 16 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f supp 0 ) e. Fin )
63		ssfi 7795	. . . . . . . . . . . . . . . 16 |- ( ( ( f supp 0 ) e. Fin /\ ( ( f oF x. f ) supp 0 ) C_ ( f supp 0 ) ) -> ( ( f oF x. f ) supp 0 ) e. Fin )
64	62, 44, 63	syl2anc 665	. . . . . . . . . . . . . . 15 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( ( f oF x. f ) supp 0 ) e. Fin )
65	54, 32	syldan 472	. . . . . . . . . . . . . . 15 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. ( ( f oF x. f ) supp 0 ) ) -> ( ( f ` x ) x. ( f ` x ) ) e. RR )
66	31	msqge0d 10183	. . . . . . . . . . . . . . . 16 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> 0 <_ ( ( f ` x ) x. ( f ` x ) ) )
67	54, 66	syldan 472	. . . . . . . . . . . . . . 15 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. ( ( f oF x. f ) supp 0 ) ) -> 0 <_ ( ( f ` x ) x. ( f ` x ) ) )
68	64, 65, 67	fsum00 13846	. . . . . . . . . . . . . 14 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( sum_ x e. ( ( f oF x. f ) supp 0 ) ( ( f ` x ) x. ( f ` x ) ) = 0 <-> A. x e. ( ( f oF x. f ) supp 0 ) ( ( f ` x ) x. ( f ` x ) ) = 0 ) )
69	68	3adant3 1025	. . . . . . . . . . . . 13 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> ( sum_ x e. ( ( f oF x. f ) supp 0 ) ( ( f ` x ) x. ( f ` x ) ) = 0 <-> A. x e. ( ( f oF x. f ) supp 0 ) ( ( f ` x ) x. ( f ` x ) ) = 0 ) )
70	61, 69	mpbid 213	. . . . . . . . . . . 12 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> A. x e. ( ( f oF x. f ) supp 0 ) ( ( f ` x ) x. ( f ` x ) ) = 0 )
71	70	r19.21bi 2794	. . . . . . . . . . 11 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. ( ( f oF x. f ) supp 0 ) ) -> ( ( f ` x ) x. ( f ` x ) ) = 0 )
72	71	adantlr 719	. . . . . . . . . 10 |- ( ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) /\ x e. ( ( f oF x. f ) supp 0 ) ) -> ( ( f ` x ) x. ( f ` x ) ) = 0 )
73	31	3adantl3 1163	. . . . . . . . . . . . 13 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> ( f ` x ) e. RR )
74	73	recnd 9670	. . . . . . . . . . . 12 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> ( f ` x ) e. CC )
75	74, 74	mul0ord 10263	. . . . . . . . . . 11 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> ( ( ( f ` x ) x. ( f ` x ) ) = 0 <-> ( ( f ` x ) = 0 \/ ( f ` x ) = 0 ) ) )
76	75	adantr 466	. . . . . . . . . 10 |- ( ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) /\ x e. ( ( f oF x. f ) supp 0 ) ) -> ( ( ( f ` x ) x. ( f ` x ) ) = 0 <-> ( ( f ` x ) = 0 \/ ( f ` x ) = 0 ) ) )
77	72, 76	mpbid 213	. . . . . . . . 9 |- ( ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) /\ x e. ( ( f oF x. f ) supp 0 ) ) -> ( ( f ` x ) = 0 \/ ( f ` x ) = 0 ) )
78		oridm 516	. . . . . . . . 9 |- ( ( ( f ` x ) = 0 \/ ( f ` x ) = 0 ) <-> ( f ` x ) = 0 )
79	77, 78	sylib 199	. . . . . . . 8 |- ( ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) /\ x e. ( ( f oF x. f ) supp 0 ) ) -> ( f ` x ) = 0 )
80	34	3adant3 1025	. . . . . . . . . . 11 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> ( f oF x. f ) : I --> RR )
81	80	adantr 466	. . . . . . . . . 10 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> ( f oF x. f ) : I --> RR )
82		ssid 3483	. . . . . . . . . . 11 |- ( ( f oF x. f ) supp 0 ) C_ ( ( f oF x. f ) supp 0 )
83	82	a1i 11	. . . . . . . . . 10 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> ( ( f oF x. f ) supp 0 ) C_ ( ( f oF x. f ) supp 0 ) )
84		simpl1 1008	. . . . . . . . . 10 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> I e. V )
85		0red 9645	. . . . . . . . . 10 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> 0 e. RR )
86	81, 83, 84, 85	suppssr 6954	. . . . . . . . 9 |- ( ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) /\ x e. ( I \ ( ( f oF x. f ) supp 0 ) ) ) -> ( ( f oF x. f ) ` x ) = 0 )
87	30	3adantl3 1163	. . . . . . . . . . . . 13 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> ( ( f oF x. f ) ` x ) = ( ( f ` x ) x. ( f ` x ) ) )
88	87	eqeq1d 2424	. . . . . . . . . . . 12 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> ( ( ( f oF x. f ) ` x ) = 0 <-> ( ( f ` x ) x. ( f ` x ) ) = 0 ) )
89	88, 75	bitrd 256	. . . . . . . . . . 11 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> ( ( ( f oF x. f ) ` x ) = 0 <-> ( ( f ` x ) = 0 \/ ( f ` x ) = 0 ) ) )
90	89, 78	syl6bb 264	. . . . . . . . . 10 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> ( ( ( f oF x. f ) ` x ) = 0 <-> ( f ` x ) = 0 ) )
91	90	biimpa 486	. . . . . . . . 9 |- ( ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) /\ ( ( f oF x. f ) ` x ) = 0 ) -> ( f ` x ) = 0 )
92	86, 91	syldan 472	. . . . . . . 8 |- ( ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) /\ x e. ( I \ ( ( f oF x. f ) supp 0 ) ) ) -> ( f ` x ) = 0 )
93		undif 3876	. . . . . . . . . . . . 13 |- ( ( ( f oF x. f ) supp 0 ) C_ I <-> ( ( ( f oF x. f ) supp 0 ) u. ( I \ ( ( f oF x. f ) supp 0 ) ) ) = I )
94	53, 93	sylib 199	. . . . . . . . . . . 12 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( ( ( f oF x. f ) supp 0 ) u. ( I \ ( ( f oF x. f ) supp 0 ) ) ) = I )
95	94	eleq2d 2492	. . . . . . . . . . 11 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( x e. ( ( ( f oF x. f ) supp 0 ) u. ( I \ ( ( f oF x. f ) supp 0 ) ) ) <-> x e. I ) )
96	95	3adant3 1025	. . . . . . . . . 10 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> ( x e. ( ( ( f oF x. f ) supp 0 ) u. ( I \ ( ( f oF x. f ) supp 0 ) ) ) <-> x e. I ) )
97	96	biimpar 487	. . . . . . . . 9 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> x e. ( ( ( f oF x. f ) supp 0 ) u. ( I \ ( ( f oF x. f ) supp 0 ) ) ) )
98		elun 3606	. . . . . . . . 9 |- ( x e. ( ( ( f oF x. f ) supp 0 ) u. ( I \ ( ( f oF x. f ) supp 0 ) ) ) <-> ( x e. ( ( f oF x. f ) supp 0 ) \/ x e. ( I \ ( ( f oF x. f ) supp 0 ) ) ) )
99	97, 98	sylib 199	. . . . . . . 8 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> ( x e. ( ( f oF x. f ) supp 0 ) \/ x e. ( I \ ( ( f oF x. f ) supp 0 ) ) ) )
100	79, 92, 99	mpjaodan 793	. . . . . . 7 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> ( f ` x ) = 0 )
101	100	ralrimiva 2839	. . . . . 6 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> A. x e. I ( f ` x ) = 0 )
102		fconstfv 6138	. . . . . . . 8 |- ( f : I --> { 0 } <-> ( f Fn I /\ A. x e. I ( f ` x ) = 0 ) )
103	102	biimpri 209	. . . . . . 7 |- ( ( f Fn I /\ A. x e. I ( f ` x ) = 0 ) -> f : I --> { 0 } )
104		c0ex 9638	. . . . . . . 8 |- 0 e. _V
105	104	fconst2 6133	. . . . . . 7 |- ( f : I --> { 0 } <-> f = ( I X. { 0 } ) )
106	103, 105	sylib 199	. . . . . 6 |- ( ( f Fn I /\ A. x e. I ( f ` x ) = 0 ) -> f = ( I X. { 0 } ) )
107	19, 101, 106	syl2anc 665	. . . . 5 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> f = ( I X. { 0 } ) )
108		isfld 17972	. . . . . . . . . . 11 |- (RRfld e. Field <-> (RRfld e. DivRing /\ RRfld e. CRing ) )
109	13, 108	mpbi 211	. . . . . . . . . 10 |- (RRfld e. DivRing /\ RRfld e. CRing )
110	109	simpli 459	. . . . . . . . 9 |- RRfld e. DivRing
111		drngring 17970	. . . . . . . . 9 |- (RRfld e. DivRing -> RRfld e. Ring )
112	110, 111	ax-mp 5	. . . . . . . 8 |- RRfld e. Ring
113	6, 11	frlm0 19304	. . . . . . . 8 |- ( (RRfld e. Ring /\ I e. V ) -> ( I X. { 0 } ) = ( 0g ` (RRfld freeLMod I ) ) )
114	112, 113	mpan 674	. . . . . . 7 |- ( I e. V -> ( I X. { 0 } ) = ( 0g ` (RRfld freeLMod I ) ) )
115	15, 114	syl5reqr 2478	. . . . . 6 |- ( I e. V -> ( 0g ` (RRfld freeLMod I ) ) = ( x e. I |-> 0 ) )
116	115	3ad2ant1 1026	. . . . 5 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> ( 0g ` (RRfld freeLMod I ) ) = ( x e. I |-> 0 ) )
117	15, 107, 116	3eqtr4a 2489	. . . 4 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> f = ( 0g ` (RRfld freeLMod I ) ) )
118		cjre 13191	. . . . 5 |- ( x e. RR -> ( * ` x ) = x )
119	118	adantl 467	. . . 4 |- ( ( I e. V /\ x e. RR ) -> ( * ` x ) = x )
120		id 23	. . . 4 |- ( I e. V -> I e. V )
121	6, 7, 8, 4, 9, 10, 11, 12, 14, 117, 119, 120	frlmphl 19326	. . 3 |- ( I e. V -> (RRfld freeLMod I ) e. PreHil )
122		df-refld 19160	. . . 4 |- RRfld = (ℂfld ↾s RR )
123	6	frlmsca 19303	. . . . 5 |- ( (RRfld e. Field /\ I e. V ) -> RRfld = (Scalar ` (RRfld freeLMod I ) ) )
124	13, 123	mpan 674	. . . 4 |- ( I e. V -> RRfld = (Scalar ` (RRfld freeLMod I ) ) )
125	122, 124	syl5reqr 2478	. . 3 |- ( I e. V -> (Scalar ` (RRfld freeLMod I ) ) = (ℂfld ↾s RR ) )
126		simpr1 1011	. . . 4 |- ( ( I e. V /\ ( f e. RR /\ f e. RR /\ 0 <_ f ) ) -> f e. RR )
127		simpr3 1013	. . . 4 |- ( ( I e. V /\ ( f e. RR /\ f e. RR /\ 0 <_ f ) ) -> 0 <_ f )
128	126, 127	resqrtcld 13468	. . 3 |- ( ( I e. V /\ ( f e. RR /\ f e. RR /\ 0 <_ f ) ) -> ( sqr ` f ) e. RR )
129	64, 65, 67	fsumge0 13843	. . . . 5 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> 0 <_ sum_ x e. ( ( f oF x. f ) supp 0 ) ( ( f ` x ) x. ( f ` x ) ) )
130	129, 57	breqtrrd 4447	. . . 4 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> 0 <_ (RRfld gsumg ( f oF x. f ) ) )
131	130, 24	breqtrrd 4447	. . 3 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> 0 <_ ( f ( .i ` (RRfld freeLMod I ) ) f ) )
132	3, 4, 5, 121, 125, 9, 128, 131	tchcph 22198	. 2 |- ( I e. V -> (toCHil ` (RRfld freeLMod I ) ) e. CPreHil )
133	2, 132	eqeltrd 2510	1 |- ( I e. V -> H e. CPreHil )

Syntax hints:   -> wi 4   <-> wb 187   \/ wo 369   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1868   A.wral 2775   _Vcvv 3081   \ cdif 3433   u. cun 3434   C_ wss 3436   {csn 3996   class class class wbr 4420   |-> cmpt 4479   X. cxp 4848   dom cdm 4850   Fun wfun 5592   Fn wfn 5593   -->wf 5594   ` cfv 5598  (class class class)co 6302   oFcof 6540   supp csupp 6922   Fincfn 7574   finSupp cfsupp 7886   RRcr 9539   0cc0 9540   x. cmul 9545   <_ cle 9677   *ccj 13148   sum_csu 13740   Basecbs 15109   ↾_s cress 15110  Scalarcsca 15181   .icip 15183   0gc0g 15326   gsum_g cgsu 15327   Ringcrg 17768   CRingccrg 17769   DivRingcdr 17963  Fieldcfield 17964  ℂ_fldccnfld 18958  RRfldcrefld 19159   freeLMod cfrlm 19296   CPreHilccph 22131  toCHilctch 22132  ℝ^crrx 22329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619  ax-mulf 9620

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-tpos 6978  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-q 11266  df-rp 11304  df-xneg 11410  df-xadd 11411  df-xmul 11412  df-ico 11642  df-fz 11786  df-fzo 11917  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-sum 13741  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-starv 15193  df-sca 15194  df-vsca 15195  df-ip 15196  df-tset 15197  df-ple 15198  df-ds 15200  df-unif 15201  df-hom 15202  df-cco 15203  df-rest 15309  df-topn 15310  df-0g 15328  df-gsum 15329  df-topgen 15330  df-prds 15334  df-pws 15336  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-mhm 16570  df-submnd 16571  df-grp 16661  df-minusg 16662  df-sbg 16663  df-subg 16802  df-ghm 16869  df-cntz 16959  df-cmn 17420  df-abl 17421  df-mgp 17712  df-ur 17724  df-ring 17770  df-cring 17771  df-oppr 17839  df-dvdsr 17857  df-unit 17858  df-invr 17888  df-dvr 17899  df-rnghom 17931  df-drng 17965  df-field 17966  df-subrg 17994  df-abv 18033  df-staf 18061  df-srng 18062  df-lmod 18081  df-lss 18144  df-lmhm 18233  df-lvec 18314  df-sra 18383  df-rgmod 18384  df-psmet 18950  df-xmet 18951  df-met 18952  df-bl 18953  df-mopn 18954  df-cnfld 18959  df-refld 19160  df-phl 19180  df-dsmm 19282  df-frlm 19297  df-top 19908  df-bases 19909  df-topon 19910  df-topsp 19911  df-xms 21322  df-ms 21323  df-nm 21584  df-ngp 21585  df-tng 21586  df-nrg 21587  df-nlm 21588  df-clm 22081  df-cph 22133  df-tch 22134  df-rrx 22331

This theorem is referenced by: (None)