Theorem rrxcph 21587

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 21582	. 2 |- ( I e. V -> H = (toCHil ` (RRfld freeLMod I ) ) )
3		eqid 2467	. . 3 |- (toCHil ` (RRfld freeLMod I ) ) = (toCHil ` (RRfld freeLMod I ) )
4		eqid 2467	. . 3 |- ( Base ` (RRfld freeLMod I ) ) = ( Base ` (RRfld freeLMod I ) )
5		eqid 2467	. . 3 |- (Scalar ` (RRfld freeLMod I ) ) = (Scalar ` (RRfld freeLMod I ) )
6		eqid 2467	. . . 4 |- (RRfld freeLMod I ) = (RRfld freeLMod I )
7		rebase 18437	. . . 4 |- RR = ( Base ` RRfld )
8		remulr 18442	. . . 4 |- x. = ( .r ` RRfld )
9		eqid 2467	. . . 4 |- ( .i ` (RRfld freeLMod I ) ) = ( .i ` (RRfld freeLMod I ) )
10		eqid 2467	. . . 4 |- ( 0g ` (RRfld freeLMod I ) ) = ( 0g ` (RRfld freeLMod I ) )
11		re0g 18443	. . . 4 |- 0 = ( 0g ` RRfld )
12		refldcj 18451	. . . 4 |- * = ( *r ` RRfld )
13		refld 18450	. . . . 5 |- RRfld e. Field
14	13	a1i 11	. . . 4 |- ( I e. V -> RRfld e. Field )
15		fconstmpt 5043	. . . . 5 |- ( I X. { 0 } ) = ( x e. I |-> 0 )
16	6, 7, 4	frlmbasf 18589	. . . . . . . 8 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> f : I --> RR )
17		ffn 5731	. . . . . . . 8 |- ( f : I --> RR -> f Fn I )
18	16, 17	syl 16	. . . . . . 7 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> f Fn I )
19	18	3adant3 1016	. . . . . 6 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> f Fn I )
20		simpl 457	. . . . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . . . 17 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> f e. ( Base ` (RRfld freeLMod I ) ) )
23	6, 7, 8, 4, 9	frlmipval 18605	. . . . . . . . . . . . . . . . 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 1229	. . . . . . . . . . . . . . . 16 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f ( .i ` (RRfld freeLMod I ) ) f ) = (RRfld gsumg ( f oF x. f ) ) )
25		ovex 6309	. . . . . . . . . . . . . . . . . . . 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 3707	. . . . . . . . . . . . . . . . . . . 20 |- ( I i^i I ) = I
28		eqidd 2468	. . . . . . . . . . . . . . . . . . . 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 6531	. . . . . . . . . . . . . . . . . . 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 6533	. . . . . . . . . . . . . . . . . . . 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 6021	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> ( f ` x ) e. RR )
32	31, 31	remulcld 9624	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> ( ( f ` x ) x. ( f ` x ) ) e. RR )
33	30, 32	eqeltrd 2555	. . . . . . . . . . . . . . . . . . 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 6051	. . . . . . . . . . . . . . . . . 18 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f oF x. f ) : I --> RR )
35		ovex 6309	. . . . . . . . . . . . . . . . . . . 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 5733	. . . . . . . . . . . . . . . . . . . 20 |- ( ( f oF x. f ) : I --> RR -> Fun ( f oF x. f ) )
38	34, 37	syl 16	. . . . . . . . . . . . . . . . . . 19 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> Fun ( f oF x. f ) )
39	6, 11, 4	frlmbasfsupp 18586	. . . . . . . . . . . . . . . . . . 19 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> f finSupp 0 )
40		0red 9597	. . . . . . . . . . . . . . . . . . . 20 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> 0 e. RR )
41		simpr 461	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. RR ) -> x e. RR )
42	41	recnd 9622	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. RR ) -> x e. CC )
43	42	mul02d 9777	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. RR ) -> ( 0 x. x ) = 0 )
44	20, 40, 16, 16, 43	suppofss1d 6937	. . . . . . . . . . . . . . . . . . 19 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( ( f oF x. f ) supp 0 ) C_ ( f supp 0 ) )
45		fsuppsssupp 7845	. . . . . . . . . . . . . . . . . . 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 1229	. . . . . . . . . . . . . . . . . 18 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f oF x. f ) finSupp 0 )
47		regsumsupp 18453	. . . . . . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . . . . 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 6914	. . . . . . . . . . . . . . . . . . . . . 22 |- ( f supp 0 ) C_ dom f
50		fdm 5735	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( f : I --> RR -> dom f = I )
51	16, 50	syl 16	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> dom f = I )
52	49, 51	syl5sseq 3552	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f supp 0 ) C_ I )
53	44, 52	sstrd 3514	. . . . . . . . . . . . . . . . . . . 20 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( ( f oF x. f ) supp 0 ) C_ I )
54	53	sselda 3504	. . . . . . . . . . . . . . . . . . 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 470	. . . . . . . . . . . . . . . . . 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 13488	. . . . . . . . . . . . . . . . 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 2508	. . . . . . . . . . . . . . . 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 2508	. . . . . . . . . . . . . . 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 1016	. . . . . . . . . . . . . 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 998	. . . . . . . . . . . . . 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 2510	. . . . . . . . . . . . 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 7836	. . . . . . . . . . . . . . . 16 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f supp 0 ) e. Fin )
63		ssfi 7740	. . . . . . . . . . . . . . . 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 661	. . . . . . . . . . . . . . 15 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( ( f oF x. f ) supp 0 ) e. Fin )
65	54, 32	syldan 470	. . . . . . . . . . . . . . 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 10121	. . . . . . . . . . . . . . . 16 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> 0 <_ ( ( f ` x ) x. ( f ` x ) ) )
67	54, 66	syldan 470	. . . . . . . . . . . . . . 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 13575	. . . . . . . . . . . . . 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 1016	. . . . . . . . . . . . 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 210	. . . . . . . . . . . 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 2833	. . . . . . . . . . 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 714	. . . . . . . . . 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 1154	. . . . . . . . . . . . 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 9622	. . . . . . . . . . . 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 10199	. . . . . . . . . . 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 465	. . . . . . . . . 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 210	. . . . . . . . 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 514	. . . . . . . . 9 |- ( ( ( f ` x ) = 0 \/ ( f ` x ) = 0 ) <-> ( f ` x ) = 0 )
79	77, 78	sylib 196	. . . . . . . 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 1016	. . . . . . . . . . 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 465	. . . . . . . . . 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 3523	. . . . . . . . . . 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 999	. . . . . . . . . 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 9597	. . . . . . . . . 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 6931	. . . . . . . . 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 1154	. . . . . . . . . . . . 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 2469	. . . . . . . . . . . 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 253	. . . . . . . . . . 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 261	. . . . . . . . . 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 484	. . . . . . . . 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 470	. . . . . . . 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 3907	. . . . . . . . . . . . 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 196	. . . . . . . . . . . 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 2537	. . . . . . . . . . 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 1016	. . . . . . . . . 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 485	. . . . . . . . 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 3645	. . . . . . . . 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 196	. . . . . . . 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 784	. . . . . . 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 2878	. . . . . 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 6123	. . . . . . . 8 |- ( f : I --> { 0 } <-> ( f Fn I /\ A. x e. I ( f ` x ) = 0 ) )
103	102	biimpri 206	. . . . . . 7 |- ( ( f Fn I /\ A. x e. I ( f ` x ) = 0 ) -> f : I --> { 0 } )
104		c0ex 9590	. . . . . . . 8 |- 0 e. _V
105	104	fconst2 6117	. . . . . . 7 |- ( f : I --> { 0 } <-> f = ( I X. { 0 } ) )
106	103, 105	sylib 196	. . . . . 6 |- ( ( f Fn I /\ A. x e. I ( f ` x ) = 0 ) -> f = ( I X. { 0 } ) )
107	19, 101, 106	syl2anc 661	. . . . 5 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> f = ( I X. { 0 } ) )
108		isfld 17205	. . . . . . . . . . 11 |- (RRfld e. Field <-> (RRfld e. DivRing /\ RRfld e. CRing ) )
109	13, 108	mpbi 208	. . . . . . . . . 10 |- (RRfld e. DivRing /\ RRfld e. CRing )
110	109	simpli 458	. . . . . . . . 9 |- RRfld e. DivRing
111		drngrng 17203	. . . . . . . . 9 |- (RRfld e. DivRing -> RRfld e. Ring )
112	110, 111	ax-mp 5	. . . . . . . 8 |- RRfld e. Ring
113	6, 11	frlm0 18580	. . . . . . . 8 |- ( (RRfld e. Ring /\ I e. V ) -> ( I X. { 0 } ) = ( 0g ` (RRfld freeLMod I ) ) )
114	112, 113	mpan 670	. . . . . . 7 |- ( I e. V -> ( I X. { 0 } ) = ( 0g ` (RRfld freeLMod I ) ) )
115	15, 114	syl5reqr 2523	. . . . . 6 |- ( I e. V -> ( 0g ` (RRfld freeLMod I ) ) = ( x e. I |-> 0 ) )
116	115	3ad2ant1 1017	. . . . 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 2534	. . . 4 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> f = ( 0g ` (RRfld freeLMod I ) ) )
118		cjre 12935	. . . . 5 |- ( x e. RR -> ( * ` x ) = x )
119	118	adantl 466	. . . 4 |- ( ( I e. V /\ x e. RR ) -> ( * ` x ) = x )
120		id 22	. . . 4 |- ( I e. V -> I e. V )
121	6, 7, 8, 4, 9, 10, 11, 12, 14, 117, 119, 120	frlmphl 18607	. . 3 |- ( I e. V -> (RRfld freeLMod I ) e. PreHil )
122		df-refld 18436	. . . 4 |- RRfld = (ℂfld ↾s RR )
123	6	frlmsca 18579	. . . . 5 |- ( (RRfld e. Field /\ I e. V ) -> RRfld = (Scalar ` (RRfld freeLMod I ) ) )
124	13, 123	mpan 670	. . . 4 |- ( I e. V -> RRfld = (Scalar ` (RRfld freeLMod I ) ) )
125	122, 124	syl5reqr 2523	. . 3 |- ( I e. V -> (Scalar ` (RRfld freeLMod I ) ) = (ℂfld ↾s RR ) )
126		simpr1 1002	. . . 4 |- ( ( I e. V /\ ( f e. RR /\ f e. RR /\ 0 <_ f ) ) -> f e. RR )
127		simpr3 1004	. . . 4 |- ( ( I e. V /\ ( f e. RR /\ f e. RR /\ 0 <_ f ) ) -> 0 <_ f )
128	126, 127	resqrtcld 13212	. . 3 |- ( ( I e. V /\ ( f e. RR /\ f e. RR /\ 0 <_ f ) ) -> ( sqr ` f ) e. RR )
129	64, 65, 67	fsumge0 13572	. . . . 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 4473	. . . 4 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> 0 <_ (RRfld gsumg ( f oF x. f ) ) )
131	130, 24	breqtrrd 4473	. . 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 21443	. 2 |- ( I e. V -> (toCHil ` (RRfld freeLMod I ) ) e. CPreHil )
133	2, 132	eqeltrd 2555	1 |- ( I e. V -> H e. CPreHil )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   A.wral 2814   _Vcvv 3113   \ cdif 3473   u. cun 3474   C_ wss 3476   {csn 4027   class class class wbr 4447   |-> cmpt 4505   X. cxp 4997   dom cdm 4999   Fun wfun 5582   Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   oFcof 6522   supp csupp 6901   Fincfn 7516   finSupp cfsupp 7829   RRcr 9491   0cc0 9492   x. cmul 9497   <_ cle 9629   *ccj 12892   sum_csu 13471   Basecbs 14490   ↾_s cress 14491  Scalarcsca 14558   .icip 14560   0gc0g 14695   gsum_g cgsu 14696   Ringcrg 17000   CRingccrg 17001   DivRingcdr 17196  Fieldcfield 17197  ℂ_fldccnfld 18219  RRfldcrefld 18435   freeLMod cfrlm 18572   CPreHilccph 21376  toCHilctch 21377  ℝ^crrx 21578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ico 11535  df-fz 11673  df-fzo 11793  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-prds 14703  df-pws 14705  df-mnd 15732  df-mhm 15786  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-subg 16003  df-ghm 16070  df-cntz 16160  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-cring 17003  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-dvr 17133  df-rnghom 17165  df-drng 17198  df-field 17199  df-subrg 17227  df-abv 17266  df-staf 17294  df-srng 17295  df-lmod 17314  df-lss 17379  df-lmhm 17468  df-lvec 17549  df-sra 17618  df-rgmod 17619  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-cnfld 18220  df-refld 18436  df-phl 18456  df-dsmm 18558  df-frlm 18573  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-xms 20586  df-ms 20587  df-nm 20866  df-ngp 20867  df-tng 20868  df-nrg 20869  df-nlm 20870  df-clm 21326  df-cph 21378  df-tch 21379  df-rrx 21580

This theorem is referenced by: (None)