Theorem rrxcph 21990

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 21985	. 2 |- ( I e. V -> H = (toCHil ` (RRfld freeLMod I ) ) )
3		eqid 2454	. . 3 |- (toCHil ` (RRfld freeLMod I ) ) = (toCHil ` (RRfld freeLMod I ) )
4		eqid 2454	. . 3 |- ( Base ` (RRfld freeLMod I ) ) = ( Base ` (RRfld freeLMod I ) )
5		eqid 2454	. . 3 |- (Scalar ` (RRfld freeLMod I ) ) = (Scalar ` (RRfld freeLMod I ) )
6		eqid 2454	. . . 4 |- (RRfld freeLMod I ) = (RRfld freeLMod I )
7		rebase 18815	. . . 4 |- RR = ( Base ` RRfld )
8		remulr 18820	. . . 4 |- x. = ( .r ` RRfld )
9		eqid 2454	. . . 4 |- ( .i ` (RRfld freeLMod I ) ) = ( .i ` (RRfld freeLMod I ) )
10		eqid 2454	. . . 4 |- ( 0g ` (RRfld freeLMod I ) ) = ( 0g ` (RRfld freeLMod I ) )
11		re0g 18821	. . . 4 |- 0 = ( 0g ` RRfld )
12		refldcj 18829	. . . 4 |- * = ( *r ` RRfld )
13		refld 18828	. . . . 5 |- RRfld e. Field
14	13	a1i 11	. . . 4 |- ( I e. V -> RRfld e. Field )
15		fconstmpt 5032	. . . . 5 |- ( I X. { 0 } ) = ( x e. I |-> 0 )
16	6, 7, 4	frlmbasf 18965	. . . . . . . 8 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> f : I --> RR )
17		ffn 5713	. . . . . . . 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 1014	. . . . . 6 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> f Fn I )
20		simpl 455	. . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . . 17 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> f e. ( Base ` (RRfld freeLMod I ) ) )
23	6, 7, 8, 4, 9	frlmipval 18981	. . . . . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . . . 16 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f ( .i ` (RRfld freeLMod I ) ) f ) = (RRfld gsumg ( f oF x. f ) ) )
25		ovex 6298	. . . . . . . . . . . . . . . . . . . 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 3693	. . . . . . . . . . . . . . . . . . . 20 |- ( I i^i I ) = I
28		eqidd 2455	. . . . . . . . . . . . . . . . . . . 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 6520	. . . . . . . . . . . . . . . . . . 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 6522	. . . . . . . . . . . . . . . . . . . 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 6007	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> ( f ` x ) e. RR )
32	31, 31	remulcld 9613	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> ( ( f ` x ) x. ( f ` x ) ) e. RR )
33	30, 32	eqeltrd 2542	. . . . . . . . . . . . . . . . . . 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 6037	. . . . . . . . . . . . . . . . . 18 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f oF x. f ) : I --> RR )
35		ovex 6298	. . . . . . . . . . . . . . . . . . . 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 5715	. . . . . . . . . . . . . . . . . . . 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 18963	. . . . . . . . . . . . . . . . . . 19 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> f finSupp 0 )
40		0red 9586	. . . . . . . . . . . . . . . . . . . 20 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> 0 e. RR )
41		simpr 459	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. RR ) -> x e. RR )
42	41	recnd 9611	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. RR ) -> x e. CC )
43	42	mul02d 9767	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. RR ) -> ( 0 x. x ) = 0 )
44	20, 40, 16, 16, 43	suppofss1d 6929	. . . . . . . . . . . . . . . . . . 19 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( ( f oF x. f ) supp 0 ) C_ ( f supp 0 ) )
45		fsuppsssupp 7837	. . . . . . . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . . . . . 18 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f oF x. f ) finSupp 0 )
47		regsumsupp 18831	. . . . . . . . . . . . . . . . . 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 1226	. . . . . . . . . . . . . . . . 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 6904	. . . . . . . . . . . . . . . . . . . . . 22 |- ( f supp 0 ) C_ dom f
50		fdm 5717	. . . . . . . . . . . . . . . . . . . . . . 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 3537	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f supp 0 ) C_ I )
53	44, 52	sstrd 3499	. . . . . . . . . . . . . . . . . . . 20 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( ( f oF x. f ) supp 0 ) C_ I )
54	53	sselda 3489	. . . . . . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . . . . . 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 13607	. . . . . . . . . . . . . . . . 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 2495	. . . . . . . . . . . . . . . 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 2495	. . . . . . . . . . . . . . 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 1014	. . . . . . . . . . . . . 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 996	. . . . . . . . . . . . . 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 2497	. . . . . . . . . . . . 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 7828	. . . . . . . . . . . . . . . 16 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f supp 0 ) e. Fin )
63		ssfi 7733	. . . . . . . . . . . . . . . 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 659	. . . . . . . . . . . . . . 15 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( ( f oF x. f ) supp 0 ) e. Fin )
65	54, 32	syldan 468	. . . . . . . . . . . . . . 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 10117	. . . . . . . . . . . . . . . 16 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> 0 <_ ( ( f ` x ) x. ( f ` x ) ) )
67	54, 66	syldan 468	. . . . . . . . . . . . . . 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 13694	. . . . . . . . . . . . . 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 1014	. . . . . . . . . . . . 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 2823	. . . . . . . . . . 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 712	. . . . . . . . . 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 1152	. . . . . . . . . . . . 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 9611	. . . . . . . . . . . 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 10195	. . . . . . . . . . 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 463	. . . . . . . . . 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 512	. . . . . . . . 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 1014	. . . . . . . . . . 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 463	. . . . . . . . . 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 3508	. . . . . . . . . . 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 997	. . . . . . . . . 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 9586	. . . . . . . . . 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 6923	. . . . . . . . 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 1152	. . . . . . . . . . . . 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 2456	. . . . . . . . . . . 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 482	. . . . . . . . 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 468	. . . . . . . 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 3896	. . . . . . . . . . . . 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 2524	. . . . . . . . . . 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 1014	. . . . . . . . . 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 483	. . . . . . . . 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 3631	. . . . . . . . 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 2868	. . . . . 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 6108	. . . . . . . 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 9579	. . . . . . . 8 |- 0 e. _V
105	104	fconst2 6104	. . . . . . 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 659	. . . . 5 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> f = ( I X. { 0 } ) )
108		isfld 17600	. . . . . . . . . . 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 456	. . . . . . . . 9 |- RRfld e. DivRing
111		drngring 17598	. . . . . . . . 9 |- (RRfld e. DivRing -> RRfld e. Ring )
112	110, 111	ax-mp 5	. . . . . . . 8 |- RRfld e. Ring
113	6, 11	frlm0 18958	. . . . . . . 8 |- ( (RRfld e. Ring /\ I e. V ) -> ( I X. { 0 } ) = ( 0g ` (RRfld freeLMod I ) ) )
114	112, 113	mpan 668	. . . . . . 7 |- ( I e. V -> ( I X. { 0 } ) = ( 0g ` (RRfld freeLMod I ) ) )
115	15, 114	syl5reqr 2510	. . . . . 6 |- ( I e. V -> ( 0g ` (RRfld freeLMod I ) ) = ( x e. I |-> 0 ) )
116	115	3ad2ant1 1015	. . . . 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 2521	. . . 4 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> f = ( 0g ` (RRfld freeLMod I ) ) )
118		cjre 13054	. . . . 5 |- ( x e. RR -> ( * ` x ) = x )
119	118	adantl 464	. . . 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 18983	. . 3 |- ( I e. V -> (RRfld freeLMod I ) e. PreHil )
122		df-refld 18814	. . . 4 |- RRfld = (ℂfld ↾s RR )
123	6	frlmsca 18957	. . . . 5 |- ( (RRfld e. Field /\ I e. V ) -> RRfld = (Scalar ` (RRfld freeLMod I ) ) )
124	13, 123	mpan 668	. . . 4 |- ( I e. V -> RRfld = (Scalar ` (RRfld freeLMod I ) ) )
125	122, 124	syl5reqr 2510	. . 3 |- ( I e. V -> (Scalar ` (RRfld freeLMod I ) ) = (ℂfld ↾s RR ) )
126		simpr1 1000	. . . 4 |- ( ( I e. V /\ ( f e. RR /\ f e. RR /\ 0 <_ f ) ) -> f e. RR )
127		simpr3 1002	. . . 4 |- ( ( I e. V /\ ( f e. RR /\ f e. RR /\ 0 <_ f ) ) -> 0 <_ f )
128	126, 127	resqrtcld 13331	. . 3 |- ( ( I e. V /\ ( f e. RR /\ f e. RR /\ 0 <_ f ) ) -> ( sqr ` f ) e. RR )
129	64, 65, 67	fsumge0 13691	. . . . 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 4465	. . . 4 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> 0 <_ (RRfld gsumg ( f oF x. f ) ) )
131	130, 24	breqtrrd 4465	. . 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 21846	. 2 |- ( I e. V -> (toCHil ` (RRfld freeLMod I ) ) e. CPreHil )
133	2, 132	eqeltrd 2542	1 |- ( I e. V -> H e. CPreHil )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 366   /\ wa 367   /\ w3a 971   = wceq 1398   e. wcel 1823   A.wral 2804   _Vcvv 3106   \ cdif 3458   u. cun 3459   C_ wss 3461   {csn 4016   class class class wbr 4439   |-> cmpt 4497   X. cxp 4986   dom cdm 4988   Fun wfun 5564   Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   oFcof 6511   supp csupp 6891   Fincfn 7509   finSupp cfsupp 7821   RRcr 9480   0cc0 9481   x. cmul 9486   <_ cle 9618   *ccj 13011   sum_csu 13590   Basecbs 14716   ↾_s cress 14717  Scalarcsca 14787   .icip 14789   0gc0g 14929   gsum_g cgsu 14930   Ringcrg 17393   CRingccrg 17394   DivRingcdr 17591  Fieldcfield 17592  ℂ_fldccnfld 18615  RRfldcrefld 18813   freeLMod cfrlm 18950   CPreHilccph 21779  toCHilctch 21780  ℝ^crrx 21981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ico 11538  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-prds 14937  df-pws 14939  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-ghm 16464  df-cntz 16554  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-dvr 17527  df-rnghom 17559  df-drng 17593  df-field 17594  df-subrg 17622  df-abv 17661  df-staf 17689  df-srng 17690  df-lmod 17709  df-lss 17774  df-lmhm 17863  df-lvec 17944  df-sra 18013  df-rgmod 18014  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-cnfld 18616  df-refld 18814  df-phl 18834  df-dsmm 18936  df-frlm 18951  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-xms 20989  df-ms 20990  df-nm 21269  df-ngp 21270  df-tng 21271  df-nrg 21272  df-nlm 21273  df-clm 21729  df-cph 21781  df-tch 21782  df-rrx 21983

This theorem is referenced by: (None)