Theorem rrxcph 23180

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 23175	. 2 |- ( I e. V -> H = (toCHil ` (RRfld freeLMod I ) ) )
3		eqid 2622	. . 3 |- (toCHil ` (RRfld freeLMod I ) ) = (toCHil ` (RRfld freeLMod I ) )
4		eqid 2622	. . 3 |- ( Base ` (RRfld freeLMod I ) ) = ( Base ` (RRfld freeLMod I ) )
5		eqid 2622	. . 3 |- (Scalar ` (RRfld freeLMod I ) ) = (Scalar ` (RRfld freeLMod I ) )
6		eqid 2622	. . . 4 |- (RRfld freeLMod I ) = (RRfld freeLMod I )
7		rebase 19952	. . . 4 |- RR = ( Base ` RRfld )
8		remulr 19957	. . . 4 |- x. = ( .r ` RRfld )
9		eqid 2622	. . . 4 |- ( .i ` (RRfld freeLMod I ) ) = ( .i ` (RRfld freeLMod I ) )
10		eqid 2622	. . . 4 |- ( 0g ` (RRfld freeLMod I ) ) = ( 0g ` (RRfld freeLMod I ) )
11		re0g 19958	. . . 4 |- 0 = ( 0g ` RRfld )
12		refldcj 19966	. . . 4 |- * = ( *r ` RRfld )
13		refld 19965	. . . . 5 |- RRfld e. Field
14	13	a1i 11	. . . 4 |- ( I e. V -> RRfld e. Field )
15		fconstmpt 5163	. . . . 5 |- ( I X. { 0 } ) = ( x e. I |-> 0 )
16	6, 7, 4	frlmbasf 20104	. . . . . . . 8 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> f : I --> RR )
17		ffn 6045	. . . . . . . 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 1081	. . . . . 6 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> f Fn I )
20		simpl 473	. . . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . . 17 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> f e. ( Base ` (RRfld freeLMod I ) ) )
23	6, 7, 8, 4, 9	frlmipval 20118	. . . . . . . . . . . . . . . . 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 1327	. . . . . . . . . . . . . . . 16 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f ( .i ` (RRfld freeLMod I ) ) f ) = (RRfld gsumg ( f oF x. f ) ) )
25		ovexd 6680	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> ( ( f ` x ) x. ( f ` x ) ) e. _V )
26		inidm 3822	. . . . . . . . . . . . . . . . . . . 20 |- ( I i^i I ) = I
27		eqidd 2623	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> ( f ` x ) = ( f ` x ) )
28	18, 18, 20, 20, 26, 27, 27	offval 6904	. . . . . . . . . . . . . . . . . . 19 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f oF x. f ) = ( x e. I |-> ( ( f ` x ) x. ( f ` x ) ) ) )
29	18, 18, 20, 20, 26, 27, 27	ofval 6906	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> ( ( f oF x. f ) ` x ) = ( ( f ` x ) x. ( f ` x ) ) )
30	16	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> ( f ` x ) e. RR )
31	30, 30	remulcld 10070	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> ( ( f ` x ) x. ( f ` x ) ) e. RR )
32	29, 31	eqeltrd 2701	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> ( ( f oF x. f ) ` x ) e. RR )
33	25, 28, 32	fmpt2d 6393	. . . . . . . . . . . . . . . . . 18 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f oF x. f ) : I --> RR )
34		ovexd 6680	. . . . . . . . . . . . . . . . . . 19 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f oF x. f ) e. _V )
35		ffun 6048	. . . . . . . . . . . . . . . . . . . 20 |- ( ( f oF x. f ) : I --> RR -> Fun ( f oF x. f ) )
36	33, 35	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> Fun ( f oF x. f ) )
37	6, 11, 4	frlmbasfsupp 20102	. . . . . . . . . . . . . . . . . . 19 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> f finSupp 0 )
38		0red 10041	. . . . . . . . . . . . . . . . . . . 20 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> 0 e. RR )
39		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. RR ) -> x e. RR )
40	39	recnd 10068	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. RR ) -> x e. CC )
41	40	mul02d 10234	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. RR ) -> ( 0 x. x ) = 0 )
42	20, 38, 16, 16, 41	suppofss1d 7332	. . . . . . . . . . . . . . . . . . 19 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( ( f oF x. f ) supp 0 ) C_ ( f supp 0 ) )
43		fsuppsssupp 8291	. . . . . . . . . . . . . . . . . . 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 )
44	34, 36, 37, 42, 43	syl22anc 1327	. . . . . . . . . . . . . . . . . 18 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f oF x. f ) finSupp 0 )
45		regsumsupp 19968	. . . . . . . . . . . . . . . . . 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 ) )
46	33, 44, 20, 45	syl3anc 1326	. . . . . . . . . . . . . . . . 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 ) )
47		suppssdm 7308	. . . . . . . . . . . . . . . . . . . . . 22 |- ( f supp 0 ) C_ dom f
48		fdm 6051	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( f : I --> RR -> dom f = I )
49	16, 48	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> dom f = I )
50	47, 49	syl5sseq 3653	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f supp 0 ) C_ I )
51	42, 50	sstrd 3613	. . . . . . . . . . . . . . . . . . . 20 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( ( f oF x. f ) supp 0 ) C_ I )
52	51	sselda 3603	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. ( ( f oF x. f ) supp 0 ) ) -> x e. I )
53	52, 29	syldan 487	. . . . . . . . . . . . . . . . . 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 ) ) )
54	53	sumeq2dv 14433	. . . . . . . . . . . . . . . . 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 ) ) )
55	46, 54	eqtrd 2656	. . . . . . . . . . . . . . . 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 ) ) )
56	24, 55	eqtrd 2656	. . . . . . . . . . . . . . 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 ) ) )
57	56	3adant3 1081	. . . . . . . . . . . . . 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 ) ) )
58		simp3 1063	. . . . . . . . . . . . . 14 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 )
59	57, 58	eqtr3d 2658	. . . . . . . . . . . . 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 )
60	37	fsuppimpd 8282	. . . . . . . . . . . . . . . 16 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( f supp 0 ) e. Fin )
61		ssfi 8180	. . . . . . . . . . . . . . . 16 |- ( ( ( f supp 0 ) e. Fin /\ ( ( f oF x. f ) supp 0 ) C_ ( f supp 0 ) ) -> ( ( f oF x. f ) supp 0 ) e. Fin )
62	60, 42, 61	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> ( ( f oF x. f ) supp 0 ) e. Fin )
63	52, 31	syldan 487	. . . . . . . . . . . . . . 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 )
64	30	msqge0d 10596	. . . . . . . . . . . . . . . 16 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. I ) -> 0 <_ ( ( f ` x ) x. ( f ` x ) ) )
65	52, 64	syldan 487	. . . . . . . . . . . . . . 15 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) /\ x e. ( ( f oF x. f ) supp 0 ) ) -> 0 <_ ( ( f ` x ) x. ( f ` x ) ) )
66	62, 63, 65	fsum00 14530	. . . . . . . . . . . . . 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 ) )
67	66	3adant3 1081	. . . . . . . . . . . . 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 ) )
68	59, 67	mpbid 222	. . . . . . . . . . . 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 )
69	68	r19.21bi 2932	. . . . . . . . . . 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 )
70	69	adantlr 751	. . . . . . . . . 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 )
71	30	3adantl3 1219	. . . . . . . . . . . . 13 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> ( f ` x ) e. RR )
72	71	recnd 10068	. . . . . . . . . . . 12 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> ( f ` x ) e. CC )
73	72, 72	mul0ord 10677	. . . . . . . . . . 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 ) ) )
74	73	adantr 481	. . . . . . . . . 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 ) ) )
75	70, 74	mpbid 222	. . . . . . . . 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 ) )
76		oridm 536	. . . . . . . . 9 |- ( ( ( f ` x ) = 0 \/ ( f ` x ) = 0 ) <-> ( f ` x ) = 0 )
77	75, 76	sylib 208	. . . . . . . 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 )
78	33	3adant3 1081	. . . . . . . . . . 11 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> ( f oF x. f ) : I --> RR )
79	78	adantr 481	. . . . . . . . . 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 )
80		ssid 3624	. . . . . . . . . . 11 |- ( ( f oF x. f ) supp 0 ) C_ ( ( f oF x. f ) supp 0 )
81	80	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 ) )
82		simpl1 1064	. . . . . . . . . 10 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> I e. V )
83		0red 10041	. . . . . . . . . 10 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> 0 e. RR )
84	79, 81, 82, 83	suppssr 7326	. . . . . . . . 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 )
85	29	3adantl3 1219	. . . . . . . . . . . . 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 ) ) )
86	85	eqeq1d 2624	. . . . . . . . . . . 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 ) )
87	86, 73	bitrd 268	. . . . . . . . . . 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 ) ) )
88	87, 76	syl6bb 276	. . . . . . . . . 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 ) )
89	88	biimpa 501	. . . . . . . . 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 )
90	84, 89	syldan 487	. . . . . . . 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 )
91		undif 4049	. . . . . . . . . . . . 13 |- ( ( ( f oF x. f ) supp 0 ) C_ I <-> ( ( ( f oF x. f ) supp 0 ) u. ( I \ ( ( f oF x. f ) supp 0 ) ) ) = I )
92	51, 91	sylib 208	. . . . . . . . . . . 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 )
93	92	eleq2d 2687	. . . . . . . . . . 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 ) )
94	93	3adant3 1081	. . . . . . . . . 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 ) )
95	94	biimpar 502	. . . . . . . . 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 ) ) ) )
96		elun 3753	. . . . . . . . 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 ) ) ) )
97	95, 96	sylib 208	. . . . . . . 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 ) ) ) )
98	77, 90, 97	mpjaodan 827	. . . . . . 7 |- ( ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) /\ x e. I ) -> ( f ` x ) = 0 )
99	98	ralrimiva 2966	. . . . . 6 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> A. x e. I ( f ` x ) = 0 )
100		fconstfv 6476	. . . . . . 7 |- ( f : I --> { 0 } <-> ( f Fn I /\ A. x e. I ( f ` x ) = 0 ) )
101		c0ex 10034	. . . . . . . 8 |- 0 e. _V
102	101	fconst2 6470	. . . . . . 7 |- ( f : I --> { 0 } <-> f = ( I X. { 0 } ) )
103	100, 102	sylbb1 227	. . . . . 6 |- ( ( f Fn I /\ A. x e. I ( f ` x ) = 0 ) -> f = ( I X. { 0 } ) )
104	19, 99, 103	syl2anc 693	. . . . 5 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> f = ( I X. { 0 } ) )
105		isfld 18756	. . . . . . . . . . 11 |- (RRfld e. Field <-> (RRfld e. DivRing /\ RRfld e. CRing ) )
106	13, 105	mpbi 220	. . . . . . . . . 10 |- (RRfld e. DivRing /\ RRfld e. CRing )
107	106	simpli 474	. . . . . . . . 9 |- RRfld e. DivRing
108		drngring 18754	. . . . . . . . 9 |- (RRfld e. DivRing -> RRfld e. Ring )
109	107, 108	ax-mp 5	. . . . . . . 8 |- RRfld e. Ring
110	6, 11	frlm0 20098	. . . . . . . 8 |- ( (RRfld e. Ring /\ I e. V ) -> ( I X. { 0 } ) = ( 0g ` (RRfld freeLMod I ) ) )
111	109, 110	mpan 706	. . . . . . 7 |- ( I e. V -> ( I X. { 0 } ) = ( 0g ` (RRfld freeLMod I ) ) )
112	15, 111	syl5reqr 2671	. . . . . 6 |- ( I e. V -> ( 0g ` (RRfld freeLMod I ) ) = ( x e. I |-> 0 ) )
113	112	3ad2ant1 1082	. . . . 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 ) )
114	15, 104, 113	3eqtr4a 2682	. . . 4 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) /\ ( f ( .i ` (RRfld freeLMod I ) ) f ) = 0 ) -> f = ( 0g ` (RRfld freeLMod I ) ) )
115		cjre 13879	. . . . 5 |- ( x e. RR -> ( * ` x ) = x )
116	115	adantl 482	. . . 4 |- ( ( I e. V /\ x e. RR ) -> ( * ` x ) = x )
117		id 22	. . . 4 |- ( I e. V -> I e. V )
118	6, 7, 8, 4, 9, 10, 11, 12, 14, 114, 116, 117	frlmphl 20120	. . 3 |- ( I e. V -> (RRfld freeLMod I ) e. PreHil )
119		df-refld 19951	. . . 4 |- RRfld = (ℂfld ↾s RR )
120	6	frlmsca 20097	. . . . 5 |- ( (RRfld e. Field /\ I e. V ) -> RRfld = (Scalar ` (RRfld freeLMod I ) ) )
121	13, 120	mpan 706	. . . 4 |- ( I e. V -> RRfld = (Scalar ` (RRfld freeLMod I ) ) )
122	119, 121	syl5reqr 2671	. . 3 |- ( I e. V -> (Scalar ` (RRfld freeLMod I ) ) = (ℂfld ↾s RR ) )
123		simpr1 1067	. . . 4 |- ( ( I e. V /\ ( f e. RR /\ f e. RR /\ 0 <_ f ) ) -> f e. RR )
124		simpr3 1069	. . . 4 |- ( ( I e. V /\ ( f e. RR /\ f e. RR /\ 0 <_ f ) ) -> 0 <_ f )
125	123, 124	resqrtcld 14156	. . 3 |- ( ( I e. V /\ ( f e. RR /\ f e. RR /\ 0 <_ f ) ) -> ( sqr ` f ) e. RR )
126	62, 63, 65	fsumge0 14527	. . . . 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 ) ) )
127	126, 55	breqtrrd 4681	. . . 4 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> 0 <_ (RRfld gsumg ( f oF x. f ) ) )
128	127, 24	breqtrrd 4681	. . 3 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> 0 <_ ( f ( .i ` (RRfld freeLMod I ) ) f ) )
129	3, 4, 5, 118, 122, 9, 125, 128	tchcph 23036	. 2 |- ( I e. V -> (toCHil ` (RRfld freeLMod I ) ) e. CPreHil )
130	2, 129	eqeltrd 2701	1 |- ( I e. V -> H e. CPreHil )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275   RRcr 9935   0cc0 9936   x. cmul 9941   <_ cle 10075   *ccj 13836   sum_csu 14416   Basecbs 15857   ↾_s cress 15858  Scalarcsca 15944   .icip 15946   0gc0g 16100   gsum_g cgsu 16101   Ringcrg 18547   CRingccrg 18548   DivRingcdr 18747  Fieldcfield 18748  ℂ_fldccnfld 19746  RRfldcrefld 19950   freeLMod cfrlm 20090   CPreHilccph 22966  toCHilctch 22967  ℝ^crrx 23171

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  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

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-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-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-inf 8349  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-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  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-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-prds 16108  df-pws 16110  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-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-field 18750  df-subrg 18778  df-abv 18817  df-staf 18845  df-srng 18846  df-lmod 18865  df-lss 18933  df-lmhm 19022  df-lvec 19103  df-sra 19172  df-rgmod 19173  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-refld 19951  df-phl 19971  df-dsmm 20076  df-frlm 20091  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-xms 22125  df-ms 22126  df-nm 22387  df-ngp 22388  df-tng 22389  df-nrg 22390  df-nlm 22391  df-clm 22863  df-cph 22968  df-tch 22969  df-rrx 23173

This theorem is referenced by:  rrxngp  40502