Theorem frlmphl 21738

Description: Conditions for a free module to be a pre-Hilbert space. (Contributed by Thierry Arnoux, 21-Jun-2019.) (Proof shortened by AV, 21-Jul-2019.)

Hypotheses

Ref	Expression
frlmphl.y	⊢ 𝑌 = (𝑅 freeLMod 𝐼)
frlmphl.b	⊢ 𝐵 = (Base‘𝑅)
frlmphl.t	⊢ · = (.r‘𝑅)
frlmphl.v	⊢ 𝑉 = (Base‘𝑌)
frlmphl.j	⊢ , = (·𝑖‘𝑌)
frlmphl.o	⊢ 𝑂 = (0g‘𝑌)
frlmphl.0	⊢ 0 = (0g‘𝑅)
frlmphl.s	⊢ ∗ = (*𝑟‘𝑅)
frlmphl.f	⊢ (𝜑 → 𝑅 ∈ Field)
frlmphl.m	⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂)
frlmphl.u	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( ∗ ‘𝑥) = 𝑥)
frlmphl.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)

Assertion

Ref	Expression
frlmphl	⊢ (𝜑 → 𝑌 ∈ PreHil)

Distinct variable groups:   𝐵,𝑔,𝑥   𝑔,𝐼,𝑥   𝑅,𝑔,𝑥   𝑔,𝑉,𝑥   𝑔,𝑊,𝑥   · ,𝑔,𝑥   𝑔,𝑌,𝑥   0 ,𝑔,𝑥   𝜑,𝑔,𝑥   , ,𝑔,𝑥   𝑔,𝑂   𝑥, ∗

Allowed substitution hints:   ∗ (𝑔)   𝑂(𝑥)

Proof of Theorem frlmphl

Dummy variables 𝑓 𝑒 ℎ 𝑖 𝑦 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frlmphl.v	. . 3 ⊢ 𝑉 = (Base‘𝑌)
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝑉 = (Base‘𝑌))
3		eqidd 2738	. 2 ⊢ (𝜑 → (+g‘𝑌) = (+g‘𝑌))
4		eqidd 2738	. 2 ⊢ (𝜑 → ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘𝑌))
5		frlmphl.j	. . 3 ⊢ , = (·𝑖‘𝑌)
6	5	a1i 11	. 2 ⊢ (𝜑 → , = (·𝑖‘𝑌))
7		frlmphl.o	. . 3 ⊢ 𝑂 = (0g‘𝑌)
8	7	a1i 11	. 2 ⊢ (𝜑 → 𝑂 = (0g‘𝑌))
9		frlmphl.f	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Field)
10		isfld 20675	. . . . 5 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
11	9, 10	sylib 218	. . . 4 ⊢ (𝜑 → (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
12	11	simpld 494	. . 3 ⊢ (𝜑 → 𝑅 ∈ DivRing)
13		frlmphl.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
14		frlmphl.y	. . . 4 ⊢ 𝑌 = (𝑅 freeLMod 𝐼)
15	14	frlmsca 21710	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘𝑌))
16	12, 13, 15	syl2anc 585	. 2 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑌))
17		frlmphl.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
18	17	a1i 11	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
19		eqidd 2738	. 2 ⊢ (𝜑 → (+g‘𝑅) = (+g‘𝑅))
20		frlmphl.t	. . 3 ⊢ · = (.r‘𝑅)
21	20	a1i 11	. 2 ⊢ (𝜑 → · = (.r‘𝑅))
22		frlmphl.s	. . 3 ⊢ ∗ = (*𝑟‘𝑅)
23	22	a1i 11	. 2 ⊢ (𝜑 → ∗ = (*𝑟‘𝑅))
24		frlmphl.0	. . 3 ⊢ 0 = (0g‘𝑅)
25	24	a1i 11	. 2 ⊢ (𝜑 → 0 = (0g‘𝑅))
26	12	drngringd 20672	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
27	14	frlmlmod 21706	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑌 ∈ LMod)
28	26, 13, 27	syl2anc 585	. . 3 ⊢ (𝜑 → 𝑌 ∈ LMod)
29	16, 12	eqeltrrd 2838	. . 3 ⊢ (𝜑 → (Scalar‘𝑌) ∈ DivRing)
30		eqid 2737	. . . 4 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌)
31	30	islvec 21058	. . 3 ⊢ (𝑌 ∈ LVec ↔ (𝑌 ∈ LMod ∧ (Scalar‘𝑌) ∈ DivRing))
32	28, 29, 31	sylanbrc 584	. 2 ⊢ (𝜑 → 𝑌 ∈ LVec)
33	9	fldcrngd 20677	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
34		frlmphl.u	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( ∗ ‘𝑥) = 𝑥)
35	17, 22, 33, 34	idsrngd 20791	. 2 ⊢ (𝜑 → 𝑅 ∈ *-Ring)
36	13	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝐼 ∈ 𝑊)
37	26	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑅 ∈ Ring)
38		simp2 1138	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑔 ∈ 𝑉)
39		simp3 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ℎ ∈ 𝑉)
40	14, 17, 20, 1, 5	frlmipval 21736	. . . . 5 ⊢ (((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring) ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉)) → (𝑔 , ℎ) = (𝑅 Σg (𝑔 ∘f · ℎ)))
41	36, 37, 38, 39, 40	syl22anc 839	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑔 , ℎ) = (𝑅 Σg (𝑔 ∘f · ℎ)))
42	14, 17, 1	frlmbasmap 21716	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑔 ∈ 𝑉) → 𝑔 ∈ (𝐵 ↑m 𝐼))
43	36, 38, 42	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑔 ∈ (𝐵 ↑m 𝐼))
44		elmapi 8787	. . . . . . . 8 ⊢ (𝑔 ∈ (𝐵 ↑m 𝐼) → 𝑔:𝐼⟶𝐵)
45	43, 44	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑔:𝐼⟶𝐵)
46	45	ffnd 6661	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑔 Fn 𝐼)
47	14, 17, 1	frlmbasmap 21716	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑊 ∧ ℎ ∈ 𝑉) → ℎ ∈ (𝐵 ↑m 𝐼))
48	36, 39, 47	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ℎ ∈ (𝐵 ↑m 𝐼))
49		elmapi 8787	. . . . . . . 8 ⊢ (ℎ ∈ (𝐵 ↑m 𝐼) → ℎ:𝐼⟶𝐵)
50	48, 49	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ℎ:𝐼⟶𝐵)
51	50	ffnd 6661	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ℎ Fn 𝐼)
52		inidm 4168	. . . . . 6 ⊢ (𝐼 ∩ 𝐼) = 𝐼
53		eqidd 2738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) = (𝑔‘𝑥))
54		eqidd 2738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → (ℎ‘𝑥) = (ℎ‘𝑥))
55	46, 51, 36, 36, 52, 53, 54	offval 7631	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑔 ∘f · ℎ) = (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))))
56	55	oveq2d 7374	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑅 Σg (𝑔 ∘f · ℎ)) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥)))))
57	41, 56	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑔 , ℎ) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥)))))
58	26	ringcmnd 20223	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CMnd)
59	58	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑅 ∈ CMnd)
60	37	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ Ring)
61	45	ffvelcdmda 7028	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ 𝐵)
62	50	ffvelcdmda 7028	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → (ℎ‘𝑥) ∈ 𝐵)
63	17, 20, 60, 61, 62	ringcld 20199	. . . . 5 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → ((𝑔‘𝑥) · (ℎ‘𝑥)) ∈ 𝐵)
64	63	fmpttd 7059	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))):𝐼⟶𝐵)
65		frlmphl.m	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂)
66	14, 17, 20, 1, 5, 7, 24, 22, 9, 65, 34, 13	frlmphllem 21737	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) finSupp 0 )
67	17, 24, 59, 36, 64, 66	gsumcl 19848	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥)))) ∈ 𝐵)
68	57, 67	eqeltrd 2837	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑔 , ℎ) ∈ 𝐵)
69		eqid 2737	. . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅)
70	58	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝑅 ∈ CMnd)
71	13	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝐼 ∈ 𝑊)
72	26	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝑅 ∈ Ring)
73	72	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ Ring)
74		simp2 1138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝑘 ∈ 𝐵)
75	74	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → 𝑘 ∈ 𝐵)
76		simp31 1211	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝑔 ∈ 𝑉)
77	71, 76, 42	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝑔 ∈ (𝐵 ↑m 𝐼))
78	77, 44	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝑔:𝐼⟶𝐵)
79	78	ffvelcdmda 7028	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ 𝐵)
80		simp33 1213	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝑖 ∈ 𝑉)
81	14, 17, 1	frlmbasmap 21716	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑖 ∈ 𝑉) → 𝑖 ∈ (𝐵 ↑m 𝐼))
82	71, 80, 81	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝑖 ∈ (𝐵 ↑m 𝐼))
83		elmapi 8787	. . . . . . . 8 ⊢ (𝑖 ∈ (𝐵 ↑m 𝐼) → 𝑖:𝐼⟶𝐵)
84	82, 83	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝑖:𝐼⟶𝐵)
85	84	ffvelcdmda 7028	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → (𝑖‘𝑥) ∈ 𝐵)
86	17, 20, 73, 79, 85	ringcld 20199	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → ((𝑔‘𝑥) · (𝑖‘𝑥)) ∈ 𝐵)
87	17, 20, 73, 75, 86	ringcld 20199	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → (𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥))) ∈ 𝐵)
88		simp32 1212	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → ℎ ∈ 𝑉)
89	71, 88, 47	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → ℎ ∈ (𝐵 ↑m 𝐼))
90	89, 49	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → ℎ:𝐼⟶𝐵)
91	90	ffvelcdmda 7028	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → (ℎ‘𝑥) ∈ 𝐵)
92	17, 20, 73, 91, 85	ringcld 20199	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥) · (𝑖‘𝑥)) ∈ 𝐵)
93		eqidd 2738	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑥 ∈ 𝐼 ↦ (𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥)))) = (𝑥 ∈ 𝐼 ↦ (𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥)))))
94		eqidd 2738	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑖‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑖‘𝑥))))
95		fveq2 6832	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑔‘𝑥) = (𝑔‘𝑦))
96	95	oveq2d 7374	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑘 · (𝑔‘𝑥)) = (𝑘 · (𝑔‘𝑦)))
97	96	cbvmptv 5190	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑥))) = (𝑦 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑦)))
98	97	oveq1i 7368	. . . . . 6 ⊢ ((𝑥 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑥))) ∘f · 𝑖) = ((𝑦 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑦))) ∘f · 𝑖)
99	17, 20, 73, 75, 79	ringcld 20199	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → (𝑘 · (𝑔‘𝑥)) ∈ 𝐵)
100	99	fmpttd 7059	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑥 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑥))):𝐼⟶𝐵)
101	100	ffnd 6661	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑥 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑥))) Fn 𝐼)
102	97	fneq1i 6587	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑥))) Fn 𝐼 ↔ (𝑦 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑦))) Fn 𝐼)
103	101, 102	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑦 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑦))) Fn 𝐼)
104	84	ffnd 6661	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝑖 Fn 𝐼)
105		eqidd 2738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑦))) = (𝑦 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑦))))
106		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
107	106	fveq2d 6836	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 = 𝑥) → (𝑔‘𝑦) = (𝑔‘𝑥))
108	107	oveq2d 7374	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 = 𝑥) → (𝑘 · (𝑔‘𝑦)) = (𝑘 · (𝑔‘𝑥)))
109		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
110		ovexd 7393	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → (𝑘 · (𝑔‘𝑥)) ∈ V)
111	105, 108, 109, 110	fvmptd 6947	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑦)))‘𝑥) = (𝑘 · (𝑔‘𝑥)))
112		eqidd 2738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → (𝑖‘𝑥) = (𝑖‘𝑥))
113	103, 104, 71, 71, 52, 111, 112	offval 7631	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → ((𝑦 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑦))) ∘f · 𝑖) = (𝑥 ∈ 𝐼 ↦ ((𝑘 · (𝑔‘𝑥)) · (𝑖‘𝑥))))
114	17, 20	ringass 20192	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ (𝑘 ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵 ∧ (𝑖‘𝑥) ∈ 𝐵)) → ((𝑘 · (𝑔‘𝑥)) · (𝑖‘𝑥)) = (𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥))))
115	73, 75, 79, 85, 114	syl13anc 1375	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → ((𝑘 · (𝑔‘𝑥)) · (𝑖‘𝑥)) = (𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥))))
116	115	mpteq2dva 5179	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑥 ∈ 𝐼 ↦ ((𝑘 · (𝑔‘𝑥)) · (𝑖‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥)))))
117	113, 116	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → ((𝑦 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑦))) ∘f · 𝑖) = (𝑥 ∈ 𝐼 ↦ (𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥)))))
118	98, 117	eqtrid 2784	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → ((𝑥 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑥))) ∘f · 𝑖) = (𝑥 ∈ 𝐼 ↦ (𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥)))))
119		ovexd 7393	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → ((𝑥 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑥))) ∘f · 𝑖) ∈ V)
120	101, 104, 71, 71	offun 7636	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → Fun ((𝑥 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑥))) ∘f · 𝑖))
121		simp3 1139	. . . . . . . . 9 ⊢ ((𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉) → 𝑖 ∈ 𝑉)
122	13, 121	anim12i 614	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝐼 ∈ 𝑊 ∧ 𝑖 ∈ 𝑉))
123	122	3adant2 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝐼 ∈ 𝑊 ∧ 𝑖 ∈ 𝑉))
124	14, 24, 1	frlmbasfsupp 21715	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑖 ∈ 𝑉) → 𝑖 finSupp 0 )
125	123, 124	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝑖 finSupp 0 )
126	17, 24	ring0cl 20206	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵)
127	72, 126	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 0 ∈ 𝐵)
128	17, 20, 24	ringrz 20233	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → (𝑦 · 0 ) = 0 )
129	72, 128	sylan 581	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑦 ∈ 𝐵) → (𝑦 · 0 ) = 0 )
130	71, 127, 100, 84, 129	suppofss2d 8146	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (((𝑥 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑥))) ∘f · 𝑖) supp 0 ) ⊆ (𝑖 supp 0 ))
131		fsuppsssupp 9285	. . . . . 6 ⊢ (((((𝑥 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑥))) ∘f · 𝑖) ∈ V ∧ Fun ((𝑥 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑥))) ∘f · 𝑖)) ∧ (𝑖 finSupp 0 ∧ (((𝑥 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑥))) ∘f · 𝑖) supp 0 ) ⊆ (𝑖 supp 0 ))) → ((𝑥 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑥))) ∘f · 𝑖) finSupp 0 )
132	119, 120, 125, 130, 131	syl22anc 839	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → ((𝑥 ∈ 𝐼 ↦ (𝑘 · (𝑔‘𝑥))) ∘f · 𝑖) finSupp 0 )
133	118, 132	eqbrtrrd 5110	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑥 ∈ 𝐼 ↦ (𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥)))) finSupp 0 )
134		simp1 1137	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝜑)
135		eleq1w 2820	. . . . . . . . 9 ⊢ (𝑔 = ℎ → (𝑔 ∈ 𝑉 ↔ ℎ ∈ 𝑉))
136		id 22	. . . . . . . . . . 11 ⊢ (𝑔 = ℎ → 𝑔 = ℎ)
137	136, 136	oveq12d 7376	. . . . . . . . . 10 ⊢ (𝑔 = ℎ → (𝑔 , 𝑔) = (ℎ , ℎ))
138	137	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑔 = ℎ → ((𝑔 , 𝑔) = 0 ↔ (ℎ , ℎ) = 0 ))
139	135, 138	3anbi23d 1442	. . . . . . . 8 ⊢ (𝑔 = ℎ → ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ (𝑔 , 𝑔) = 0 ) ↔ (𝜑 ∧ ℎ ∈ 𝑉 ∧ (ℎ , ℎ) = 0 )))
140		eqeq1 2741	. . . . . . . 8 ⊢ (𝑔 = ℎ → (𝑔 = 𝑂 ↔ ℎ = 𝑂))
141	139, 140	imbi12d 344	. . . . . . 7 ⊢ (𝑔 = ℎ → (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂) ↔ ((𝜑 ∧ ℎ ∈ 𝑉 ∧ (ℎ , ℎ) = 0 ) → ℎ = 𝑂)))
142	141, 65	chvarvv 1991	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ 𝑉 ∧ (ℎ , ℎ) = 0 ) → ℎ = 𝑂)
143	14, 17, 20, 1, 5, 7, 24, 22, 9, 142, 34, 13	frlmphllem 21737	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉) → (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑖‘𝑥))) finSupp 0 )
144	134, 88, 80, 143	syl3anc 1374	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑖‘𝑥))) finSupp 0 )
145	17, 24, 69, 70, 71, 87, 92, 93, 94, 133, 144	gsummptfsadd 19857	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥)))(+g‘𝑅)((ℎ‘𝑥) · (𝑖‘𝑥))))) = ((𝑅 Σg (𝑥 ∈ 𝐼 ↦ (𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥)))))(+g‘𝑅)(𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑖‘𝑥))))))
146	14, 17, 20	frlmip 21735	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ DivRing) → (𝑔 ∈ (𝐵 ↑m 𝐼), ℎ ∈ (𝐵 ↑m 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))))) = (·𝑖‘𝑌))
147	13, 12, 146	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (𝑔 ∈ (𝐵 ↑m 𝐼), ℎ ∈ (𝐵 ↑m 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))))) = (·𝑖‘𝑌))
148	5, 147	eqtr4id 2791	. . . . . . 7 ⊢ (𝜑 → , = (𝑔 ∈ (𝐵 ↑m 𝐼), ℎ ∈ (𝐵 ↑m 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))))))
149		fveq1 6831	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑔 → (𝑒‘𝑥) = (𝑔‘𝑥))
150	149	oveq1d 7373	. . . . . . . . . 10 ⊢ (𝑒 = 𝑔 → ((𝑒‘𝑥) · (𝑓‘𝑥)) = ((𝑔‘𝑥) · (𝑓‘𝑥)))
151	150	mpteq2dv 5180	. . . . . . . . 9 ⊢ (𝑒 = 𝑔 → (𝑥 ∈ 𝐼 ↦ ((𝑒‘𝑥) · (𝑓‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (𝑓‘𝑥))))
152	151	oveq2d 7374	. . . . . . . 8 ⊢ (𝑒 = 𝑔 → (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑒‘𝑥) · (𝑓‘𝑥)))) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (𝑓‘𝑥)))))
153		fveq1 6831	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑓‘𝑥) = (ℎ‘𝑥))
154	153	oveq2d 7374	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑔‘𝑥) · (𝑓‘𝑥)) = ((𝑔‘𝑥) · (ℎ‘𝑥)))
155	154	mpteq2dv 5180	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (𝑓‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))))
156	155	oveq2d 7374	. . . . . . . 8 ⊢ (𝑓 = ℎ → (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (𝑓‘𝑥)))) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥)))))
157	152, 156	cbvmpov 7453	. . . . . . 7 ⊢ (𝑒 ∈ (𝐵 ↑m 𝐼), 𝑓 ∈ (𝐵 ↑m 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑒‘𝑥) · (𝑓‘𝑥))))) = (𝑔 ∈ (𝐵 ↑m 𝐼), ℎ ∈ (𝐵 ↑m 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥)))))
158	148, 157	eqtr4di 2790	. . . . . 6 ⊢ (𝜑 → , = (𝑒 ∈ (𝐵 ↑m 𝐼), 𝑓 ∈ (𝐵 ↑m 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑒‘𝑥) · (𝑓‘𝑥))))))
159	158	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → , = (𝑒 ∈ (𝐵 ↑m 𝐼), 𝑓 ∈ (𝐵 ↑m 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑒‘𝑥) · (𝑓‘𝑥))))))
160		simprl 771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = ((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) ∧ 𝑓 = 𝑖)) → 𝑒 = ((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ))
161	160	fveq1d 6834	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = ((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) ∧ 𝑓 = 𝑖)) → (𝑒‘𝑥) = (((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ)‘𝑥))
162		simprr 773	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = ((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) ∧ 𝑓 = 𝑖)) → 𝑓 = 𝑖)
163	162	fveq1d 6834	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = ((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) ∧ 𝑓 = 𝑖)) → (𝑓‘𝑥) = (𝑖‘𝑥))
164	161, 163	oveq12d 7376	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = ((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) ∧ 𝑓 = 𝑖)) → ((𝑒‘𝑥) · (𝑓‘𝑥)) = ((((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ)‘𝑥) · (𝑖‘𝑥)))
165	164	mpteq2dv 5180	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = ((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) ∧ 𝑓 = 𝑖)) → (𝑥 ∈ 𝐼 ↦ ((𝑒‘𝑥) · (𝑓‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ)‘𝑥) · (𝑖‘𝑥))))
166	165	oveq2d 7374	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = ((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) ∧ 𝑓 = 𝑖)) → (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑒‘𝑥) · (𝑓‘𝑥)))) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ)‘𝑥) · (𝑖‘𝑥)))))
167	28	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝑌 ∈ LMod)
168	16	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝑅 = (Scalar‘𝑌))
169	168	fveq2d 6836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (Base‘𝑅) = (Base‘(Scalar‘𝑌)))
170	17, 169	eqtrid 2784	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝐵 = (Base‘(Scalar‘𝑌)))
171	74, 170	eleqtrd 2839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → 𝑘 ∈ (Base‘(Scalar‘𝑌)))
172		eqid 2737	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘𝑌)
173		eqid 2737	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌))
174	1, 30, 172, 173	lmodvscl 20831	. . . . . . . 8 ⊢ ((𝑌 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑔 ∈ 𝑉) → (𝑘( ·𝑠 ‘𝑌)𝑔) ∈ 𝑉)
175	167, 171, 76, 174	syl3anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑘( ·𝑠 ‘𝑌)𝑔) ∈ 𝑉)
176		eqid 2737	. . . . . . . 8 ⊢ (+g‘𝑌) = (+g‘𝑌)
177	1, 176	lmodvacl 20828	. . . . . . 7 ⊢ ((𝑌 ∈ LMod ∧ (𝑘( ·𝑠 ‘𝑌)𝑔) ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) ∈ 𝑉)
178	167, 175, 88, 177	syl3anc 1374	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → ((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) ∈ 𝑉)
179	14, 17, 1	frlmbasmap 21716	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ ((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) ∈ 𝑉) → ((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) ∈ (𝐵 ↑m 𝐼))
180	71, 178, 179	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → ((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) ∈ (𝐵 ↑m 𝐼))
181		ovexd 7393	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ)‘𝑥) · (𝑖‘𝑥)))) ∈ V)
182	159, 166, 180, 82, 181	ovmpod 7510	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) , 𝑖) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ)‘𝑥) · (𝑖‘𝑥)))))
183	14, 1, 72, 71, 175, 88, 69, 176	frlmplusgval 21721	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → ((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) = ((𝑘( ·𝑠 ‘𝑌)𝑔) ∘f (+g‘𝑅)ℎ))
184	14, 17, 1	frlmbasmap 21716	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑊 ∧ (𝑘( ·𝑠 ‘𝑌)𝑔) ∈ 𝑉) → (𝑘( ·𝑠 ‘𝑌)𝑔) ∈ (𝐵 ↑m 𝐼))
185	71, 175, 184	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑘( ·𝑠 ‘𝑌)𝑔) ∈ (𝐵 ↑m 𝐼))
186		elmapi 8787	. . . . . . . . . . . 12 ⊢ ((𝑘( ·𝑠 ‘𝑌)𝑔) ∈ (𝐵 ↑m 𝐼) → (𝑘( ·𝑠 ‘𝑌)𝑔):𝐼⟶𝐵)
187		ffn 6660	. . . . . . . . . . . 12 ⊢ ((𝑘( ·𝑠 ‘𝑌)𝑔):𝐼⟶𝐵 → (𝑘( ·𝑠 ‘𝑌)𝑔) Fn 𝐼)
188	185, 186, 187	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑘( ·𝑠 ‘𝑌)𝑔) Fn 𝐼)
189	90	ffnd 6661	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → ℎ Fn 𝐼)
190	71	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑊)
191	76	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → 𝑔 ∈ 𝑉)
192	14, 1, 17, 190, 75, 191, 109, 172, 20	frlmvscaval 21725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → ((𝑘( ·𝑠 ‘𝑌)𝑔)‘𝑥) = (𝑘 · (𝑔‘𝑥)))
193		eqidd 2738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → (ℎ‘𝑥) = (ℎ‘𝑥))
194	188, 189, 71, 71, 52, 192, 193	offval 7631	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → ((𝑘( ·𝑠 ‘𝑌)𝑔) ∘f (+g‘𝑅)ℎ) = (𝑥 ∈ 𝐼 ↦ ((𝑘 · (𝑔‘𝑥))(+g‘𝑅)(ℎ‘𝑥))))
195	183, 194	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → ((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) = (𝑥 ∈ 𝐼 ↦ ((𝑘 · (𝑔‘𝑥))(+g‘𝑅)(ℎ‘𝑥))))
196		ovexd 7393	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → ((𝑘 · (𝑔‘𝑥))(+g‘𝑅)(ℎ‘𝑥)) ∈ V)
197	195, 196	fvmpt2d 6953	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → (((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ)‘𝑥) = ((𝑘 · (𝑔‘𝑥))(+g‘𝑅)(ℎ‘𝑥)))
198	197	oveq1d 7373	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → ((((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ)‘𝑥) · (𝑖‘𝑥)) = (((𝑘 · (𝑔‘𝑥))(+g‘𝑅)(ℎ‘𝑥)) · (𝑖‘𝑥)))
199	17, 69, 20	ringdir 20201	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ ((𝑘 · (𝑔‘𝑥)) ∈ 𝐵 ∧ (ℎ‘𝑥) ∈ 𝐵 ∧ (𝑖‘𝑥) ∈ 𝐵)) → (((𝑘 · (𝑔‘𝑥))(+g‘𝑅)(ℎ‘𝑥)) · (𝑖‘𝑥)) = (((𝑘 · (𝑔‘𝑥)) · (𝑖‘𝑥))(+g‘𝑅)((ℎ‘𝑥) · (𝑖‘𝑥))))
200	73, 99, 91, 85, 199	syl13anc 1375	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → (((𝑘 · (𝑔‘𝑥))(+g‘𝑅)(ℎ‘𝑥)) · (𝑖‘𝑥)) = (((𝑘 · (𝑔‘𝑥)) · (𝑖‘𝑥))(+g‘𝑅)((ℎ‘𝑥) · (𝑖‘𝑥))))
201	115	oveq1d 7373	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → (((𝑘 · (𝑔‘𝑥)) · (𝑖‘𝑥))(+g‘𝑅)((ℎ‘𝑥) · (𝑖‘𝑥))) = ((𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥)))(+g‘𝑅)((ℎ‘𝑥) · (𝑖‘𝑥))))
202	198, 200, 201	3eqtrd 2776	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ 𝑥 ∈ 𝐼) → ((((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ)‘𝑥) · (𝑖‘𝑥)) = ((𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥)))(+g‘𝑅)((ℎ‘𝑥) · (𝑖‘𝑥))))
203	202	mpteq2dva 5179	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑥 ∈ 𝐼 ↦ ((((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ)‘𝑥) · (𝑖‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥)))(+g‘𝑅)((ℎ‘𝑥) · (𝑖‘𝑥)))))
204	203	oveq2d 7374	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ)‘𝑥) · (𝑖‘𝑥)))) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥)))(+g‘𝑅)((ℎ‘𝑥) · (𝑖‘𝑥))))))
205	182, 204	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) , 𝑖) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥)))(+g‘𝑅)((ℎ‘𝑥) · (𝑖‘𝑥))))))
206		simprl 771	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = 𝑔 ∧ 𝑓 = 𝑖)) → 𝑒 = 𝑔)
207	206	fveq1d 6834	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = 𝑔 ∧ 𝑓 = 𝑖)) → (𝑒‘𝑥) = (𝑔‘𝑥))
208		simprr 773	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = 𝑔 ∧ 𝑓 = 𝑖)) → 𝑓 = 𝑖)
209	208	fveq1d 6834	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = 𝑔 ∧ 𝑓 = 𝑖)) → (𝑓‘𝑥) = (𝑖‘𝑥))
210	207, 209	oveq12d 7376	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = 𝑔 ∧ 𝑓 = 𝑖)) → ((𝑒‘𝑥) · (𝑓‘𝑥)) = ((𝑔‘𝑥) · (𝑖‘𝑥)))
211	210	mpteq2dv 5180	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = 𝑔 ∧ 𝑓 = 𝑖)) → (𝑥 ∈ 𝐼 ↦ ((𝑒‘𝑥) · (𝑓‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (𝑖‘𝑥))))
212	211	oveq2d 7374	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = 𝑔 ∧ 𝑓 = 𝑖)) → (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑒‘𝑥) · (𝑓‘𝑥)))) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (𝑖‘𝑥)))))
213		ovexd 7393	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (𝑖‘𝑥)))) ∈ V)
214	159, 212, 77, 82, 213	ovmpod 7510	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑔 , 𝑖) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (𝑖‘𝑥)))))
215	214	oveq2d 7374	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑘 · (𝑔 , 𝑖)) = (𝑘 · (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (𝑖‘𝑥))))))
216	14, 17, 20, 1, 5, 7, 24, 22, 9, 65, 34, 13	frlmphllem 21737	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ 𝑖 ∈ 𝑉) → (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (𝑖‘𝑥))) finSupp 0 )
217	134, 76, 80, 216	syl3anc 1374	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (𝑖‘𝑥))) finSupp 0 )
218	17, 24, 20, 72, 71, 74, 86, 217	gsummulc2 20254	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑅 Σg (𝑥 ∈ 𝐼 ↦ (𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥))))) = (𝑘 · (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (𝑖‘𝑥))))))
219	215, 218	eqtr4d 2775	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑘 · (𝑔 , 𝑖)) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ (𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥))))))
220		simprl 771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = ℎ ∧ 𝑓 = 𝑖)) → 𝑒 = ℎ)
221	220	fveq1d 6834	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = ℎ ∧ 𝑓 = 𝑖)) → (𝑒‘𝑥) = (ℎ‘𝑥))
222		simprr 773	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = ℎ ∧ 𝑓 = 𝑖)) → 𝑓 = 𝑖)
223	222	fveq1d 6834	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = ℎ ∧ 𝑓 = 𝑖)) → (𝑓‘𝑥) = (𝑖‘𝑥))
224	221, 223	oveq12d 7376	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = ℎ ∧ 𝑓 = 𝑖)) → ((𝑒‘𝑥) · (𝑓‘𝑥)) = ((ℎ‘𝑥) · (𝑖‘𝑥)))
225	224	mpteq2dv 5180	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = ℎ ∧ 𝑓 = 𝑖)) → (𝑥 ∈ 𝐼 ↦ ((𝑒‘𝑥) · (𝑓‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑖‘𝑥))))
226	225	oveq2d 7374	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) ∧ (𝑒 = ℎ ∧ 𝑓 = 𝑖)) → (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑒‘𝑥) · (𝑓‘𝑥)))) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑖‘𝑥)))))
227		ovexd 7393	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑖‘𝑥)))) ∈ V)
228	159, 226, 89, 82, 227	ovmpod 7510	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (ℎ , 𝑖) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑖‘𝑥)))))
229	219, 228	oveq12d 7376	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → ((𝑘 · (𝑔 , 𝑖))(+g‘𝑅)(ℎ , 𝑖)) = ((𝑅 Σg (𝑥 ∈ 𝐼 ↦ (𝑘 · ((𝑔‘𝑥) · (𝑖‘𝑥)))))(+g‘𝑅)(𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑖‘𝑥))))))
230	145, 205, 229	3eqtr4d 2782	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ (𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉 ∧ 𝑖 ∈ 𝑉)) → (((𝑘( ·𝑠 ‘𝑌)𝑔)(+g‘𝑌)ℎ) , 𝑖) = ((𝑘 · (𝑔 , 𝑖))(+g‘𝑅)(ℎ , 𝑖)))
231	33	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑅 ∈ CRing)
232	231	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ CRing)
233	17, 20	crngcom 20190	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (ℎ‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) → ((ℎ‘𝑥) · (𝑔‘𝑥)) = ((𝑔‘𝑥) · (ℎ‘𝑥)))
234	232, 62, 61, 233	syl3anc 1374	. . . . 5 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥) · (𝑔‘𝑥)) = ((𝑔‘𝑥) · (ℎ‘𝑥)))
235	234	mpteq2dva 5179	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))))
236	235	oveq2d 7374	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑔‘𝑥)))) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥)))))
237	158	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → , = (𝑒 ∈ (𝐵 ↑m 𝐼), 𝑓 ∈ (𝐵 ↑m 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑒‘𝑥) · (𝑓‘𝑥))))))
238		simprl 771	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ (𝑒 = ℎ ∧ 𝑓 = 𝑔)) → 𝑒 = ℎ)
239	238	fveq1d 6834	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ (𝑒 = ℎ ∧ 𝑓 = 𝑔)) → (𝑒‘𝑥) = (ℎ‘𝑥))
240		simprr 773	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ (𝑒 = ℎ ∧ 𝑓 = 𝑔)) → 𝑓 = 𝑔)
241	240	fveq1d 6834	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ (𝑒 = ℎ ∧ 𝑓 = 𝑔)) → (𝑓‘𝑥) = (𝑔‘𝑥))
242	239, 241	oveq12d 7376	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ (𝑒 = ℎ ∧ 𝑓 = 𝑔)) → ((𝑒‘𝑥) · (𝑓‘𝑥)) = ((ℎ‘𝑥) · (𝑔‘𝑥)))
243	242	mpteq2dv 5180	. . . . 5 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ (𝑒 = ℎ ∧ 𝑓 = 𝑔)) → (𝑥 ∈ 𝐼 ↦ ((𝑒‘𝑥) · (𝑓‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑔‘𝑥))))
244	243	oveq2d 7374	. . . 4 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ (𝑒 = ℎ ∧ 𝑓 = 𝑔)) → (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑒‘𝑥) · (𝑓‘𝑥)))) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑔‘𝑥)))))
245		ovexd 7393	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑔‘𝑥)))) ∈ V)
246	237, 244, 48, 43, 245	ovmpod 7510	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (ℎ , 𝑔) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑔‘𝑥)))))
247		fveq2 6832	. . . . . 6 ⊢ (𝑥 = (𝑔 , ℎ) → ( ∗ ‘𝑥) = ( ∗ ‘(𝑔 , ℎ)))
248		id 22	. . . . . 6 ⊢ (𝑥 = (𝑔 , ℎ) → 𝑥 = (𝑔 , ℎ))
249	247, 248	eqeq12d 2753	. . . . 5 ⊢ (𝑥 = (𝑔 , ℎ) → (( ∗ ‘𝑥) = 𝑥 ↔ ( ∗ ‘(𝑔 , ℎ)) = (𝑔 , ℎ)))
250	34	ralrimiva 3130	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( ∗ ‘𝑥) = 𝑥)
251	250	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ∀𝑥 ∈ 𝐵 ( ∗ ‘𝑥) = 𝑥)
252	249, 251, 68	rspcdva 3566	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ( ∗ ‘(𝑔 , ℎ)) = (𝑔 , ℎ))
253	252, 57	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ( ∗ ‘(𝑔 , ℎ)) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥)))))
254	236, 246, 253	3eqtr4rd 2783	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ( ∗ ‘(𝑔 , ℎ)) = (ℎ , 𝑔))
255	2, 3, 4, 6, 8, 16, 18, 19, 21, 23, 25, 32, 35, 68, 230, 65, 254	isphld 21611	1 ⊢ (𝜑 → 𝑌 ∈ PreHil)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ⊆ wss 3890   class class class wbr 5086   ↦ cmpt 5167  Fun wfun 6484   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490  (class class class)co 7358   ∈ cmpo 7360   ∘_f cof 7620   supp csupp 8101   ↑_m cmap 8764   finSupp cfsupp 9265  Basecbs 17137  +_gcplusg 17178  ._rcmulr 17179  *_𝑟cstv 17180  Scalarcsca 17181   ·_𝑠 cvsca 17182  ·_𝑖cip 17183  0_gc0g 17360   Σ_g cgsu 17361  CMndccmn 19713  Ringcrg 20172  CRingccrg 20173  DivRingcdr 20664  Fieldcfield 20665  LModclmod 20813  LVecclvec 21056  PreHilcphl 21581   freeLMod cfrlm 21703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8102  df-tpos 8167  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-map 8766  df-ixp 8837  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-fsupp 9266  df-sup 9346  df-oi 9416  df-card 9852  df-pnf 11169  df-mnf 11170  df-xr 11171  df-ltxr 11172  df-le 11173  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-7 12214  df-8 12215  df-9 12216  df-n0 12403  df-z 12490  df-dec 12609  df-uz 12753  df-fz 13425  df-fzo 13572  df-seq 13926  df-hash 14255  df-struct 17075  df-sets 17092  df-slot 17110  df-ndx 17122  df-base 17138  df-ress 17159  df-plusg 17191  df-mulr 17192  df-sca 17194  df-vsca 17195  df-ip 17196  df-tset 17197  df-ple 17198  df-ds 17200  df-hom 17202  df-cco 17203  df-0g 17362  df-gsum 17363  df-prds 17368  df-pws 17370  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-mhm 18709  df-submnd 18710  df-grp 18870  df-minusg 18871  df-sbg 18872  df-subg 19057  df-ghm 19146  df-cntz 19250  df-cmn 19715  df-abl 19716  df-mgp 20080  df-rng 20092  df-ur 20121  df-ring 20174  df-cring 20175  df-oppr 20275  df-rhm 20410  df-subrg 20505  df-drng 20666  df-field 20667  df-staf 20774  df-srng 20775  df-lmod 20815  df-lss 20885  df-lmhm 20976  df-lvec 21057  df-sra 21127  df-rgmod 21128  df-phl 21583  df-dsmm 21689  df-frlm 21704

This theorem is referenced by:  rrxcph  25337