Theorem frlmip 21899

Description: The inner product of a free module. (Contributed by Thierry Arnoux, 20-Jun-2019.)

Hypotheses

Ref	Expression
frlmphl.y	⊢ 𝑌 = (𝑅 freeLMod 𝐼)
frlmphl.b	⊢ 𝐵 = (Base‘𝑅)
frlmphl.t	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
frlmip	⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑓 ∈ (𝐵 ↑m 𝐼), 𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥))))) = (·𝑖‘𝑌))

Distinct variable groups:   𝐵,𝑓,𝑔,𝑥   𝑓,𝐼,𝑔,𝑥   𝑅,𝑓,𝑔,𝑥   𝑓,𝑉,𝑔,𝑥   𝑓,𝑊,𝑔,𝑥

Allowed substitution hints:   · (𝑥,𝑓,𝑔)   𝑌(𝑥,𝑓,𝑔)

Proof of Theorem frlmip

Step	Hyp	Ref	Expression
1		frlmphl.y	. . . 4 ⊢ 𝑌 = (𝑅 freeLMod 𝐼)
2		eqid 2769	. . . . . . 7 ⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
3		eqid 2769	. . . . . . 7 ⊢ (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
4	2, 3	frlmpws 21871	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 freeLMod 𝐼) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘(𝑅 freeLMod 𝐼))))
5	4	ancoms 463	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑅 freeLMod 𝐼) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘(𝑅 freeLMod 𝐼))))
6		frlmphl.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
7	6	ressid 17306	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾s 𝐵) = 𝑅)
8		eqidd 2770	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘𝐵))
9	6	eqimssi 4005	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ (Base‘𝑅)
10	9	a1i 11	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → 𝐵 ⊆ (Base‘𝑅))
11	8, 10	srasca 21281	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾s 𝐵) = (Scalar‘((subringAlg ‘𝑅)‘𝐵)))
12	7, 11	eqtr3d 2806	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → 𝑅 = (Scalar‘((subringAlg ‘𝑅)‘𝐵)))
13	12	oveq1d 7428	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
14	13	adantl 486	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
15		fvex 6897	. . . . . . . . 9 ⊢ ((subringAlg ‘𝑅)‘𝐵) ∈ V
16		rlmval 21292	. . . . . . . . . . . 12 ⊢ (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘(Base‘𝑅))
17	6	fveq2i 6887	. . . . . . . . . . . 12 ⊢ ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘(Base‘𝑅))
18	16, 17	eqtr4i 2795	. . . . . . . . . . 11 ⊢ (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘𝐵)
19	18	oveq1i 7423	. . . . . . . . . 10 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = (((subringAlg ‘𝑅)‘𝐵) ↑s 𝐼)
20		eqid 2769	. . . . . . . . . 10 ⊢ (Scalar‘((subringAlg ‘𝑅)‘𝐵)) = (Scalar‘((subringAlg ‘𝑅)‘𝐵))
21	19, 20	pwsval 17541	. . . . . . . . 9 ⊢ ((((subringAlg ‘𝑅)‘𝐵) ∈ V ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
22	15, 21	mpan 702	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑊 → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
23	22	adantr 485	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
24	14, 23	eqtr4d 2807	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((ringLMod‘𝑅) ↑s 𝐼))
25	1	fveq2i 6887	. . . . . . 7 ⊢ (Base‘𝑌) = (Base‘(𝑅 freeLMod 𝐼))
26	25	a1i 11	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (Base‘𝑌) = (Base‘(𝑅 freeLMod 𝐼)))
27	24, 26	oveq12d 7431	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘(𝑅 freeLMod 𝐼))))
28	5, 27	eqtr4d 2807	. . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑅 freeLMod 𝐼) = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
29	1, 28	eqtrid 2816	. . 3 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → 𝑌 = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
30	29	fveq2d 6888	. 2 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (·𝑖‘𝑌) = (·𝑖‘((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))))
31		fvex 6897	. . . 4 ⊢ (Base‘𝑌) ∈ V
32		eqid 2769	. . . . 5 ⊢ ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)) = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))
33		eqid 2769	. . . . 5 ⊢ (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
34	32, 33	ressip 17400	. . . 4 ⊢ ((Base‘𝑌) ∈ V → (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (·𝑖‘((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))))
35	31, 34	ax-mp 5	. . 3 ⊢ (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (·𝑖‘((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
36		eqid 2769	. . . . 5 ⊢ (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))
37		simpr 489	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
38		snex 5413	. . . . . . 7 ⊢ {((subringAlg ‘𝑅)‘𝐵)} ∈ V
39		xpexg 7751	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ {((subringAlg ‘𝑅)‘𝐵)} ∈ V) → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
40	38, 39	mpan2 703	. . . . . 6 ⊢ (𝐼 ∈ 𝑊 → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
41	40	adantr 485	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
42		eqid 2769	. . . . 5 ⊢ (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
43	15	snnz 4747	. . . . . 6 ⊢ {((subringAlg ‘𝑅)‘𝐵)} ≠ ∅
44		dmxp 5922	. . . . . 6 ⊢ ({((subringAlg ‘𝑅)‘𝐵)} ≠ ∅ → dom (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) = 𝐼)
45	43, 44	mp1i 14	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → dom (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) = 𝐼)
46	36, 37, 41, 42, 45, 33	prdsip 17516	. . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (𝑓 ∈ (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))), 𝑔 ∈ (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔‘𝑥))))))
47	36, 37, 41, 42, 45	prdsbas 17512	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = X𝑥 ∈ 𝐼 (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)))
48		eqidd 2770	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘𝐵))
49	9	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → 𝐵 ⊆ (Base‘𝑅))
50	48, 49	srabase 21278	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (Base‘𝑅) = (Base‘((subringAlg ‘𝑅)‘𝐵)))
51	6	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → 𝐵 = (Base‘𝑅))
52	15	fvconst2 7205	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → ((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥) = ((subringAlg ‘𝑅)‘𝐵))
53	52	fveq2d 6888	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = (Base‘((subringAlg ‘𝑅)‘𝐵)))
54	50, 51, 53	3eqtr4rd 2815	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = 𝐵)
55	54	adantl 486	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = 𝐵)
56	55	ixpeq2dva 8912	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → X𝑥 ∈ 𝐼 (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = X𝑥 ∈ 𝐼 𝐵)
57	6	fvexi 6898	. . . . . . . 8 ⊢ 𝐵 ∈ V
58		ixpconstg 8906	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝐵 ∈ V) → X𝑥 ∈ 𝐼 𝐵 = (𝐵 ↑m 𝐼))
59	57, 58	mpan2 703	. . . . . . 7 ⊢ (𝐼 ∈ 𝑊 → X𝑥 ∈ 𝐼 𝐵 = (𝐵 ↑m 𝐼))
60	59	adantr 485	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → X𝑥 ∈ 𝐼 𝐵 = (𝐵 ↑m 𝐼))
61	47, 56, 60	3eqtrd 2808	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (𝐵 ↑m 𝐼))
62		frlmphl.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
63	52, 49	sraip 21283	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → (.r‘𝑅) = (·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)))
64	62, 63	eqtr2id 2817	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = · )
65	64	oveqd 7430	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 → ((𝑓‘𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔‘𝑥)) = ((𝑓‘𝑥) · (𝑔‘𝑥)))
66	65	mpteq2ia 5210	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥)))
67	66	oveq2i 7424	. . . . . 6 ⊢ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔‘𝑥)))) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥))))
68	67	a1i 11	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔‘𝑥)))) = (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥)))))
69	61, 61, 68	mpoeq123dv 7488	. . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑓 ∈ (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))), 𝑔 ∈ (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔‘𝑥))))) = (𝑓 ∈ (𝐵 ↑m 𝐼), 𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥))))))
70	46, 69	eqtrd 2804	. . 3 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (𝑓 ∈ (𝐵 ↑m 𝐼), 𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥))))))
71	35, 70	eqtr3id 2818	. 2 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (·𝑖‘((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))) = (𝑓 ∈ (𝐵 ↑m 𝐼), 𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥))))))
72	30, 71	eqtr2d 2805	1 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑓 ∈ (𝐵 ↑m 𝐼), 𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥))))) = (·𝑖‘𝑌))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  {csn 4594   ↦ cmpt 5196   × cxp 5662  dom cdm 5664  ‘cfv 6539  (class class class)co 7413   ∈ cmpo 7415   ↑_m cmap 8826  Xcixp 8897  Basecbs 17271   ↾_s cress 17292  ._rcmulr 17313  Scalarcsca 17315  ·_𝑖cip 17317   Σ_g cgsu 17495  X_scprds 17500   ↑_s cpws 17501  subringAlg csra 21272  ringLModcrglmod 21273   freeLMod cfrlm 21867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735  ax-cnex 11158  ax-resscn 11159  ax-1cn 11160  ax-icn 11161  ax-addcl 11162  ax-addrcl 11163  ax-mulcl 11164  ax-mulrcl 11165  ax-mulcom 11166  ax-addass 11167  ax-mulass 11168  ax-distr 11169  ax-i2m1 11170  ax-1ne0 11171  ax-1rid 11172  ax-rnegex 11173  ax-rrecex 11174  ax-cnre 11175  ax-pre-lttri 11176  ax-pre-lttrn 11177  ax-pre-ltadd 11178  ax-pre-mulgt0 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-map 8828  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9404  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11445  df-neg 11446  df-nn 12236  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12865  df-fz 13538  df-struct 17209  df-sets 17226  df-slot 17244  df-ndx 17256  df-base 17272  df-ress 17293  df-plusg 17325  df-mulr 17326  df-sca 17328  df-vsca 17329  df-ip 17330  df-tset 17331  df-ple 17332  df-ds 17334  df-hom 17336  df-cco 17337  df-prds 17502  df-pws 17504  df-sra 21274  df-rgmod 21275  df-dsmm 21853  df-frlm 21868

This theorem is referenced by:  frlmipval  21900  frlmphl  21902