Theorem frlmip 21685

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 2729	. . . . . . 7 ⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
3		eqid 2729	. . . . . . 7 ⊢ (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
4	2, 3	frlmpws 21657	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 freeLMod 𝐼) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘(𝑅 freeLMod 𝐼))))
5	4	ancoms 458	. . . . 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 17155	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾s 𝐵) = 𝑅)
8		eqidd 2730	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘𝐵))
9	6	eqimssi 3996	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ (Base‘𝑅)
10	9	a1i 11	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → 𝐵 ⊆ (Base‘𝑅))
11	8, 10	srasca 21084	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾s 𝐵) = (Scalar‘((subringAlg ‘𝑅)‘𝐵)))
12	7, 11	eqtr3d 2766	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → 𝑅 = (Scalar‘((subringAlg ‘𝑅)‘𝐵)))
13	12	oveq1d 7364	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
14	13	adantl 481	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
15		fvex 6835	. . . . . . . . 9 ⊢ ((subringAlg ‘𝑅)‘𝐵) ∈ V
16		rlmval 21095	. . . . . . . . . . . 12 ⊢ (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘(Base‘𝑅))
17	6	fveq2i 6825	. . . . . . . . . . . 12 ⊢ ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘(Base‘𝑅))
18	16, 17	eqtr4i 2755	. . . . . . . . . . 11 ⊢ (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘𝐵)
19	18	oveq1i 7359	. . . . . . . . . 10 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = (((subringAlg ‘𝑅)‘𝐵) ↑s 𝐼)
20		eqid 2729	. . . . . . . . . 10 ⊢ (Scalar‘((subringAlg ‘𝑅)‘𝐵)) = (Scalar‘((subringAlg ‘𝑅)‘𝐵))
21	19, 20	pwsval 17390	. . . . . . . . 9 ⊢ ((((subringAlg ‘𝑅)‘𝐵) ∈ V ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
22	15, 21	mpan 690	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑊 → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
23	22	adantr 480	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
24	14, 23	eqtr4d 2767	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((ringLMod‘𝑅) ↑s 𝐼))
25	1	fveq2i 6825	. . . . . . 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 7367	. . . . 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 2767	. . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑅 freeLMod 𝐼) = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
29	1, 28	eqtrid 2776	. . 3 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → 𝑌 = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
30	29	fveq2d 6826	. 2 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (·𝑖‘𝑌) = (·𝑖‘((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))))
31		fvex 6835	. . . 4 ⊢ (Base‘𝑌) ∈ V
32		eqid 2729	. . . . 5 ⊢ ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)) = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))
33		eqid 2729	. . . . 5 ⊢ (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
34	32, 33	ressip 17249	. . . 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 2729	. . . . 5 ⊢ (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))
37		simpr 484	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
38		snex 5375	. . . . . . 7 ⊢ {((subringAlg ‘𝑅)‘𝐵)} ∈ V
39		xpexg 7686	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ {((subringAlg ‘𝑅)‘𝐵)} ∈ V) → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
40	38, 39	mpan2 691	. . . . . 6 ⊢ (𝐼 ∈ 𝑊 → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
41	40	adantr 480	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
42		eqid 2729	. . . . 5 ⊢ (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
43	15	snnz 4728	. . . . . 6 ⊢ {((subringAlg ‘𝑅)‘𝐵)} ≠ ∅
44		dmxp 5871	. . . . . 6 ⊢ ({((subringAlg ‘𝑅)‘𝐵)} ≠ ∅ → dom (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) = 𝐼)
45	43, 44	mp1i 13	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → dom (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) = 𝐼)
46	36, 37, 41, 42, 45, 33	prdsip 17365	. . . 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 17361	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = X𝑥 ∈ 𝐼 (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)))
48		eqidd 2730	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘𝐵))
49	9	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → 𝐵 ⊆ (Base‘𝑅))
50	48, 49	srabase 21081	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (Base‘𝑅) = (Base‘((subringAlg ‘𝑅)‘𝐵)))
51	6	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → 𝐵 = (Base‘𝑅))
52	15	fvconst2 7140	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → ((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥) = ((subringAlg ‘𝑅)‘𝐵))
53	52	fveq2d 6826	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = (Base‘((subringAlg ‘𝑅)‘𝐵)))
54	50, 51, 53	3eqtr4rd 2775	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = 𝐵)
55	54	adantl 481	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = 𝐵)
56	55	ixpeq2dva 8839	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → X𝑥 ∈ 𝐼 (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = X𝑥 ∈ 𝐼 𝐵)
57	6	fvexi 6836	. . . . . . . 8 ⊢ 𝐵 ∈ V
58		ixpconstg 8833	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝐵 ∈ V) → X𝑥 ∈ 𝐼 𝐵 = (𝐵 ↑m 𝐼))
59	57, 58	mpan2 691	. . . . . . 7 ⊢ (𝐼 ∈ 𝑊 → X𝑥 ∈ 𝐼 𝐵 = (𝐵 ↑m 𝐼))
60	59	adantr 480	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → X𝑥 ∈ 𝐼 𝐵 = (𝐵 ↑m 𝐼))
61	47, 56, 60	3eqtrd 2768	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (𝐵 ↑m 𝐼))
62		frlmphl.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
63	52, 49	sraip 21086	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → (.r‘𝑅) = (·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)))
64	62, 63	eqtr2id 2777	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = · )
65	64	oveqd 7366	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 → ((𝑓‘𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔‘𝑥)) = ((𝑓‘𝑥) · (𝑔‘𝑥)))
66	65	mpteq2ia 5187	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥)))
67	66	oveq2i 7360	. . . . . 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 7424	. . . 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 2764	. . 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 2778	. 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 2765	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 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3436   ⊆ wss 3903  ∅c0 4284  {csn 4577   ↦ cmpt 5173   × cxp 5617  dom cdm 5619  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351   ↑_m cmap 8753  Xcixp 8824  Basecbs 17120   ↾_s cress 17141  ._rcmulr 17162  Scalarcsca 17164  ·_𝑖cip 17166   Σ_g cgsu 17344  X_scprds 17349   ↑_s cpws 17350  subringAlg csra 21075  ringLModcrglmod 21076   freeLMod cfrlm 21653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-map 8755  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-hom 17185  df-cco 17186  df-prds 17351  df-pws 17353  df-sra 21077  df-rgmod 21078  df-dsmm 21639  df-frlm 21654

This theorem is referenced by:  frlmipval  21686  frlmphl  21688