Theorem frlmip 22084

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 2760	. . . . . . 7 ⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
3		eqid 2760	. . . . . . 7 ⊢ (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
4	2, 3	frlmpws 22056	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 freeLMod 𝐼) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘(𝑅 freeLMod 𝐼))))
5	4	ancoms 464	. . . . 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 17422	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾s 𝐵) = 𝑅)
8		eqidd 2761	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘𝐵))
9	6	eqimssi 3968	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ (Base‘𝑅)
10	9	a1i 11	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → 𝐵 ⊆ (Base‘𝑅))
11	8, 10	srasca 21455	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾s 𝐵) = (Scalar‘((subringAlg ‘𝑅)‘𝐵)))
12	7, 11	eqtr3d 2797	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → 𝑅 = (Scalar‘((subringAlg ‘𝑅)‘𝐵)))
13	12	oveq1d 7435	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
14	13	adantl 487	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
15		fvex 6898	. . . . . . . . 9 ⊢ ((subringAlg ‘𝑅)‘𝐵) ∈ V
16		rlmval 21466	. . . . . . . . . . . 12 ⊢ (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘(Base‘𝑅))
17	6	fveq2i 6888	. . . . . . . . . . . 12 ⊢ ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘(Base‘𝑅))
18	16, 17	eqtr4i 2786	. . . . . . . . . . 11 ⊢ (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘𝐵)
19	18	oveq1i 7430	. . . . . . . . . 10 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = (((subringAlg ‘𝑅)‘𝐵) ↑s 𝐼)
20		eqid 2760	. . . . . . . . . 10 ⊢ (Scalar‘((subringAlg ‘𝑅)‘𝐵)) = (Scalar‘((subringAlg ‘𝑅)‘𝐵))
21	19, 20	pwsval 17657	. . . . . . . . 9 ⊢ ((((subringAlg ‘𝑅)‘𝐵) ∈ V ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
22	15, 21	mpan 703	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑊 → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
23	22	adantr 486	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
24	14, 23	eqtr4d 2798	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((ringLMod‘𝑅) ↑s 𝐼))
25	1	fveq2i 6888	. . . . . . 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 7438	. . . . 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 2798	. . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑅 freeLMod 𝐼) = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
29	1, 28	eqtrid 2807	. . 3 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → 𝑌 = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
30	29	fveq2d 6889	. 2 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (·𝑖‘𝑌) = (·𝑖‘((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))))
31		fvex 6898	. . . 4 ⊢ (Base‘𝑌) ∈ V
32		eqid 2760	. . . . 5 ⊢ ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)) = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))
33		eqid 2760	. . . . 5 ⊢ (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
34	32, 33	ressip 17516	. . . 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 2760	. . . . 5 ⊢ (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))
37		simpr 490	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
38		snex 5396	. . . . . . 7 ⊢ {((subringAlg ‘𝑅)‘𝐵)} ∈ V
39		xpexg 7764	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ {((subringAlg ‘𝑅)‘𝐵)} ∈ V) → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
40	38, 39	mpan2 704	. . . . . 6 ⊢ (𝐼 ∈ 𝑊 → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
41	40	adantr 486	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
42		eqid 2760	. . . . 5 ⊢ (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
43	15	snnz 4736	. . . . . 6 ⊢ {((subringAlg ‘𝑅)‘𝐵)} ≠ ∅
44		dmxp 5910	. . . . . 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 17632	. . . 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 17628	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = X𝑥 ∈ 𝐼 (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)))
48		eqidd 2761	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘𝐵))
49	9	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → 𝐵 ⊆ (Base‘𝑅))
50	48, 49	srabase 21452	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (Base‘𝑅) = (Base‘((subringAlg ‘𝑅)‘𝐵)))
51	6	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → 𝐵 = (Base‘𝑅))
52	15	fvconst2 7210	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → ((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥) = ((subringAlg ‘𝑅)‘𝐵))
53	52	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = (Base‘((subringAlg ‘𝑅)‘𝐵)))
54	50, 51, 53	3eqtr4rd 2806	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = 𝐵)
55	54	adantl 487	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = 𝐵)
56	55	ixpeq2dva 8940	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → X𝑥 ∈ 𝐼 (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = X𝑥 ∈ 𝐼 𝐵)
57	6	fvexi 6899	. . . . . . . 8 ⊢ 𝐵 ∈ V
58		ixpconstg 8934	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝐵 ∈ V) → X𝑥 ∈ 𝐼 𝐵 = (𝐵 ↑m 𝐼))
59	57, 58	mpan2 704	. . . . . . 7 ⊢ (𝐼 ∈ 𝑊 → X𝑥 ∈ 𝐼 𝐵 = (𝐵 ↑m 𝐼))
60	59	adantr 486	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → X𝑥 ∈ 𝐼 𝐵 = (𝐵 ↑m 𝐼))
61	47, 56, 60	3eqtrd 2799	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (𝐵 ↑m 𝐼))
62		frlmphl.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
63	52, 49	sraip 21457	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → (.r‘𝑅) = (·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)))
64	62, 63	eqtr2id 2808	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = · )
65	64	oveqd 7437	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 → ((𝑓‘𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔‘𝑥)) = ((𝑓‘𝑥) · (𝑔‘𝑥)))
66	65	mpteq2ia 5199	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥)))
67	66	oveq2i 7431	. . . . . 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 7495	. . . 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 2795	. . 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 2809	. 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 2796	1 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑓 ∈ (𝐵 ↑m 𝐼), 𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥))))) = (·𝑖‘𝑌))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  Vcvv 3450   ⊆ wss 3895  ∅c0 4278  {csn 4583   ↦ cmpt 5185   × cxp 5648  dom cdm 5650  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422   ↑_m cmap 8847  Xcixp 8925  Basecbs 17387   ↾_s cress 17408  ._rcmulr 17429  Scalarcsca 17431  ·_𝑖cip 17433   Σ_g cgsu 17611  X_scprds 17616   ↑_s cpws 17617  subringAlg csra 21446  ringLModcrglmod 21447   freeLMod cfrlm 22052

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-hom 17452  df-cco 17453  df-prds 17618  df-pws 17620  df-sra 21448  df-rgmod 21449  df-dsmm 22038  df-frlm 22053

This theorem is used by:  frlmipval  22085  frlmphl  22087