Theorem frlmip 20543

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 2758	. . . . . . 7 ⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
3		eqid 2758	. . . . . . 7 ⊢ (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
4	2, 3	frlmpws 20515	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 freeLMod 𝐼) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘(𝑅 freeLMod 𝐼))))
5	4	ancoms 462	. . . . 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 16617	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾s 𝐵) = 𝑅)
8		eqidd 2759	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘𝐵))
9	6	eqimssi 3950	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ (Base‘𝑅)
10	9	a1i 11	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → 𝐵 ⊆ (Base‘𝑅))
11	8, 10	srasca 20021	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾s 𝐵) = (Scalar‘((subringAlg ‘𝑅)‘𝐵)))
12	7, 11	eqtr3d 2795	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → 𝑅 = (Scalar‘((subringAlg ‘𝑅)‘𝐵)))
13	12	oveq1d 7165	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
14	13	adantl 485	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
15		fvex 6671	. . . . . . . . 9 ⊢ ((subringAlg ‘𝑅)‘𝐵) ∈ V
16		rlmval 20031	. . . . . . . . . . . 12 ⊢ (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘(Base‘𝑅))
17	6	fveq2i 6661	. . . . . . . . . . . 12 ⊢ ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘(Base‘𝑅))
18	16, 17	eqtr4i 2784	. . . . . . . . . . 11 ⊢ (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘𝐵)
19	18	oveq1i 7160	. . . . . . . . . 10 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = (((subringAlg ‘𝑅)‘𝐵) ↑s 𝐼)
20		eqid 2758	. . . . . . . . . 10 ⊢ (Scalar‘((subringAlg ‘𝑅)‘𝐵)) = (Scalar‘((subringAlg ‘𝑅)‘𝐵))
21	19, 20	pwsval 16817	. . . . . . . . 9 ⊢ ((((subringAlg ‘𝑅)‘𝐵) ∈ V ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
22	15, 21	mpan 689	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑊 → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
23	22	adantr 484	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
24	14, 23	eqtr4d 2796	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((ringLMod‘𝑅) ↑s 𝐼))
25	1	fveq2i 6661	. . . . . . 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 7168	. . . . 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 2796	. . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑅 freeLMod 𝐼) = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
29	1, 28	syl5eq 2805	. . 3 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → 𝑌 = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
30	29	fveq2d 6662	. 2 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (·𝑖‘𝑌) = (·𝑖‘((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))))
31		fvex 6671	. . . 4 ⊢ (Base‘𝑌) ∈ V
32		eqid 2758	. . . . 5 ⊢ ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)) = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))
33		eqid 2758	. . . . 5 ⊢ (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
34	32, 33	ressip 16710	. . . 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 2758	. . . . 5 ⊢ (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))
37		simpr 488	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
38		snex 5300	. . . . . . 7 ⊢ {((subringAlg ‘𝑅)‘𝐵)} ∈ V
39		xpexg 7471	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ {((subringAlg ‘𝑅)‘𝐵)} ∈ V) → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
40	38, 39	mpan2 690	. . . . . 6 ⊢ (𝐼 ∈ 𝑊 → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
41	40	adantr 484	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
42		eqid 2758	. . . . 5 ⊢ (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
43	15	snnz 4669	. . . . . 6 ⊢ {((subringAlg ‘𝑅)‘𝐵)} ≠ ∅
44		dmxp 5770	. . . . . 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 16792	. . . 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 16788	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = X𝑥 ∈ 𝐼 (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)))
48		eqidd 2759	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘𝐵))
49	9	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → 𝐵 ⊆ (Base‘𝑅))
50	48, 49	srabase 20018	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (Base‘𝑅) = (Base‘((subringAlg ‘𝑅)‘𝐵)))
51	6	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → 𝐵 = (Base‘𝑅))
52	15	fvconst2 6957	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → ((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥) = ((subringAlg ‘𝑅)‘𝐵))
53	52	fveq2d 6662	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = (Base‘((subringAlg ‘𝑅)‘𝐵)))
54	50, 51, 53	3eqtr4rd 2804	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = 𝐵)
55	54	adantl 485	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = 𝐵)
56	55	ixpeq2dva 8494	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → X𝑥 ∈ 𝐼 (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = X𝑥 ∈ 𝐼 𝐵)
57	6	fvexi 6672	. . . . . . . 8 ⊢ 𝐵 ∈ V
58		ixpconstg 8488	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝐵 ∈ V) → X𝑥 ∈ 𝐼 𝐵 = (𝐵 ↑m 𝐼))
59	57, 58	mpan2 690	. . . . . . 7 ⊢ (𝐼 ∈ 𝑊 → X𝑥 ∈ 𝐼 𝐵 = (𝐵 ↑m 𝐼))
60	59	adantr 484	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → X𝑥 ∈ 𝐼 𝐵 = (𝐵 ↑m 𝐼))
61	47, 56, 60	3eqtrd 2797	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (𝐵 ↑m 𝐼))
62		frlmphl.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
63	52, 49	sraip 20023	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → (.r‘𝑅) = (·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)))
64	62, 63	syl5req 2806	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = · )
65	64	oveqd 7167	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 → ((𝑓‘𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔‘𝑥)) = ((𝑓‘𝑥) · (𝑔‘𝑥)))
66	65	mpteq2ia 5123	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥)))
67	66	oveq2i 7161	. . . . . 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 7223	. . . 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 2793	. . 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	syl5eqr 2807	. 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 2794	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 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  Vcvv 3409   ⊆ wss 3858  ∅c0 4225  {csn 4522   ↦ cmpt 5112   × cxp 5522  dom cdm 5524  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152   ↑_m cmap 8416  Xcixp 8479  Basecbs 16541   ↾_s cress 16542  ._rcmulr 16624  Scalarcsca 16626  ·_𝑖cip 16628   Σ_g cgsu 16772  X_scprds 16777   ↑_s cpws 16778  subringAlg csra 20008  ringLModcrglmod 20009   freeLMod cfrlm 20511

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-map 8418  df-ixp 8480  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-sup 8939  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-3 11738  df-4 11739  df-5 11740  df-6 11741  df-7 11742  df-8 11743  df-9 11744  df-n0 11935  df-z 12021  df-dec 12138  df-uz 12283  df-fz 12940  df-struct 16543  df-ndx 16544  df-slot 16545  df-base 16547  df-sets 16548  df-ress 16549  df-plusg 16636  df-mulr 16637  df-sca 16639  df-vsca 16640  df-ip 16641  df-tset 16642  df-ple 16643  df-ds 16645  df-hom 16647  df-cco 16648  df-prds 16779  df-pws 16781  df-sra 20012  df-rgmod 20013  df-dsmm 20497  df-frlm 20512

This theorem is referenced by:  frlmipval  20544  frlmphl  20546