Theorem frlmip 21750

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 2737	. . . . . . 7 ⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
3		eqid 2737	. . . . . . 7 ⊢ (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
4	2, 3	frlmpws 21722	. . . . . 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 17185	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾s 𝐵) = 𝑅)
8		eqidd 2738	. . . . . . . . . . 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 21149	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾s 𝐵) = (Scalar‘((subringAlg ‘𝑅)‘𝐵)))
12	7, 11	eqtr3d 2774	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → 𝑅 = (Scalar‘((subringAlg ‘𝑅)‘𝐵)))
13	12	oveq1d 7385	. . . . . . . 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 6857	. . . . . . . . 9 ⊢ ((subringAlg ‘𝑅)‘𝐵) ∈ V
16		rlmval 21160	. . . . . . . . . . . 12 ⊢ (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘(Base‘𝑅))
17	6	fveq2i 6847	. . . . . . . . . . . 12 ⊢ ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘(Base‘𝑅))
18	16, 17	eqtr4i 2763	. . . . . . . . . . 11 ⊢ (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘𝐵)
19	18	oveq1i 7380	. . . . . . . . . 10 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = (((subringAlg ‘𝑅)‘𝐵) ↑s 𝐼)
20		eqid 2737	. . . . . . . . . 10 ⊢ (Scalar‘((subringAlg ‘𝑅)‘𝐵)) = (Scalar‘((subringAlg ‘𝑅)‘𝐵))
21	19, 20	pwsval 17420	. . . . . . . . 9 ⊢ ((((subringAlg ‘𝑅)‘𝐵) ∈ V ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
22	15, 21	mpan 691	. . . . . . . 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 2775	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((ringLMod‘𝑅) ↑s 𝐼))
25	1	fveq2i 6847	. . . . . . 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 7388	. . . . 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 2775	. . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑅 freeLMod 𝐼) = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
29	1, 28	eqtrid 2784	. . 3 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → 𝑌 = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
30	29	fveq2d 6848	. 2 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (·𝑖‘𝑌) = (·𝑖‘((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))))
31		fvex 6857	. . . 4 ⊢ (Base‘𝑌) ∈ V
32		eqid 2737	. . . . 5 ⊢ ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)) = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))
33		eqid 2737	. . . . 5 ⊢ (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
34	32, 33	ressip 17279	. . . 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 2737	. . . . 5 ⊢ (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))
37		simpr 484	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
38		snex 5387	. . . . . . 7 ⊢ {((subringAlg ‘𝑅)‘𝐵)} ∈ V
39		xpexg 7707	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ {((subringAlg ‘𝑅)‘𝐵)} ∈ V) → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
40	38, 39	mpan2 692	. . . . . 6 ⊢ (𝐼 ∈ 𝑊 → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
41	40	adantr 480	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
42		eqid 2737	. . . . 5 ⊢ (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
43	15	snnz 4735	. . . . . 6 ⊢ {((subringAlg ‘𝑅)‘𝐵)} ≠ ∅
44		dmxp 5888	. . . . . 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 17395	. . . 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 17391	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = X𝑥 ∈ 𝐼 (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)))
48		eqidd 2738	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘𝐵))
49	9	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → 𝐵 ⊆ (Base‘𝑅))
50	48, 49	srabase 21146	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (Base‘𝑅) = (Base‘((subringAlg ‘𝑅)‘𝐵)))
51	6	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → 𝐵 = (Base‘𝑅))
52	15	fvconst2 7162	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → ((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥) = ((subringAlg ‘𝑅)‘𝐵))
53	52	fveq2d 6848	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = (Base‘((subringAlg ‘𝑅)‘𝐵)))
54	50, 51, 53	3eqtr4rd 2783	. . . . . . . 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 8864	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → X𝑥 ∈ 𝐼 (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = X𝑥 ∈ 𝐼 𝐵)
57	6	fvexi 6858	. . . . . . . 8 ⊢ 𝐵 ∈ V
58		ixpconstg 8858	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝐵 ∈ V) → X𝑥 ∈ 𝐼 𝐵 = (𝐵 ↑m 𝐼))
59	57, 58	mpan2 692	. . . . . . 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 2776	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (𝐵 ↑m 𝐼))
62		frlmphl.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
63	52, 49	sraip 21151	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 → (.r‘𝑅) = (·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)))
64	62, 63	eqtr2id 2785	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = · )
65	64	oveqd 7387	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 → ((𝑓‘𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔‘𝑥)) = ((𝑓‘𝑥) · (𝑔‘𝑥)))
66	65	mpteq2ia 5195	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥)))
67	66	oveq2i 7381	. . . . . 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 7445	. . . 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 2772	. . 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 2786	. 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 2773	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 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442   ⊆ wss 3903  ∅c0 4287  {csn 4582   ↦ cmpt 5181   × cxp 5632  dom cdm 5634  ‘cfv 6502  (class class class)co 7370   ∈ cmpo 7372   ↑_m cmap 8777  Xcixp 8849  Basecbs 17150   ↾_s cress 17171  ._rcmulr 17192  Scalarcsca 17194  ·_𝑖cip 17196   Σ_g cgsu 17374  X_scprds 17379   ↑_s cpws 17380  subringAlg csra 21140  ringLModcrglmod 21141   freeLMod cfrlm 21718

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 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

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-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-er 8647  df-map 8779  df-ixp 8850  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-sup 9359  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-3 12223  df-4 12224  df-5 12225  df-6 12226  df-7 12227  df-8 12228  df-9 12229  df-n0 12416  df-z 12503  df-dec 12622  df-uz 12766  df-fz 13438  df-struct 17088  df-sets 17105  df-slot 17123  df-ndx 17135  df-base 17151  df-ress 17172  df-plusg 17204  df-mulr 17205  df-sca 17207  df-vsca 17208  df-ip 17209  df-tset 17210  df-ple 17211  df-ds 17213  df-hom 17215  df-cco 17216  df-prds 17381  df-pws 17383  df-sra 21142  df-rgmod 21143  df-dsmm 21704  df-frlm 21719

This theorem is referenced by:  frlmipval  21751  frlmphl  21753