Theorem frlmbas 20099

Description: Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.)

Hypotheses

Ref	Expression
frlmval.f	⊢ 𝐹 = (𝑅 freeLMod 𝐼)
frlmbas.n	⊢ 𝑁 = (Base‘𝑅)
frlmbas.z	⊢ 0 = (0g‘𝑅)
frlmbas.b	⊢ 𝐵 = {𝑘 ∈ (𝑁 ↑𝑚 𝐼) ∣ 𝑘 finSupp 0 }

Assertion

Ref	Expression
frlmbas	⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐵 = (Base‘𝐹))

Distinct variable groups:   𝑘,𝑁   𝑅,𝑘   𝑘,𝐼   𝑘,𝑊   𝑘,𝑉   0 ,𝑘

Allowed substitution hints:   𝐵(𝑘)   𝐹(𝑘)

Proof of Theorem frlmbas

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . 5 ⊢ (ringLMod‘𝑅) ∈ V
2		fnconstg 6093	. . . . 5 ⊢ ((ringLMod‘𝑅) ∈ V → (𝐼 × {(ringLMod‘𝑅)}) Fn 𝐼)
3	1, 2	ax-mp 5	. . . 4 ⊢ (𝐼 × {(ringLMod‘𝑅)}) Fn 𝐼
4		eqid 2622	. . . . 5 ⊢ (𝑅Xs(𝐼 × {(ringLMod‘𝑅)})) = (𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))
5		eqid 2622	. . . . 5 ⊢ {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin} = {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin}
6	4, 5	dsmmbas2 20081	. . . 4 ⊢ (((𝐼 × {(ringLMod‘𝑅)}) Fn 𝐼 ∧ 𝐼 ∈ 𝑊) → {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin} = (Base‘(𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))))
7	3, 6	mpan 706	. . 3 ⊢ (𝐼 ∈ 𝑊 → {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin} = (Base‘(𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))))
8	7	adantl 482	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin} = (Base‘(𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))))
9		frlmbas.b	. . 3 ⊢ 𝐵 = {𝑘 ∈ (𝑁 ↑𝑚 𝐼) ∣ 𝑘 finSupp 0 }
10		fvco2 6273	. . . . . . . . . . . . 13 ⊢ (((𝐼 × {(ringLMod‘𝑅)}) Fn 𝐼 ∧ 𝑥 ∈ 𝐼) → ((0g ∘ (𝐼 × {(ringLMod‘𝑅)}))‘𝑥) = (0g‘((𝐼 × {(ringLMod‘𝑅)})‘𝑥)))
11	3, 10	mpan 706	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐼 → ((0g ∘ (𝐼 × {(ringLMod‘𝑅)}))‘𝑥) = (0g‘((𝐼 × {(ringLMod‘𝑅)})‘𝑥)))
12	11	adantl 482	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((0g ∘ (𝐼 × {(ringLMod‘𝑅)}))‘𝑥) = (0g‘((𝐼 × {(ringLMod‘𝑅)})‘𝑥)))
13	1	fvconst2 6469	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐼 → ((𝐼 × {(ringLMod‘𝑅)})‘𝑥) = (ringLMod‘𝑅))
14	13	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {(ringLMod‘𝑅)})‘𝑥) = (ringLMod‘𝑅))
15	14	fveq2d 6195	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) ∧ 𝑥 ∈ 𝐼) → (0g‘((𝐼 × {(ringLMod‘𝑅)})‘𝑥)) = (0g‘(ringLMod‘𝑅)))
16		frlmbas.z	. . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝑅)
17		rlm0 19197	. . . . . . . . . . . . 13 ⊢ (0g‘𝑅) = (0g‘(ringLMod‘𝑅))
18	16, 17	eqtri 2644	. . . . . . . . . . . 12 ⊢ 0 = (0g‘(ringLMod‘𝑅))
19	15, 18	syl6eqr 2674	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) ∧ 𝑥 ∈ 𝐼) → (0g‘((𝐼 × {(ringLMod‘𝑅)})‘𝑥)) = 0 )
20	12, 19	eqtrd 2656	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((0g ∘ (𝐼 × {(ringLMod‘𝑅)}))‘𝑥) = 0 )
21	20	neeq2d 2854	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((𝑘‘𝑥) ≠ ((0g ∘ (𝐼 × {(ringLMod‘𝑅)}))‘𝑥) ↔ (𝑘‘𝑥) ≠ 0 ))
22	21	rabbidva 3188	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) → {𝑥 ∈ 𝐼 ∣ (𝑘‘𝑥) ≠ ((0g ∘ (𝐼 × {(ringLMod‘𝑅)}))‘𝑥)} = {𝑥 ∈ 𝐼 ∣ (𝑘‘𝑥) ≠ 0 })
23		elmapfn 7880	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑁 ↑𝑚 𝐼) → 𝑘 Fn 𝐼)
24	23	adantl 482	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) → 𝑘 Fn 𝐼)
25		fn0g 17262	. . . . . . . . . 10 ⊢ 0g Fn V
26		ssv 3625	. . . . . . . . . 10 ⊢ ran (𝐼 × {(ringLMod‘𝑅)}) ⊆ V
27		fnco 5999	. . . . . . . . . 10 ⊢ ((0g Fn V ∧ (𝐼 × {(ringLMod‘𝑅)}) Fn 𝐼 ∧ ran (𝐼 × {(ringLMod‘𝑅)}) ⊆ V) → (0g ∘ (𝐼 × {(ringLMod‘𝑅)})) Fn 𝐼)
28	25, 3, 26, 27	mp3an 1424	. . . . . . . . 9 ⊢ (0g ∘ (𝐼 × {(ringLMod‘𝑅)})) Fn 𝐼
29		fndmdif 6321	. . . . . . . . 9 ⊢ ((𝑘 Fn 𝐼 ∧ (0g ∘ (𝐼 × {(ringLMod‘𝑅)})) Fn 𝐼) → dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) = {𝑥 ∈ 𝐼 ∣ (𝑘‘𝑥) ≠ ((0g ∘ (𝐼 × {(ringLMod‘𝑅)}))‘𝑥)})
30	24, 28, 29	sylancl 694	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) → dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) = {𝑥 ∈ 𝐼 ∣ (𝑘‘𝑥) ≠ ((0g ∘ (𝐼 × {(ringLMod‘𝑅)}))‘𝑥)})
31		simplr 792	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) → 𝐼 ∈ 𝑊)
32		fvex 6201	. . . . . . . . . . 11 ⊢ (0g‘𝑅) ∈ V
33	16, 32	eqeltri 2697	. . . . . . . . . 10 ⊢ 0 ∈ V
34	33	a1i 11	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) → 0 ∈ V)
35		suppvalfn 7302	. . . . . . . . 9 ⊢ ((𝑘 Fn 𝐼 ∧ 𝐼 ∈ 𝑊 ∧ 0 ∈ V) → (𝑘 supp 0 ) = {𝑥 ∈ 𝐼 ∣ (𝑘‘𝑥) ≠ 0 })
36	24, 31, 34, 35	syl3anc 1326	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) → (𝑘 supp 0 ) = {𝑥 ∈ 𝐼 ∣ (𝑘‘𝑥) ≠ 0 })
37	22, 30, 36	3eqtr4d 2666	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) → dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) = (𝑘 supp 0 ))
38	37	eleq1d 2686	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) → (dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin ↔ (𝑘 supp 0 ) ∈ Fin))
39		elmapfun 7881	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑁 ↑𝑚 𝐼) → Fun 𝑘)
40		id 22	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑁 ↑𝑚 𝐼) → 𝑘 ∈ (𝑁 ↑𝑚 𝐼))
41	33	a1i 11	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑁 ↑𝑚 𝐼) → 0 ∈ V)
42	39, 40, 41	3jca 1242	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑁 ↑𝑚 𝐼) → (Fun 𝑘 ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼) ∧ 0 ∈ V))
43	42	adantl 482	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) → (Fun 𝑘 ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼) ∧ 0 ∈ V))
44		funisfsupp 8280	. . . . . . 7 ⊢ ((Fun 𝑘 ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼) ∧ 0 ∈ V) → (𝑘 finSupp 0 ↔ (𝑘 supp 0 ) ∈ Fin))
45	43, 44	syl 17	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) → (𝑘 finSupp 0 ↔ (𝑘 supp 0 ) ∈ Fin))
46	38, 45	bitr4d 271	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝑘 ∈ (𝑁 ↑𝑚 𝐼)) → (dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin ↔ 𝑘 finSupp 0 ))
47	46	rabbidva 3188	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → {𝑘 ∈ (𝑁 ↑𝑚 𝐼) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin} = {𝑘 ∈ (𝑁 ↑𝑚 𝐼) ∣ 𝑘 finSupp 0 })
48		eqid 2622	. . . . . . . . 9 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = ((ringLMod‘𝑅) ↑s 𝐼)
49		frlmbas.n	. . . . . . . . . 10 ⊢ 𝑁 = (Base‘𝑅)
50		rlmbas 19195	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅))
51	49, 50	eqtri 2644	. . . . . . . . 9 ⊢ 𝑁 = (Base‘(ringLMod‘𝑅))
52	48, 51	pwsbas 16147	. . . . . . . 8 ⊢ (((ringLMod‘𝑅) ∈ V ∧ 𝐼 ∈ 𝑊) → (𝑁 ↑𝑚 𝐼) = (Base‘((ringLMod‘𝑅) ↑s 𝐼)))
53	1, 52	mpan 706	. . . . . . 7 ⊢ (𝐼 ∈ 𝑊 → (𝑁 ↑𝑚 𝐼) = (Base‘((ringLMod‘𝑅) ↑s 𝐼)))
54	53	adantl 482	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑁 ↑𝑚 𝐼) = (Base‘((ringLMod‘𝑅) ↑s 𝐼)))
55		eqid 2622	. . . . . . . . . . 11 ⊢ (Scalar‘(ringLMod‘𝑅)) = (Scalar‘(ringLMod‘𝑅))
56	48, 55	pwsval 16146	. . . . . . . . . 10 ⊢ (((ringLMod‘𝑅) ∈ V ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)})))
57	1, 56	mpan 706	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑊 → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)})))
58	57	adantl 482	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)})))
59		rlmsca 19200	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → 𝑅 = (Scalar‘(ringLMod‘𝑅)))
60	59	adantr 481	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘(ringLMod‘𝑅)))
61	60	oveq1d 6665	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅Xs(𝐼 × {(ringLMod‘𝑅)})) = ((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)})))
62	58, 61	eqtr4d 2659	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = (𝑅Xs(𝐼 × {(ringLMod‘𝑅)})))
63	62	fveq2d 6195	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (Base‘((ringLMod‘𝑅) ↑s 𝐼)) = (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))))
64	54, 63	eqtrd 2656	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑁 ↑𝑚 𝐼) = (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))))
65		rabeq 3192	. . . . 5 ⊢ ((𝑁 ↑𝑚 𝐼) = (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) → {𝑘 ∈ (𝑁 ↑𝑚 𝐼) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin} = {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin})
66	64, 65	syl 17	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → {𝑘 ∈ (𝑁 ↑𝑚 𝐼) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin} = {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin})
67	47, 66	eqtr3d 2658	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → {𝑘 ∈ (𝑁 ↑𝑚 𝐼) ∣ 𝑘 finSupp 0 } = {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin})
68	9, 67	syl5eq 2668	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐵 = {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin})
69		frlmval.f	. . . 4 ⊢ 𝐹 = (𝑅 freeLMod 𝐼)
70	69	frlmval 20092	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))
71	70	fveq2d 6195	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (Base‘𝐹) = (Base‘(𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))))
72	8, 68, 71	3eqtr4d 2666	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐵 = (Base‘𝐹))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  {crab 2916  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  {csn 4177   class class class wbr 4653   × cxp 5112  dom cdm 5114  ran crn 5115   ∘ ccom 5118  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650   supp csupp 7295   ↑_𝑚 cmap 7857  Fincfn 7955   finSupp cfsupp 8275  Basecbs 15857  Scalarcsca 15944  0_gc0g 16100  X_scprds 16106   ↑_s cpws 16107  ringLModcrglmod 19169   ⊕_m cdsmm 20075   freeLMod cfrlm 20090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-pws 16110  df-sra 19172  df-rgmod 19173  df-dsmm 20076  df-frlm 20091

This theorem is referenced by:  frlmelbas  20100  frlmfibas  20105  ellspd  20141  islindf4  20177  rrxbase  23176  rrxds  23181  frlmpwfi  37668