Theorem mplsubglem 21915

Description: If 𝐴 is an ideal of sets (a nonempty collection closed under subset and binary union) of the set 𝐷 of finite bags (the primary applications being 𝐴 = Fin and 𝐴 = 𝒫 𝐵 for some 𝐵), then the set of all power series whose coefficient functions are supported on an element of 𝐴 is a subgroup of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 16-Jul-2019.)

Hypotheses

Ref	Expression
mplsubglem.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
mplsubglem.b	⊢ 𝐵 = (Base‘𝑆)
mplsubglem.z	⊢ 0 = (0g‘𝑅)
mplsubglem.d	⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
mplsubglem.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mplsubglem.0	⊢ (𝜑 → ∅ ∈ 𝐴)
mplsubglem.a	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∪ 𝑦) ∈ 𝐴)
mplsubglem.y	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ 𝐴)
mplsubglem.u	⊢ (𝜑 → 𝑈 = {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴})
mplsubglem.r	⊢ (𝜑 → 𝑅 ∈ Grp)

Assertion

Ref	Expression
mplsubglem	⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑆))

Distinct variable groups:   𝑓,𝑔,𝑥,𝑦, 0   𝐴,𝑓,𝑔,𝑥,𝑦   𝐵,𝑓,𝑔   𝐷,𝑔   𝑓,𝐼   𝜑,𝑥,𝑦   𝑆,𝑓,𝑔,𝑦

Allowed substitution hints:   𝜑(𝑓,𝑔)   𝐵(𝑥,𝑦)   𝐷(𝑥,𝑦,𝑓)   𝑅(𝑥,𝑦,𝑓,𝑔)   𝑆(𝑥)   𝑈(𝑥,𝑦,𝑓,𝑔)   𝐼(𝑥,𝑦,𝑔)   𝑊(𝑥,𝑦,𝑓,𝑔)

Proof of Theorem mplsubglem

Dummy variables 𝑘 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mplsubglem.u	. . 3 ⊢ (𝜑 → 𝑈 = {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴})
2		ssrab2 4046	. . 3 ⊢ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ⊆ 𝐵
3	1, 2	eqsstrdi 3994	. 2 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
4		mplsubglem.s	. . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
5		mplsubglem.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
6		mplsubglem.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp)
7		mplsubglem.d	. . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
8		mplsubglem.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
9		mplsubglem.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑆)
10	4, 5, 6, 7, 8, 9	psr0cl 21868	. . . 4 ⊢ (𝜑 → (𝐷 × { 0 }) ∈ 𝐵)
11		eqid 2730	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	11, 8	grpidcl 18904	. . . . . . . 8 ⊢ (𝑅 ∈ Grp → 0 ∈ (Base‘𝑅))
13		fconst6g 6752	. . . . . . . 8 ⊢ ( 0 ∈ (Base‘𝑅) → (𝐷 × { 0 }):𝐷⟶(Base‘𝑅))
14	6, 12, 13	3syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐷 × { 0 }):𝐷⟶(Base‘𝑅))
15		eldifi 4097	. . . . . . . . 9 ⊢ (𝑢 ∈ (𝐷 ∖ ∅) → 𝑢 ∈ 𝐷)
16	8	fvexi 6875	. . . . . . . . . 10 ⊢ 0 ∈ V
17	16	fvconst2 7181	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐷 → ((𝐷 × { 0 })‘𝑢) = 0 )
18	15, 17	syl 17	. . . . . . . 8 ⊢ (𝑢 ∈ (𝐷 ∖ ∅) → ((𝐷 × { 0 })‘𝑢) = 0 )
19	18	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐷 ∖ ∅)) → ((𝐷 × { 0 })‘𝑢) = 0 )
20	14, 19	suppss 8176	. . . . . 6 ⊢ (𝜑 → ((𝐷 × { 0 }) supp 0 ) ⊆ ∅)
21		ss0 4368	. . . . . 6 ⊢ (((𝐷 × { 0 }) supp 0 ) ⊆ ∅ → ((𝐷 × { 0 }) supp 0 ) = ∅)
22	20, 21	syl 17	. . . . 5 ⊢ (𝜑 → ((𝐷 × { 0 }) supp 0 ) = ∅)
23		mplsubglem.0	. . . . 5 ⊢ (𝜑 → ∅ ∈ 𝐴)
24	22, 23	eqeltrd 2829	. . . 4 ⊢ (𝜑 → ((𝐷 × { 0 }) supp 0 ) ∈ 𝐴)
25	1	eleq2d 2815	. . . . 5 ⊢ (𝜑 → ((𝐷 × { 0 }) ∈ 𝑈 ↔ (𝐷 × { 0 }) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
26		oveq1 7397	. . . . . . 7 ⊢ (𝑔 = (𝐷 × { 0 }) → (𝑔 supp 0 ) = ((𝐷 × { 0 }) supp 0 ))
27	26	eleq1d 2814	. . . . . 6 ⊢ (𝑔 = (𝐷 × { 0 }) → ((𝑔 supp 0 ) ∈ 𝐴 ↔ ((𝐷 × { 0 }) supp 0 ) ∈ 𝐴))
28	27	elrab 3662	. . . . 5 ⊢ ((𝐷 × { 0 }) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ↔ ((𝐷 × { 0 }) ∈ 𝐵 ∧ ((𝐷 × { 0 }) supp 0 ) ∈ 𝐴))
29	25, 28	bitrdi 287	. . . 4 ⊢ (𝜑 → ((𝐷 × { 0 }) ∈ 𝑈 ↔ ((𝐷 × { 0 }) ∈ 𝐵 ∧ ((𝐷 × { 0 }) supp 0 ) ∈ 𝐴)))
30	10, 24, 29	mpbir2and 713	. . 3 ⊢ (𝜑 → (𝐷 × { 0 }) ∈ 𝑈)
31	30	ne0d 4308	. 2 ⊢ (𝜑 → 𝑈 ≠ ∅)
32		eqid 2730	. . . . . . 7 ⊢ (+g‘𝑆) = (+g‘𝑆)
33	6	grpmgmd 18900	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Mgm)
34	33	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑅 ∈ Mgm)
35	1	eleq2d 2815	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ 𝑈 ↔ 𝑢 ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
36		oveq1 7397	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑢 → (𝑔 supp 0 ) = (𝑢 supp 0 ))
37	36	eleq1d 2814	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑢 → ((𝑔 supp 0 ) ∈ 𝐴 ↔ (𝑢 supp 0 ) ∈ 𝐴))
38	37	elrab 3662	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ↔ (𝑢 ∈ 𝐵 ∧ (𝑢 supp 0 ) ∈ 𝐴))
39	35, 38	bitrdi 287	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 ∈ 𝑈 ↔ (𝑢 ∈ 𝐵 ∧ (𝑢 supp 0 ) ∈ 𝐴)))
40	39	biimpa 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 ∈ 𝐵 ∧ (𝑢 supp 0 ) ∈ 𝐴))
41	40	simpld 494	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ 𝐵)
42	41	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑢 ∈ 𝐵)
43	1	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 = {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴})
44	43	eleq2d 2815	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑣 ∈ 𝑈 ↔ 𝑣 ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
45		oveq1 7397	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑣 → (𝑔 supp 0 ) = (𝑣 supp 0 ))
46	45	eleq1d 2814	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑣 → ((𝑔 supp 0 ) ∈ 𝐴 ↔ (𝑣 supp 0 ) ∈ 𝐴))
47	46	elrab 3662	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ↔ (𝑣 ∈ 𝐵 ∧ (𝑣 supp 0 ) ∈ 𝐴))
48	44, 47	bitrdi 287	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑣 ∈ 𝑈 ↔ (𝑣 ∈ 𝐵 ∧ (𝑣 supp 0 ) ∈ 𝐴)))
49	48	biimpa 476	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → (𝑣 ∈ 𝐵 ∧ (𝑣 supp 0 ) ∈ 𝐴))
50	49	simpld 494	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ 𝐵)
51	4, 9, 32, 34, 42, 50	psraddcl 21854	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → (𝑢(+g‘𝑆)𝑣) ∈ 𝐵)
52		ovexd 7425	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ((𝑢(+g‘𝑆)𝑣) supp 0 ) ∈ V)
53		sseq2 3976	. . . . . . . . . 10 ⊢ (𝑥 = ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))))
54	53	imbi1d 341	. . . . . . . . 9 ⊢ (𝑥 = ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → ((𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴) ↔ (𝑦 ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → 𝑦 ∈ 𝐴)))
55	54	albidv 1920	. . . . . . . 8 ⊢ (𝑥 = ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → (∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴) ↔ ∀𝑦(𝑦 ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → 𝑦 ∈ 𝐴)))
56		mplsubglem.y	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ 𝐴)
57	56	expr 456	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
58	57	alrimiv 1927	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
59	58	ralrimiva 3126	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
60	59	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
61	40	simprd 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 supp 0 ) ∈ 𝐴)
62	61	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → (𝑢 supp 0 ) ∈ 𝐴)
63	49	simprd 495	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → (𝑣 supp 0 ) ∈ 𝐴)
64		mplsubglem.a	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∪ 𝑦) ∈ 𝐴)
65	64	ralrimivva 3181	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴)
66	65	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴)
67		uneq1 4127	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑢 supp 0 ) → (𝑥 ∪ 𝑦) = ((𝑢 supp 0 ) ∪ 𝑦))
68	67	eleq1d 2814	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 supp 0 ) → ((𝑥 ∪ 𝑦) ∈ 𝐴 ↔ ((𝑢 supp 0 ) ∪ 𝑦) ∈ 𝐴))
69		uneq2 4128	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑣 supp 0 ) → ((𝑢 supp 0 ) ∪ 𝑦) = ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))
70	69	eleq1d 2814	. . . . . . . . . 10 ⊢ (𝑦 = (𝑣 supp 0 ) → (((𝑢 supp 0 ) ∪ 𝑦) ∈ 𝐴 ↔ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) ∈ 𝐴))
71	68, 70	rspc2va 3603	. . . . . . . . 9 ⊢ ((((𝑢 supp 0 ) ∈ 𝐴 ∧ (𝑣 supp 0 ) ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴) → ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) ∈ 𝐴)
72	62, 63, 66, 71	syl21anc 837	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) ∈ 𝐴)
73	55, 60, 72	rspcdva 3592	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ∀𝑦(𝑦 ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → 𝑦 ∈ 𝐴))
74	4, 11, 7, 9, 51	psrelbas 21850	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → (𝑢(+g‘𝑆)𝑣):𝐷⟶(Base‘𝑅))
75		eqid 2730	. . . . . . . . . . . 12 ⊢ (+g‘𝑅) = (+g‘𝑅)
76	4, 9, 75, 32, 42, 50	psradd 21853	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → (𝑢(+g‘𝑆)𝑣) = (𝑢 ∘f (+g‘𝑅)𝑣))
77	76	fveq1d 6863	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ((𝑢(+g‘𝑆)𝑣)‘𝑘) = ((𝑢 ∘f (+g‘𝑅)𝑣)‘𝑘))
78	77	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))) → ((𝑢(+g‘𝑆)𝑣)‘𝑘) = ((𝑢 ∘f (+g‘𝑅)𝑣)‘𝑘))
79		eldifi 4097	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))) → 𝑘 ∈ 𝐷)
80	4, 11, 7, 9, 41	psrelbas 21850	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢:𝐷⟶(Base‘𝑅))
81	80	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑢:𝐷⟶(Base‘𝑅))
82	81	ffnd 6692	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑢 Fn 𝐷)
83	4, 11, 7, 9, 50	psrelbas 21850	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑣:𝐷⟶(Base‘𝑅))
84	83	ffnd 6692	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑣 Fn 𝐷)
85		ovex 7423	. . . . . . . . . . . . 13 ⊢ (ℕ0 ↑m 𝐼) ∈ V
86	7, 85	rabex2 5299	. . . . . . . . . . . 12 ⊢ 𝐷 ∈ V
87	86	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝐷 ∈ V)
88		inidm 4193	. . . . . . . . . . 11 ⊢ (𝐷 ∩ 𝐷) = 𝐷
89		eqidd 2731	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ 𝐷) → (𝑢‘𝑘) = (𝑢‘𝑘))
90		eqidd 2731	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ 𝐷) → (𝑣‘𝑘) = (𝑣‘𝑘))
91	82, 84, 87, 87, 88, 89, 90	ofval 7667	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ 𝐷) → ((𝑢 ∘f (+g‘𝑅)𝑣)‘𝑘) = ((𝑢‘𝑘)(+g‘𝑅)(𝑣‘𝑘)))
92	79, 91	sylan2 593	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))) → ((𝑢 ∘f (+g‘𝑅)𝑣)‘𝑘) = ((𝑢‘𝑘)(+g‘𝑅)(𝑣‘𝑘)))
93		ssun1 4144	. . . . . . . . . . . . . 14 ⊢ (𝑢 supp 0 ) ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))
94		sscon 4109	. . . . . . . . . . . . . 14 ⊢ ((𝑢 supp 0 ) ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))) ⊆ (𝐷 ∖ (𝑢 supp 0 )))
95	93, 94	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))) ⊆ (𝐷 ∖ (𝑢 supp 0 ))
96	95	sseli 3945	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))) → 𝑘 ∈ (𝐷 ∖ (𝑢 supp 0 )))
97		ssidd 3973	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 supp 0 ) ⊆ (𝑢 supp 0 ))
98	86	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝐷 ∈ V)
99	16	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 0 ∈ V)
100	80, 97, 98, 99	suppssr 8177	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ (𝑢 supp 0 ))) → (𝑢‘𝑘) = 0 )
101	100	adantlr 715	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ (𝑢 supp 0 ))) → (𝑢‘𝑘) = 0 )
102	96, 101	sylan2 593	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))) → (𝑢‘𝑘) = 0 )
103		ssun2 4145	. . . . . . . . . . . . . 14 ⊢ (𝑣 supp 0 ) ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))
104		sscon 4109	. . . . . . . . . . . . . 14 ⊢ ((𝑣 supp 0 ) ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))) ⊆ (𝐷 ∖ (𝑣 supp 0 )))
105	103, 104	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))) ⊆ (𝐷 ∖ (𝑣 supp 0 ))
106	105	sseli 3945	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))) → 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 )))
107		ssidd 3973	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → (𝑣 supp 0 ) ⊆ (𝑣 supp 0 ))
108	16	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 0 ∈ V)
109	83, 107, 87, 108	suppssr 8177	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → (𝑣‘𝑘) = 0 )
110	106, 109	sylan2 593	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))) → (𝑣‘𝑘) = 0 )
111	102, 110	oveq12d 7408	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))) → ((𝑢‘𝑘)(+g‘𝑅)(𝑣‘𝑘)) = ( 0 (+g‘𝑅) 0 ))
112	6	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑅 ∈ Grp)
113	11, 75, 8	grplid 18906	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Grp ∧ 0 ∈ (Base‘𝑅)) → ( 0 (+g‘𝑅) 0 ) = 0 )
114	112, 12, 113	syl2anc2 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ( 0 (+g‘𝑅) 0 ) = 0 )
115	114	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))) → ( 0 (+g‘𝑅) 0 ) = 0 )
116	111, 115	eqtrd 2765	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))) → ((𝑢‘𝑘)(+g‘𝑅)(𝑣‘𝑘)) = 0 )
117	78, 92, 116	3eqtrd 2769	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))) → ((𝑢(+g‘𝑆)𝑣)‘𝑘) = 0 )
118	74, 117	suppss 8176	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ((𝑢(+g‘𝑆)𝑣) supp 0 ) ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))
119		sseq1 3975	. . . . . . . . 9 ⊢ (𝑦 = ((𝑢(+g‘𝑆)𝑣) supp 0 ) → (𝑦 ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) ↔ ((𝑢(+g‘𝑆)𝑣) supp 0 ) ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))))
120		eleq1 2817	. . . . . . . . 9 ⊢ (𝑦 = ((𝑢(+g‘𝑆)𝑣) supp 0 ) → (𝑦 ∈ 𝐴 ↔ ((𝑢(+g‘𝑆)𝑣) supp 0 ) ∈ 𝐴))
121	119, 120	imbi12d 344	. . . . . . . 8 ⊢ (𝑦 = ((𝑢(+g‘𝑆)𝑣) supp 0 ) → ((𝑦 ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → 𝑦 ∈ 𝐴) ↔ (((𝑢(+g‘𝑆)𝑣) supp 0 ) ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → ((𝑢(+g‘𝑆)𝑣) supp 0 ) ∈ 𝐴)))
122	121	spcgv 3565	. . . . . . 7 ⊢ (((𝑢(+g‘𝑆)𝑣) supp 0 ) ∈ V → (∀𝑦(𝑦 ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → 𝑦 ∈ 𝐴) → (((𝑢(+g‘𝑆)𝑣) supp 0 ) ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → ((𝑢(+g‘𝑆)𝑣) supp 0 ) ∈ 𝐴)))
123	52, 73, 118, 122	syl3c 66	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ((𝑢(+g‘𝑆)𝑣) supp 0 ) ∈ 𝐴)
124	1	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑈 = {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴})
125	124	eleq2d 2815	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ((𝑢(+g‘𝑆)𝑣) ∈ 𝑈 ↔ (𝑢(+g‘𝑆)𝑣) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
126		oveq1 7397	. . . . . . . . 9 ⊢ (𝑔 = (𝑢(+g‘𝑆)𝑣) → (𝑔 supp 0 ) = ((𝑢(+g‘𝑆)𝑣) supp 0 ))
127	126	eleq1d 2814	. . . . . . . 8 ⊢ (𝑔 = (𝑢(+g‘𝑆)𝑣) → ((𝑔 supp 0 ) ∈ 𝐴 ↔ ((𝑢(+g‘𝑆)𝑣) supp 0 ) ∈ 𝐴))
128	127	elrab 3662	. . . . . . 7 ⊢ ((𝑢(+g‘𝑆)𝑣) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ↔ ((𝑢(+g‘𝑆)𝑣) ∈ 𝐵 ∧ ((𝑢(+g‘𝑆)𝑣) supp 0 ) ∈ 𝐴))
129	125, 128	bitrdi 287	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ((𝑢(+g‘𝑆)𝑣) ∈ 𝑈 ↔ ((𝑢(+g‘𝑆)𝑣) ∈ 𝐵 ∧ ((𝑢(+g‘𝑆)𝑣) supp 0 ) ∈ 𝐴)))
130	51, 123, 129	mpbir2and 713	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → (𝑢(+g‘𝑆)𝑣) ∈ 𝑈)
131	130	ralrimiva 3126	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∀𝑣 ∈ 𝑈 (𝑢(+g‘𝑆)𝑣) ∈ 𝑈)
132	4, 5, 6	psrgrp 21872	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Grp)
133		eqid 2730	. . . . . . 7 ⊢ (invg‘𝑆) = (invg‘𝑆)
134	9, 133	grpinvcl 18926	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑢 ∈ 𝐵) → ((invg‘𝑆)‘𝑢) ∈ 𝐵)
135	132, 41, 134	syl2an2r 685	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((invg‘𝑆)‘𝑢) ∈ 𝐵)
136		ovexd 7425	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (((invg‘𝑆)‘𝑢) supp 0 ) ∈ V)
137		sseq2 3976	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 supp 0 ) → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ (𝑢 supp 0 )))
138	137	imbi1d 341	. . . . . . . 8 ⊢ (𝑥 = (𝑢 supp 0 ) → ((𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴) ↔ (𝑦 ⊆ (𝑢 supp 0 ) → 𝑦 ∈ 𝐴)))
139	138	albidv 1920	. . . . . . 7 ⊢ (𝑥 = (𝑢 supp 0 ) → (∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴) ↔ ∀𝑦(𝑦 ⊆ (𝑢 supp 0 ) → 𝑦 ∈ 𝐴)))
140	59	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
141	139, 140, 61	rspcdva 3592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∀𝑦(𝑦 ⊆ (𝑢 supp 0 ) → 𝑦 ∈ 𝐴))
142	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝐼 ∈ 𝑊)
143	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑅 ∈ Grp)
144		eqid 2730	. . . . . . . . 9 ⊢ (invg‘𝑅) = (invg‘𝑅)
145	4, 142, 143, 7, 144, 9, 133, 41	psrneg 21875	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((invg‘𝑆)‘𝑢) = ((invg‘𝑅) ∘ 𝑢))
146	145	oveq1d 7405	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (((invg‘𝑆)‘𝑢) supp 0 ) = (((invg‘𝑅) ∘ 𝑢) supp 0 ))
147	11, 144	grpinvfn 18920	. . . . . . . . 9 ⊢ (invg‘𝑅) Fn (Base‘𝑅)
148	147	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (invg‘𝑅) Fn (Base‘𝑅))
149	8, 144	grpinvid 18938	. . . . . . . . 9 ⊢ (𝑅 ∈ Grp → ((invg‘𝑅)‘ 0 ) = 0 )
150	143, 149	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((invg‘𝑅)‘ 0 ) = 0 )
151	148, 80, 98, 99, 150	suppcoss 8189	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (((invg‘𝑅) ∘ 𝑢) supp 0 ) ⊆ (𝑢 supp 0 ))
152	146, 151	eqsstrd 3984	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (((invg‘𝑆)‘𝑢) supp 0 ) ⊆ (𝑢 supp 0 ))
153		sseq1 3975	. . . . . . . 8 ⊢ (𝑦 = (((invg‘𝑆)‘𝑢) supp 0 ) → (𝑦 ⊆ (𝑢 supp 0 ) ↔ (((invg‘𝑆)‘𝑢) supp 0 ) ⊆ (𝑢 supp 0 )))
154		eleq1 2817	. . . . . . . 8 ⊢ (𝑦 = (((invg‘𝑆)‘𝑢) supp 0 ) → (𝑦 ∈ 𝐴 ↔ (((invg‘𝑆)‘𝑢) supp 0 ) ∈ 𝐴))
155	153, 154	imbi12d 344	. . . . . . 7 ⊢ (𝑦 = (((invg‘𝑆)‘𝑢) supp 0 ) → ((𝑦 ⊆ (𝑢 supp 0 ) → 𝑦 ∈ 𝐴) ↔ ((((invg‘𝑆)‘𝑢) supp 0 ) ⊆ (𝑢 supp 0 ) → (((invg‘𝑆)‘𝑢) supp 0 ) ∈ 𝐴)))
156	155	spcgv 3565	. . . . . 6 ⊢ ((((invg‘𝑆)‘𝑢) supp 0 ) ∈ V → (∀𝑦(𝑦 ⊆ (𝑢 supp 0 ) → 𝑦 ∈ 𝐴) → ((((invg‘𝑆)‘𝑢) supp 0 ) ⊆ (𝑢 supp 0 ) → (((invg‘𝑆)‘𝑢) supp 0 ) ∈ 𝐴)))
157	136, 141, 152, 156	syl3c 66	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (((invg‘𝑆)‘𝑢) supp 0 ) ∈ 𝐴)
158	43	eleq2d 2815	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (((invg‘𝑆)‘𝑢) ∈ 𝑈 ↔ ((invg‘𝑆)‘𝑢) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
159		oveq1 7397	. . . . . . . 8 ⊢ (𝑔 = ((invg‘𝑆)‘𝑢) → (𝑔 supp 0 ) = (((invg‘𝑆)‘𝑢) supp 0 ))
160	159	eleq1d 2814	. . . . . . 7 ⊢ (𝑔 = ((invg‘𝑆)‘𝑢) → ((𝑔 supp 0 ) ∈ 𝐴 ↔ (((invg‘𝑆)‘𝑢) supp 0 ) ∈ 𝐴))
161	160	elrab 3662	. . . . . 6 ⊢ (((invg‘𝑆)‘𝑢) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ↔ (((invg‘𝑆)‘𝑢) ∈ 𝐵 ∧ (((invg‘𝑆)‘𝑢) supp 0 ) ∈ 𝐴))
162	158, 161	bitrdi 287	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (((invg‘𝑆)‘𝑢) ∈ 𝑈 ↔ (((invg‘𝑆)‘𝑢) ∈ 𝐵 ∧ (((invg‘𝑆)‘𝑢) supp 0 ) ∈ 𝐴)))
163	135, 157, 162	mpbir2and 713	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((invg‘𝑆)‘𝑢) ∈ 𝑈)
164	131, 163	jca 511	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (∀𝑣 ∈ 𝑈 (𝑢(+g‘𝑆)𝑣) ∈ 𝑈 ∧ ((invg‘𝑆)‘𝑢) ∈ 𝑈))
165	164	ralrimiva 3126	. 2 ⊢ (𝜑 → ∀𝑢 ∈ 𝑈 (∀𝑣 ∈ 𝑈 (𝑢(+g‘𝑆)𝑣) ∈ 𝑈 ∧ ((invg‘𝑆)‘𝑢) ∈ 𝑈))
166	9, 32, 133	issubg2 19080	. . 3 ⊢ (𝑆 ∈ Grp → (𝑈 ∈ (SubGrp‘𝑆) ↔ (𝑈 ⊆ 𝐵 ∧ 𝑈 ≠ ∅ ∧ ∀𝑢 ∈ 𝑈 (∀𝑣 ∈ 𝑈 (𝑢(+g‘𝑆)𝑣) ∈ 𝑈 ∧ ((invg‘𝑆)‘𝑢) ∈ 𝑈))))
167	132, 166	syl 17	. 2 ⊢ (𝜑 → (𝑈 ∈ (SubGrp‘𝑆) ↔ (𝑈 ⊆ 𝐵 ∧ 𝑈 ≠ ∅ ∧ ∀𝑢 ∈ 𝑈 (∀𝑣 ∈ 𝑈 (𝑢(+g‘𝑆)𝑣) ∈ 𝑈 ∧ ((invg‘𝑆)‘𝑢) ∈ 𝑈))))
168	3, 31, 165, 167	mpbir3and 1343	1 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  {crab 3408  Vcvv 3450   ∖ cdif 3914   ∪ cun 3915   ⊆ wss 3917  ∅c0 4299  {csn 4592   × cxp 5639  ^◡ccnv 5640   “ cima 5644   ∘ ccom 5645   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∘_f cof 7654   supp csupp 8142   ↑_m cmap 8802  Fincfn 8921  ℕcn 12193  ℕ₀cn0 12449  Basecbs 17186  +_gcplusg 17227  0_gc0g 17409  Mgmcmgm 18572  Grpcgrp 18872  inv_gcminusg 18873  SubGrpcsubg 19059   mPwSer cmps 21820

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-sup 9400  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-hom 17251  df-cco 17252  df-0g 17411  df-prds 17417  df-pws 17419  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-grp 18875  df-minusg 18876  df-subg 19062  df-psr 21825

This theorem is referenced by:  mpllsslem  21916  mplsubg  21918