Theorem mplsubglem 20208

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 4055	. . 3 ⊢ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ⊆ 𝐵
3	1, 2	eqsstrdi 4020	. 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 20168	. . . 4 ⊢ (𝜑 → (𝐷 × { 0 }) ∈ 𝐵)
11		eqid 2821	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	11, 8	grpidcl 18125	. . . . . . . 8 ⊢ (𝑅 ∈ Grp → 0 ∈ (Base‘𝑅))
13		fconst6g 6562	. . . . . . . 8 ⊢ ( 0 ∈ (Base‘𝑅) → (𝐷 × { 0 }):𝐷⟶(Base‘𝑅))
14	6, 12, 13	3syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐷 × { 0 }):𝐷⟶(Base‘𝑅))
15		eldifi 4102	. . . . . . . . 9 ⊢ (𝑢 ∈ (𝐷 ∖ ∅) → 𝑢 ∈ 𝐷)
16	8	fvexi 6678	. . . . . . . . . 10 ⊢ 0 ∈ V
17	16	fvconst2 6960	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐷 → ((𝐷 × { 0 })‘𝑢) = 0 )
18	15, 17	syl 17	. . . . . . . 8 ⊢ (𝑢 ∈ (𝐷 ∖ ∅) → ((𝐷 × { 0 })‘𝑢) = 0 )
19	18	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐷 ∖ ∅)) → ((𝐷 × { 0 })‘𝑢) = 0 )
20	14, 19	suppss 7854	. . . . . 6 ⊢ (𝜑 → ((𝐷 × { 0 }) supp 0 ) ⊆ ∅)
21		ss0 4351	. . . . . 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 2913	. . . 4 ⊢ (𝜑 → ((𝐷 × { 0 }) supp 0 ) ∈ 𝐴)
25	1	eleq2d 2898	. . . . 5 ⊢ (𝜑 → ((𝐷 × { 0 }) ∈ 𝑈 ↔ (𝐷 × { 0 }) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
26		oveq1 7157	. . . . . . 7 ⊢ (𝑔 = (𝐷 × { 0 }) → (𝑔 supp 0 ) = ((𝐷 × { 0 }) supp 0 ))
27	26	eleq1d 2897	. . . . . 6 ⊢ (𝑔 = (𝐷 × { 0 }) → ((𝑔 supp 0 ) ∈ 𝐴 ↔ ((𝐷 × { 0 }) supp 0 ) ∈ 𝐴))
28	27	elrab 3679	. . . . 5 ⊢ ((𝐷 × { 0 }) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ↔ ((𝐷 × { 0 }) ∈ 𝐵 ∧ ((𝐷 × { 0 }) supp 0 ) ∈ 𝐴))
29	25, 28	syl6bb 289	. . . 4 ⊢ (𝜑 → ((𝐷 × { 0 }) ∈ 𝑈 ↔ ((𝐷 × { 0 }) ∈ 𝐵 ∧ ((𝐷 × { 0 }) supp 0 ) ∈ 𝐴)))
30	10, 24, 29	mpbir2and 711	. . 3 ⊢ (𝜑 → (𝐷 × { 0 }) ∈ 𝑈)
31	30	ne0d 4300	. 2 ⊢ (𝜑 → 𝑈 ≠ ∅)
32		eqid 2821	. . . . . . 7 ⊢ (+g‘𝑆) = (+g‘𝑆)
33	6	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑅 ∈ Grp)
34	1	eleq2d 2898	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ 𝑈 ↔ 𝑢 ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
35		oveq1 7157	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑢 → (𝑔 supp 0 ) = (𝑢 supp 0 ))
36	35	eleq1d 2897	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑢 → ((𝑔 supp 0 ) ∈ 𝐴 ↔ (𝑢 supp 0 ) ∈ 𝐴))
37	36	elrab 3679	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ↔ (𝑢 ∈ 𝐵 ∧ (𝑢 supp 0 ) ∈ 𝐴))
38	34, 37	syl6bb 289	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 ∈ 𝑈 ↔ (𝑢 ∈ 𝐵 ∧ (𝑢 supp 0 ) ∈ 𝐴)))
39	38	biimpa 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 ∈ 𝐵 ∧ (𝑢 supp 0 ) ∈ 𝐴))
40	39	simpld 497	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ 𝐵)
41	40	adantr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑢 ∈ 𝐵)
42	1	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 = {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴})
43	42	eleq2d 2898	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑣 ∈ 𝑈 ↔ 𝑣 ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
44		oveq1 7157	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑣 → (𝑔 supp 0 ) = (𝑣 supp 0 ))
45	44	eleq1d 2897	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑣 → ((𝑔 supp 0 ) ∈ 𝐴 ↔ (𝑣 supp 0 ) ∈ 𝐴))
46	45	elrab 3679	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ↔ (𝑣 ∈ 𝐵 ∧ (𝑣 supp 0 ) ∈ 𝐴))
47	43, 46	syl6bb 289	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑣 ∈ 𝑈 ↔ (𝑣 ∈ 𝐵 ∧ (𝑣 supp 0 ) ∈ 𝐴)))
48	47	biimpa 479	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → (𝑣 ∈ 𝐵 ∧ (𝑣 supp 0 ) ∈ 𝐴))
49	48	simpld 497	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ 𝐵)
50	4, 9, 32, 33, 41, 49	psraddcl 20157	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → (𝑢(+g‘𝑆)𝑣) ∈ 𝐵)
51		ovexd 7185	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ((𝑢(+g‘𝑆)𝑣) supp 0 ) ∈ V)
52		sseq2 3992	. . . . . . . . . 10 ⊢ (𝑥 = ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))))
53	52	imbi1d 344	. . . . . . . . 9 ⊢ (𝑥 = ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → ((𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴) ↔ (𝑦 ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → 𝑦 ∈ 𝐴)))
54	53	albidv 1917	. . . . . . . 8 ⊢ (𝑥 = ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → (∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴) ↔ ∀𝑦(𝑦 ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → 𝑦 ∈ 𝐴)))
55		mplsubglem.y	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ 𝐴)
56	55	expr 459	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
57	56	alrimiv 1924	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
58	57	ralrimiva 3182	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
59	58	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
60	39	simprd 498	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 supp 0 ) ∈ 𝐴)
61	60	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → (𝑢 supp 0 ) ∈ 𝐴)
62	48	simprd 498	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → (𝑣 supp 0 ) ∈ 𝐴)
63		mplsubglem.a	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∪ 𝑦) ∈ 𝐴)
64	63	ralrimivva 3191	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴)
65	64	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴)
66		uneq1 4131	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑢 supp 0 ) → (𝑥 ∪ 𝑦) = ((𝑢 supp 0 ) ∪ 𝑦))
67	66	eleq1d 2897	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 supp 0 ) → ((𝑥 ∪ 𝑦) ∈ 𝐴 ↔ ((𝑢 supp 0 ) ∪ 𝑦) ∈ 𝐴))
68		uneq2 4132	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑣 supp 0 ) → ((𝑢 supp 0 ) ∪ 𝑦) = ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))
69	68	eleq1d 2897	. . . . . . . . . 10 ⊢ (𝑦 = (𝑣 supp 0 ) → (((𝑢 supp 0 ) ∪ 𝑦) ∈ 𝐴 ↔ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) ∈ 𝐴))
70	67, 69	rspc2va 3633	. . . . . . . . 9 ⊢ ((((𝑢 supp 0 ) ∈ 𝐴 ∧ (𝑣 supp 0 ) ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴) → ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) ∈ 𝐴)
71	61, 62, 65, 70	syl21anc 835	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) ∈ 𝐴)
72	54, 59, 71	rspcdva 3624	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ∀𝑦(𝑦 ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → 𝑦 ∈ 𝐴))
73	4, 11, 7, 9, 50	psrelbas 20153	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → (𝑢(+g‘𝑆)𝑣):𝐷⟶(Base‘𝑅))
74		eqid 2821	. . . . . . . . . . . 12 ⊢ (+g‘𝑅) = (+g‘𝑅)
75	4, 9, 74, 32, 41, 49	psradd 20156	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → (𝑢(+g‘𝑆)𝑣) = (𝑢 ∘f (+g‘𝑅)𝑣))
76	75	fveq1d 6666	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ((𝑢(+g‘𝑆)𝑣)‘𝑘) = ((𝑢 ∘f (+g‘𝑅)𝑣)‘𝑘))
77	76	adantr 483	. . . . . . . . 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))
78		eldifi 4102	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))) → 𝑘 ∈ 𝐷)
79	4, 11, 7, 9, 40	psrelbas 20153	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢:𝐷⟶(Base‘𝑅))
80	79	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑢:𝐷⟶(Base‘𝑅))
81	80	ffnd 6509	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑢 Fn 𝐷)
82	4, 11, 7, 9, 49	psrelbas 20153	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑣:𝐷⟶(Base‘𝑅))
83	82	ffnd 6509	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑣 Fn 𝐷)
84		ovex 7183	. . . . . . . . . . . . 13 ⊢ (ℕ0 ↑m 𝐼) ∈ V
85	7, 84	rabex2 5229	. . . . . . . . . . . 12 ⊢ 𝐷 ∈ V
86	85	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝐷 ∈ V)
87		inidm 4194	. . . . . . . . . . 11 ⊢ (𝐷 ∩ 𝐷) = 𝐷
88		eqidd 2822	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ 𝐷) → (𝑢‘𝑘) = (𝑢‘𝑘))
89		eqidd 2822	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ 𝐷) → (𝑣‘𝑘) = (𝑣‘𝑘))
90	81, 83, 86, 86, 87, 88, 89	ofval 7412	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ 𝐷) → ((𝑢 ∘f (+g‘𝑅)𝑣)‘𝑘) = ((𝑢‘𝑘)(+g‘𝑅)(𝑣‘𝑘)))
91	78, 90	sylan2 594	. . . . . . . . 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)))
92		ssun1 4147	. . . . . . . . . . . . . 14 ⊢ (𝑢 supp 0 ) ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))
93		sscon 4114	. . . . . . . . . . . . . 14 ⊢ ((𝑢 supp 0 ) ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))) ⊆ (𝐷 ∖ (𝑢 supp 0 )))
94	92, 93	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))) ⊆ (𝐷 ∖ (𝑢 supp 0 ))
95	94	sseli 3962	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))) → 𝑘 ∈ (𝐷 ∖ (𝑢 supp 0 )))
96		ssidd 3989	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 supp 0 ) ⊆ (𝑢 supp 0 ))
97	85	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝐷 ∈ V)
98	16	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 0 ∈ V)
99	79, 96, 97, 98	suppssr 7855	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ (𝑢 supp 0 ))) → (𝑢‘𝑘) = 0 )
100	99	adantlr 713	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ (𝑢 supp 0 ))) → (𝑢‘𝑘) = 0 )
101	95, 100	sylan2 594	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))) → (𝑢‘𝑘) = 0 )
102		ssun2 4148	. . . . . . . . . . . . . 14 ⊢ (𝑣 supp 0 ) ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))
103		sscon 4114	. . . . . . . . . . . . . 14 ⊢ ((𝑣 supp 0 ) ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )) → (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))) ⊆ (𝐷 ∖ (𝑣 supp 0 )))
104	102, 103	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))) ⊆ (𝐷 ∖ (𝑣 supp 0 ))
105	104	sseli 3962	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 ))) → 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 )))
106		ssidd 3989	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → (𝑣 supp 0 ) ⊆ (𝑣 supp 0 ))
107	16	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 0 ∈ V)
108	82, 106, 86, 107	suppssr 7855	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → (𝑣‘𝑘) = 0 )
109	105, 108	sylan2 594	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))) → (𝑣‘𝑘) = 0 )
110	101, 109	oveq12d 7168	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))) → ((𝑢‘𝑘)(+g‘𝑅)(𝑣‘𝑘)) = ( 0 (+g‘𝑅) 0 ))
111	11, 74, 8	grplid 18127	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Grp ∧ 0 ∈ (Base‘𝑅)) → ( 0 (+g‘𝑅) 0 ) = 0 )
112	33, 12, 111	syl2anc2 587	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ( 0 (+g‘𝑅) 0 ) = 0 )
113	112	adantr 483	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))) → ( 0 (+g‘𝑅) 0 ) = 0 )
114	110, 113	eqtrd 2856	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))) → ((𝑢‘𝑘)(+g‘𝑅)(𝑣‘𝑘)) = 0 )
115	77, 91, 114	3eqtrd 2860	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))) → ((𝑢(+g‘𝑆)𝑣)‘𝑘) = 0 )
116	73, 115	suppss 7854	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ((𝑢(+g‘𝑆)𝑣) supp 0 ) ⊆ ((𝑢 supp 0 ) ∪ (𝑣 supp 0 )))
117		sseq1 3991	. . . . . . . . 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 ))))
118		eleq1 2900	. . . . . . . . 9 ⊢ (𝑦 = ((𝑢(+g‘𝑆)𝑣) supp 0 ) → (𝑦 ∈ 𝐴 ↔ ((𝑢(+g‘𝑆)𝑣) supp 0 ) ∈ 𝐴))
119	117, 118	imbi12d 347	. . . . . . . 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)))
120	119	spcgv 3594	. . . . . . 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)))
121	51, 72, 116, 120	syl3c 66	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ((𝑢(+g‘𝑆)𝑣) supp 0 ) ∈ 𝐴)
122	1	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → 𝑈 = {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴})
123	122	eleq2d 2898	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ((𝑢(+g‘𝑆)𝑣) ∈ 𝑈 ↔ (𝑢(+g‘𝑆)𝑣) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
124		oveq1 7157	. . . . . . . . 9 ⊢ (𝑔 = (𝑢(+g‘𝑆)𝑣) → (𝑔 supp 0 ) = ((𝑢(+g‘𝑆)𝑣) supp 0 ))
125	124	eleq1d 2897	. . . . . . . 8 ⊢ (𝑔 = (𝑢(+g‘𝑆)𝑣) → ((𝑔 supp 0 ) ∈ 𝐴 ↔ ((𝑢(+g‘𝑆)𝑣) supp 0 ) ∈ 𝐴))
126	125	elrab 3679	. . . . . . 7 ⊢ ((𝑢(+g‘𝑆)𝑣) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ↔ ((𝑢(+g‘𝑆)𝑣) ∈ 𝐵 ∧ ((𝑢(+g‘𝑆)𝑣) supp 0 ) ∈ 𝐴))
127	123, 126	syl6bb 289	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → ((𝑢(+g‘𝑆)𝑣) ∈ 𝑈 ↔ ((𝑢(+g‘𝑆)𝑣) ∈ 𝐵 ∧ ((𝑢(+g‘𝑆)𝑣) supp 0 ) ∈ 𝐴)))
128	50, 121, 127	mpbir2and 711	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) → (𝑢(+g‘𝑆)𝑣) ∈ 𝑈)
129	128	ralrimiva 3182	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∀𝑣 ∈ 𝑈 (𝑢(+g‘𝑆)𝑣) ∈ 𝑈)
130	4, 5, 6	psrgrp 20172	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Grp)
131		eqid 2821	. . . . . . 7 ⊢ (invg‘𝑆) = (invg‘𝑆)
132	9, 131	grpinvcl 18145	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑢 ∈ 𝐵) → ((invg‘𝑆)‘𝑢) ∈ 𝐵)
133	130, 40, 132	syl2an2r 683	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((invg‘𝑆)‘𝑢) ∈ 𝐵)
134		ovexd 7185	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (((invg‘𝑆)‘𝑢) supp 0 ) ∈ V)
135		sseq2 3992	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 supp 0 ) → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ (𝑢 supp 0 )))
136	135	imbi1d 344	. . . . . . . 8 ⊢ (𝑥 = (𝑢 supp 0 ) → ((𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴) ↔ (𝑦 ⊆ (𝑢 supp 0 ) → 𝑦 ∈ 𝐴)))
137	136	albidv 1917	. . . . . . 7 ⊢ (𝑥 = (𝑢 supp 0 ) → (∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴) ↔ ∀𝑦(𝑦 ⊆ (𝑢 supp 0 ) → 𝑦 ∈ 𝐴)))
138	58	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
139	137, 138, 60	rspcdva 3624	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∀𝑦(𝑦 ⊆ (𝑢 supp 0 ) → 𝑦 ∈ 𝐴))
140	4, 11, 7, 9, 133	psrelbas 20153	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((invg‘𝑆)‘𝑢):𝐷⟶(Base‘𝑅))
141	5	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝐼 ∈ 𝑊)
142	6	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑅 ∈ Grp)
143		eqid 2821	. . . . . . . . . . 11 ⊢ (invg‘𝑅) = (invg‘𝑅)
144	4, 141, 142, 7, 143, 9, 131, 40	psrneg 20174	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((invg‘𝑆)‘𝑢) = ((invg‘𝑅) ∘ 𝑢))
145	144	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ (𝑢 supp 0 ))) → ((invg‘𝑆)‘𝑢) = ((invg‘𝑅) ∘ 𝑢))
146	145	fveq1d 6666	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ (𝑢 supp 0 ))) → (((invg‘𝑆)‘𝑢)‘𝑘) = (((invg‘𝑅) ∘ 𝑢)‘𝑘))
147		eldifi 4102	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝐷 ∖ (𝑢 supp 0 )) → 𝑘 ∈ 𝐷)
148		fvco3 6754	. . . . . . . . 9 ⊢ ((𝑢:𝐷⟶(Base‘𝑅) ∧ 𝑘 ∈ 𝐷) → (((invg‘𝑅) ∘ 𝑢)‘𝑘) = ((invg‘𝑅)‘(𝑢‘𝑘)))
149	79, 147, 148	syl2an 597	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ (𝑢 supp 0 ))) → (((invg‘𝑅) ∘ 𝑢)‘𝑘) = ((invg‘𝑅)‘(𝑢‘𝑘)))
150	99	fveq2d 6668	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ (𝑢 supp 0 ))) → ((invg‘𝑅)‘(𝑢‘𝑘)) = ((invg‘𝑅)‘ 0 ))
151	8, 143	grpinvid 18154	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Grp → ((invg‘𝑅)‘ 0 ) = 0 )
152	142, 151	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((invg‘𝑅)‘ 0 ) = 0 )
153	152	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ (𝑢 supp 0 ))) → ((invg‘𝑅)‘ 0 ) = 0 )
154	150, 153	eqtrd 2856	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ (𝑢 supp 0 ))) → ((invg‘𝑅)‘(𝑢‘𝑘)) = 0 )
155	146, 149, 154	3eqtrd 2860	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑘 ∈ (𝐷 ∖ (𝑢 supp 0 ))) → (((invg‘𝑆)‘𝑢)‘𝑘) = 0 )
156	140, 155	suppss 7854	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (((invg‘𝑆)‘𝑢) supp 0 ) ⊆ (𝑢 supp 0 ))
157		sseq1 3991	. . . . . . . 8 ⊢ (𝑦 = (((invg‘𝑆)‘𝑢) supp 0 ) → (𝑦 ⊆ (𝑢 supp 0 ) ↔ (((invg‘𝑆)‘𝑢) supp 0 ) ⊆ (𝑢 supp 0 )))
158		eleq1 2900	. . . . . . . 8 ⊢ (𝑦 = (((invg‘𝑆)‘𝑢) supp 0 ) → (𝑦 ∈ 𝐴 ↔ (((invg‘𝑆)‘𝑢) supp 0 ) ∈ 𝐴))
159	157, 158	imbi12d 347	. . . . . . 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)))
160	159	spcgv 3594	. . . . . 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)))
161	134, 139, 156, 160	syl3c 66	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (((invg‘𝑆)‘𝑢) supp 0 ) ∈ 𝐴)
162	42	eleq2d 2898	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (((invg‘𝑆)‘𝑢) ∈ 𝑈 ↔ ((invg‘𝑆)‘𝑢) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
163		oveq1 7157	. . . . . . . 8 ⊢ (𝑔 = ((invg‘𝑆)‘𝑢) → (𝑔 supp 0 ) = (((invg‘𝑆)‘𝑢) supp 0 ))
164	163	eleq1d 2897	. . . . . . 7 ⊢ (𝑔 = ((invg‘𝑆)‘𝑢) → ((𝑔 supp 0 ) ∈ 𝐴 ↔ (((invg‘𝑆)‘𝑢) supp 0 ) ∈ 𝐴))
165	164	elrab 3679	. . . . . 6 ⊢ (((invg‘𝑆)‘𝑢) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ↔ (((invg‘𝑆)‘𝑢) ∈ 𝐵 ∧ (((invg‘𝑆)‘𝑢) supp 0 ) ∈ 𝐴))
166	162, 165	syl6bb 289	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (((invg‘𝑆)‘𝑢) ∈ 𝑈 ↔ (((invg‘𝑆)‘𝑢) ∈ 𝐵 ∧ (((invg‘𝑆)‘𝑢) supp 0 ) ∈ 𝐴)))
167	133, 161, 166	mpbir2and 711	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((invg‘𝑆)‘𝑢) ∈ 𝑈)
168	129, 167	jca 514	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (∀𝑣 ∈ 𝑈 (𝑢(+g‘𝑆)𝑣) ∈ 𝑈 ∧ ((invg‘𝑆)‘𝑢) ∈ 𝑈))
169	168	ralrimiva 3182	. 2 ⊢ (𝜑 → ∀𝑢 ∈ 𝑈 (∀𝑣 ∈ 𝑈 (𝑢(+g‘𝑆)𝑣) ∈ 𝑈 ∧ ((invg‘𝑆)‘𝑢) ∈ 𝑈))
170	9, 32, 131	issubg2 18288	. . 3 ⊢ (𝑆 ∈ Grp → (𝑈 ∈ (SubGrp‘𝑆) ↔ (𝑈 ⊆ 𝐵 ∧ 𝑈 ≠ ∅ ∧ ∀𝑢 ∈ 𝑈 (∀𝑣 ∈ 𝑈 (𝑢(+g‘𝑆)𝑣) ∈ 𝑈 ∧ ((invg‘𝑆)‘𝑢) ∈ 𝑈))))
171	130, 170	syl 17	. 2 ⊢ (𝜑 → (𝑈 ∈ (SubGrp‘𝑆) ↔ (𝑈 ⊆ 𝐵 ∧ 𝑈 ≠ ∅ ∧ ∀𝑢 ∈ 𝑈 (∀𝑣 ∈ 𝑈 (𝑢(+g‘𝑆)𝑣) ∈ 𝑈 ∧ ((invg‘𝑆)‘𝑢) ∈ 𝑈))))
172	3, 31, 169, 171	mpbir3and 1338	1 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑆))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083  ∀wal 1531   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  {crab 3142  Vcvv 3494   ∖ cdif 3932   ∪ cun 3933   ⊆ wss 3935  ∅c0 4290  {csn 4560   × cxp 5547  ^◡ccnv 5548   “ cima 5552   ∘ ccom 5553  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150   ∘_f cof 7401   supp csupp 7824   ↑_m cmap 8400  Fincfn 8503  ℕcn 11632  ℕ₀cn0 11891  Basecbs 16477  +_gcplusg 16559  0_gc0g 16707  Grpcgrp 18097  inv_gcminusg 18098  SubGrpcsubg 18267   mPwSer cmps 20125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-om 7575  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fsupp 8828  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12887  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-mulr 16573  df-sca 16575  df-vsca 16576  df-tset 16578  df-0g 16709  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-grp 18100  df-minusg 18101  df-subg 18270  df-psr 20130

This theorem is referenced by:  mpllsslem  20209  mplsubg  20211