Theorem dprdss 20017

Description: Create a direct product by finding subgroups inside each factor of another direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)

Hypotheses

Ref	Expression
dprdss.1	⊢ (𝜑 → 𝐺dom DProd 𝑇)
dprdss.2	⊢ (𝜑 → dom 𝑇 = 𝐼)
dprdss.3	⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
dprdss.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑆‘𝑘) ⊆ (𝑇‘𝑘))

Assertion

Ref	Expression
dprdss	⊢ (𝜑 → (𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) ⊆ (𝐺 DProd 𝑇)))

Distinct variable groups:   𝑘,𝐺   𝜑,𝑘   𝑆,𝑘   𝑇,𝑘   𝑘,𝐼

Proof of Theorem dprdss

Dummy variables 𝑓 𝑎 ℎ 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2736	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		eqid 2736	. . 3 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		dprdss.1	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑇)
5		dprdgrp 19993	. . . 4 ⊢ (𝐺dom DProd 𝑇 → 𝐺 ∈ Grp)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		dprdss.2	. . . 4 ⊢ (𝜑 → dom 𝑇 = 𝐼)
8	4, 7	dprddomcld 19989	. . 3 ⊢ (𝜑 → 𝐼 ∈ V)
9		dprdss.3	. . 3 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
10		dprdss.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑆‘𝑘) ⊆ (𝑇‘𝑘))
11	10	ralrimiva 3133	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘))
12		fveq2 6881	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝑆‘𝑘) = (𝑆‘𝑥))
13		fveq2 6881	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝑇‘𝑘) = (𝑇‘𝑥))
14	12, 13	sseq12d 3997	. . . . . . 7 ⊢ (𝑘 = 𝑥 → ((𝑆‘𝑘) ⊆ (𝑇‘𝑘) ↔ (𝑆‘𝑥) ⊆ (𝑇‘𝑥)))
15	14	rspcv 3602	. . . . . 6 ⊢ (𝑥 ∈ 𝐼 → (∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘) → (𝑆‘𝑥) ⊆ (𝑇‘𝑥)))
16	11, 15	mpan9 506	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ⊆ (𝑇‘𝑥))
17	16	3ad2antr1 1189	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘𝑥) ⊆ (𝑇‘𝑥))
18	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝐺dom DProd 𝑇)
19	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → dom 𝑇 = 𝐼)
20		simpr1 1195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐼)
21		simpr2 1196	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐼)
22		simpr3 1197	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
23	18, 19, 20, 21, 22, 1	dprdcntz 19996	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑇‘𝑦)))
24	4, 7	dprdf2 19995	. . . . . . . . 9 ⊢ (𝜑 → 𝑇:𝐼⟶(SubGrp‘𝐺))
25	24	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑇:𝐼⟶(SubGrp‘𝐺))
26	25, 21	ffvelcdmd 7080	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑦) ∈ (SubGrp‘𝐺))
27		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
28	27	subgss 19115	. . . . . . 7 ⊢ ((𝑇‘𝑦) ∈ (SubGrp‘𝐺) → (𝑇‘𝑦) ⊆ (Base‘𝐺))
29	26, 28	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑦) ⊆ (Base‘𝐺))
30		fveq2 6881	. . . . . . . 8 ⊢ (𝑘 = 𝑦 → (𝑆‘𝑘) = (𝑆‘𝑦))
31		fveq2 6881	. . . . . . . 8 ⊢ (𝑘 = 𝑦 → (𝑇‘𝑘) = (𝑇‘𝑦))
32	30, 31	sseq12d 3997	. . . . . . 7 ⊢ (𝑘 = 𝑦 → ((𝑆‘𝑘) ⊆ (𝑇‘𝑘) ↔ (𝑆‘𝑦) ⊆ (𝑇‘𝑦)))
33	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → ∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘))
34	32, 33, 21	rspcdva 3607	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘𝑦) ⊆ (𝑇‘𝑦))
35	27, 1	cntz2ss 19323	. . . . . 6 ⊢ (((𝑇‘𝑦) ⊆ (Base‘𝐺) ∧ (𝑆‘𝑦) ⊆ (𝑇‘𝑦)) → ((Cntz‘𝐺)‘(𝑇‘𝑦)) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)))
36	29, 34, 35	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → ((Cntz‘𝐺)‘(𝑇‘𝑦)) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)))
37	23, 36	sstrd 3974	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)))
38	17, 37	sstrd 3974	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)))
39	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Grp)
40	27	subgacs 19149	. . . . . . 7 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
41		acsmre 17669	. . . . . . 7 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
42	39, 40, 41	3syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
43		difss 4116	. . . . . . . . 9 ⊢ (𝐼 ∖ {𝑥}) ⊆ 𝐼
44	11	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘))
45		ssralv 4032	. . . . . . . . 9 ⊢ ((𝐼 ∖ {𝑥}) ⊆ 𝐼 → (∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘) → ∀𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) ⊆ (𝑇‘𝑘)))
46	43, 44, 45	mpsyl 68	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∀𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) ⊆ (𝑇‘𝑘))
47		ss2iun 4991	. . . . . . . 8 ⊢ (∀𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) ⊆ (𝑇‘𝑘) → ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) ⊆ ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑇‘𝑘))
48	46, 47	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) ⊆ ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑇‘𝑘))
49	9	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺))
50		ffun 6714	. . . . . . . 8 ⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → Fun 𝑆)
51		funiunfv 7245	. . . . . . . 8 ⊢ (Fun 𝑆 → ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) = ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
52	49, 50, 51	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) = ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
53	24	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑇:𝐼⟶(SubGrp‘𝐺))
54		ffun 6714	. . . . . . . 8 ⊢ (𝑇:𝐼⟶(SubGrp‘𝐺) → Fun 𝑇)
55		funiunfv 7245	. . . . . . . 8 ⊢ (Fun 𝑇 → ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑇‘𝑘) = ∪ (𝑇 “ (𝐼 ∖ {𝑥})))
56	53, 54, 55	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑇‘𝑘) = ∪ (𝑇 “ (𝐼 ∖ {𝑥})))
57	48, 52, 56	3sstr3d 4018	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ∪ (𝑇 “ (𝐼 ∖ {𝑥})))
58		imassrn 6063	. . . . . . . 8 ⊢ (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑇
59	53	frnd 6719	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ran 𝑇 ⊆ (SubGrp‘𝐺))
60		mresspw 17609	. . . . . . . . . 10 ⊢ ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
61	42, 60	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
62	59, 61	sstrd 3974	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ran 𝑇 ⊆ 𝒫 (Base‘𝐺))
63	58, 62	sstrid 3975	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
64		sspwuni 5081	. . . . . . 7 ⊢ ((𝑇 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
65	63, 64	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
66	42, 3, 57, 65	mrcssd 17641	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇 “ (𝐼 ∖ {𝑥}))))
67		ss2in 4225	. . . . 5 ⊢ (((𝑆‘𝑥) ⊆ (𝑇‘𝑥) ∧ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇 “ (𝐼 ∖ {𝑥})))) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ ((𝑇‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇 “ (𝐼 ∖ {𝑥})))))
68	16, 66, 67	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ ((𝑇‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇 “ (𝐼 ∖ {𝑥})))))
69	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd 𝑇)
70	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → dom 𝑇 = 𝐼)
71		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
72	69, 70, 71, 2, 3	dprddisj 19997	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑇‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)})
73	68, 72	sseqtrd 4000	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ {(0g‘𝐺)})
74	1, 2, 3, 6, 8, 9, 38, 73	dmdprdd 19987	. 2 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
75	4	a1d 25	. . . . 5 ⊢ (𝜑 → (𝐺dom DProd 𝑆 → 𝐺dom DProd 𝑇))
76		ss2ixp 8929	. . . . . . 7 ⊢ (∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘) → X𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ X𝑘 ∈ 𝐼 (𝑇‘𝑘))
77	11, 76	syl 17	. . . . . 6 ⊢ (𝜑 → X𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ X𝑘 ∈ 𝐼 (𝑇‘𝑘))
78		rabss2 4058	. . . . . 6 ⊢ (X𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ X𝑘 ∈ 𝐼 (𝑇‘𝑘) → {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)} ⊆ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)})
79		ssrexv 4033	. . . . . 6 ⊢ ({ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)} ⊆ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)} → (∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓) → ∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓)))
80	77, 78, 79	3syl 18	. . . . 5 ⊢ (𝜑 → (∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓) → ∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓)))
81	75, 80	anim12d 609	. . . 4 ⊢ (𝜑 → ((𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓)) → (𝐺dom DProd 𝑇 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓))))
82		fdm 6720	. . . . 5 ⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → dom 𝑆 = 𝐼)
83		eqid 2736	. . . . . 6 ⊢ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)} = {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}
84	2, 83	eldprd 19992	. . . . 5 ⊢ (dom 𝑆 = 𝐼 → (𝑎 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓))))
85	9, 82, 84	3syl 18	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓))))
86		eqid 2736	. . . . . 6 ⊢ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)} = {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}
87	2, 86	eldprd 19992	. . . . 5 ⊢ (dom 𝑇 = 𝐼 → (𝑎 ∈ (𝐺 DProd 𝑇) ↔ (𝐺dom DProd 𝑇 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓))))
88	7, 87	syl 17	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝐺 DProd 𝑇) ↔ (𝐺dom DProd 𝑇 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓))))
89	81, 85, 88	3imtr4d 294	. . 3 ⊢ (𝜑 → (𝑎 ∈ (𝐺 DProd 𝑆) → 𝑎 ∈ (𝐺 DProd 𝑇)))
90	89	ssrdv 3969	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ (𝐺 DProd 𝑇))
91	74, 90	jca 511	1 ⊢ (𝜑 → (𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) ⊆ (𝐺 DProd 𝑇)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  {crab 3420  Vcvv 3464   ∖ cdif 3928   ∩ cin 3930   ⊆ wss 3931  𝒫 cpw 4580  {csn 4606  ∪ cuni 4888  ∪ ciun 4972   class class class wbr 5124  dom cdm 5659  ran crn 5660   “ cima 5662  Fun wfun 6530  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Xcixp 8916   finSupp cfsupp 9378  Basecbs 17233  0_gc0g 17458   Σ_g cgsu 17459  Moorecmre 17599  mrClscmrc 17600  ACScacs 17602  Grpcgrp 18921  SubGrpcsubg 19108  Cntzccntz 19303   DProd cdprd 19981

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8724  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-ress 17257  df-plusg 17289  df-0g 17460  df-mre 17603  df-mrc 17604  df-acs 17606  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-submnd 18767  df-grp 18924  df-minusg 18925  df-subg 19111  df-cntz 19305  df-dprd 19983

This theorem is referenced by: (None)