Theorem dprdss 19808

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 19784	. . . 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 19780	. . 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 3143	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘))
12		fveq2 6842	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝑆‘𝑘) = (𝑆‘𝑥))
13		fveq2 6842	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝑇‘𝑘) = (𝑇‘𝑥))
14	12, 13	sseq12d 3977	. . . . . . 7 ⊢ (𝑘 = 𝑥 → ((𝑆‘𝑘) ⊆ (𝑇‘𝑘) ↔ (𝑆‘𝑥) ⊆ (𝑇‘𝑥)))
15	14	rspcv 3577	. . . . . 6 ⊢ (𝑥 ∈ 𝐼 → (∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘) → (𝑆‘𝑥) ⊆ (𝑇‘𝑥)))
16	11, 15	mpan9 507	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ⊆ (𝑇‘𝑥))
17	16	3ad2antr1 1188	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘𝑥) ⊆ (𝑇‘𝑥))
18	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝐺dom DProd 𝑇)
19	7	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → dom 𝑇 = 𝐼)
20		simpr1 1194	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐼)
21		simpr2 1195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐼)
22		simpr3 1196	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
23	18, 19, 20, 21, 22, 1	dprdcntz 19787	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑇‘𝑦)))
24	4, 7	dprdf2 19786	. . . . . . . . 9 ⊢ (𝜑 → 𝑇:𝐼⟶(SubGrp‘𝐺))
25	24	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑇:𝐼⟶(SubGrp‘𝐺))
26	25, 21	ffvelcdmd 7036	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑦) ∈ (SubGrp‘𝐺))
27		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
28	27	subgss 18929	. . . . . . 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 6842	. . . . . . . 8 ⊢ (𝑘 = 𝑦 → (𝑆‘𝑘) = (𝑆‘𝑦))
31		fveq2 6842	. . . . . . . 8 ⊢ (𝑘 = 𝑦 → (𝑇‘𝑘) = (𝑇‘𝑦))
32	30, 31	sseq12d 3977	. . . . . . 7 ⊢ (𝑘 = 𝑦 → ((𝑆‘𝑘) ⊆ (𝑇‘𝑘) ↔ (𝑆‘𝑦) ⊆ (𝑇‘𝑦)))
33	11	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → ∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘))
34	32, 33, 21	rspcdva 3582	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘𝑦) ⊆ (𝑇‘𝑦))
35	27, 1	cntz2ss 19113	. . . . . 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 3954	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)))
38	17, 37	sstrd 3954	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)))
39	6	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Grp)
40	27	subgacs 18963	. . . . . . 7 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
41		acsmre 17532	. . . . . . 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 4091	. . . . . . . . 9 ⊢ (𝐼 ∖ {𝑥}) ⊆ 𝐼
44	11	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘))
45		ssralv 4010	. . . . . . . . 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 4972	. . . . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺))
50		ffun 6671	. . . . . . . 8 ⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → Fun 𝑆)
51		funiunfv 7195	. . . . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑇:𝐼⟶(SubGrp‘𝐺))
54		ffun 6671	. . . . . . . 8 ⊢ (𝑇:𝐼⟶(SubGrp‘𝐺) → Fun 𝑇)
55		funiunfv 7195	. . . . . . . 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 3990	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ∪ (𝑇 “ (𝐼 ∖ {𝑥})))
58		imassrn 6024	. . . . . . . 8 ⊢ (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑇
59	53	frnd 6676	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ran 𝑇 ⊆ (SubGrp‘𝐺))
60		mresspw 17472	. . . . . . . . . 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 3954	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ran 𝑇 ⊆ 𝒫 (Base‘𝐺))
63	58, 62	sstrid 3955	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
64		sspwuni 5060	. . . . . . 7 ⊢ ((𝑇 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
65	63, 64	sylib 217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
66	42, 3, 57, 65	mrcssd 17504	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇 “ (𝐼 ∖ {𝑥}))))
67		ss2in 4196	. . . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd 𝑇)
70	7	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → dom 𝑇 = 𝐼)
71		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
72	69, 70, 71, 2, 3	dprddisj 19788	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑇‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)})
73	68, 72	sseqtrd 3984	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ {(0g‘𝐺)})
74	1, 2, 3, 6, 8, 9, 38, 73	dmdprdd 19778	. 2 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
75	4	a1d 25	. . . . 5 ⊢ (𝜑 → (𝐺dom DProd 𝑆 → 𝐺dom DProd 𝑇))
76		ss2ixp 8848	. . . . . . 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 4035	. . . . . 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 4011	. . . . . 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 6677	. . . . 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 19783	. . . . 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 19783	. . . . 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 293	. . 3 ⊢ (𝜑 → (𝑎 ∈ (𝐺 DProd 𝑆) → 𝑎 ∈ (𝐺 DProd 𝑇)))
90	89	ssrdv 3950	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ (𝐺 DProd 𝑇))
91	74, 90	jca 512	1 ⊢ (𝜑 → (𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) ⊆ (𝐺 DProd 𝑇)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  {crab 3407  Vcvv 3445   ∖ cdif 3907   ∩ cin 3909   ⊆ wss 3910  𝒫 cpw 4560  {csn 4586  ∪ cuni 4865  ∪ ciun 4954   class class class wbr 5105  dom cdm 5633  ran crn 5634   “ cima 5636  Fun wfun 6490  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  Xcixp 8835   finSupp cfsupp 9305  Basecbs 17083  0_gc0g 17321   Σ_g cgsu 17322  Moorecmre 17462  mrClscmrc 17463  ACScacs 17465  Grpcgrp 18748  SubGrpcsubg 18922  Cntzccntz 19095   DProd cdprd 19772

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-subg 18925  df-cntz 19097  df-dprd 19774

This theorem is referenced by: (None)