Theorem dprdss 18872

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 2797	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2797	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		eqid 2797	. . 3 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		dprdss.1	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑇)
5		dprdgrp 18848	. . . 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 18844	. . 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 3151	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘))
12		fveq2 6545	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝑆‘𝑘) = (𝑆‘𝑥))
13		fveq2 6545	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝑇‘𝑘) = (𝑇‘𝑥))
14	12, 13	sseq12d 3927	. . . . . . 7 ⊢ (𝑘 = 𝑥 → ((𝑆‘𝑘) ⊆ (𝑇‘𝑘) ↔ (𝑆‘𝑥) ⊆ (𝑇‘𝑥)))
15	14	rspcv 3557	. . . . . 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 1181	. . . 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 1187	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐼)
21		simpr2 1188	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐼)
22		simpr3 1189	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
23	18, 19, 20, 21, 22, 1	dprdcntz 18851	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑇‘𝑦)))
24	4, 7	dprdf2 18850	. . . . . . . . 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	ffvelrnd 6724	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑦) ∈ (SubGrp‘𝐺))
27		eqid 2797	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
28	27	subgss 18038	. . . . . . 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 6545	. . . . . . . 8 ⊢ (𝑘 = 𝑦 → (𝑆‘𝑘) = (𝑆‘𝑦))
31		fveq2 6545	. . . . . . . 8 ⊢ (𝑘 = 𝑦 → (𝑇‘𝑘) = (𝑇‘𝑦))
32	30, 31	sseq12d 3927	. . . . . . 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 3567	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘𝑦) ⊆ (𝑇‘𝑦))
35	27, 1	cntz2ss 18208	. . . . . 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 3905	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)))
38	17, 37	sstrd 3905	. . 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 18072	. . . . . . 7 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
41		acsmre 16756	. . . . . . 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 4035	. . . . . . . . 9 ⊢ (𝐼 ∖ {𝑥}) ⊆ 𝐼
44	11	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘))
45		ssralv 3960	. . . . . . . . 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 4848	. . . . . . . 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 6392	. . . . . . . 8 ⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → Fun 𝑆)
51		funiunfv 6879	. . . . . . . 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 6392	. . . . . . . 8 ⊢ (𝑇:𝐼⟶(SubGrp‘𝐺) → Fun 𝑇)
55		funiunfv 6879	. . . . . . . 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 3940	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ∪ (𝑇 “ (𝐼 ∖ {𝑥})))
58		imassrn 5824	. . . . . . . 8 ⊢ (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑇
59	53	frnd 6396	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ran 𝑇 ⊆ (SubGrp‘𝐺))
60		mresspw 16696	. . . . . . . . . 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 3905	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ran 𝑇 ⊆ 𝒫 (Base‘𝐺))
63	58, 62	sstrid 3906	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
64		sspwuni 4927	. . . . . . 7 ⊢ ((𝑇 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
65	63, 64	sylib 219	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
66	42, 3, 57, 65	mrcssd 16728	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇 “ (𝐼 ∖ {𝑥}))))
67		ss2in 4139	. . . . 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 18852	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑇‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)})
73	68, 72	sseqtrd 3934	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ {(0g‘𝐺)})
74	1, 2, 3, 6, 8, 9, 38, 73	dmdprdd 18842	. 2 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
75	4	a1d 25	. . . . 5 ⊢ (𝜑 → (𝐺dom DProd 𝑆 → 𝐺dom DProd 𝑇))
76		ss2ixp 8330	. . . . . . 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 3981	. . . . . 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 3961	. . . . . 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 608	. . . 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 6397	. . . . 5 ⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → dom 𝑆 = 𝐼)
83		eqid 2797	. . . . . 6 ⊢ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)} = {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}
84	2, 83	eldprd 18847	. . . . 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 2797	. . . . . 6 ⊢ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)} = {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}
87	2, 86	eldprd 18847	. . . . 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 295	. . 3 ⊢ (𝜑 → (𝑎 ∈ (𝐺 DProd 𝑆) → 𝑎 ∈ (𝐺 DProd 𝑇)))
90	89	ssrdv 3901	. 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 207   ∧ wa 396   ∧ w3a 1080   = wceq 1525   ∈ wcel 2083   ≠ wne 2986  ∀wral 3107  ∃wrex 3108  {crab 3111  Vcvv 3440   ∖ cdif 3862   ∩ cin 3864   ⊆ wss 3865  𝒫 cpw 4459  {csn 4478  ∪ cuni 4751  ∪ ciun 4831   class class class wbr 4968  dom cdm 5450  ran crn 5451   “ cima 5453  Fun wfun 6226  ⟶wf 6228  ‘cfv 6232  (class class class)co 7023  Xcixp 8317   finSupp cfsupp 8686  Basecbs 16316  0_gc0g 16546   Σ_g cgsu 16547  Moorecmre 16686  mrClscmrc 16687  ACScacs 16689  Grpcgrp 17865  SubGrpcsubg 18031  Cntzccntz 18190   DProd cdprd 18836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-iin 4834  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-ixp 8318  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-2 11554  df-ndx 16319  df-slot 16320  df-base 16322  df-sets 16323  df-ress 16324  df-plusg 16411  df-0g 16548  df-mre 16690  df-mrc 16691  df-acs 16693  df-mgm 17685  df-sgrp 17727  df-mnd 17738  df-submnd 17779  df-grp 17868  df-minusg 17869  df-subg 18034  df-cntz 18192  df-dprd 18838

This theorem is referenced by: (None)