Theorem dprdss 18428

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 2622	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2622	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		eqid 2622	. . 3 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		dprdss.1	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑇)
5		dprdgrp 18404	. . . 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 18400	. . 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 2966	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘))
12		fveq2 6191	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝑆‘𝑘) = (𝑆‘𝑥))
13		fveq2 6191	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝑇‘𝑘) = (𝑇‘𝑥))
14	12, 13	sseq12d 3634	. . . . . . 7 ⊢ (𝑘 = 𝑥 → ((𝑆‘𝑘) ⊆ (𝑇‘𝑘) ↔ (𝑆‘𝑥) ⊆ (𝑇‘𝑥)))
15	14	rspcv 3305	. . . . . 6 ⊢ (𝑥 ∈ 𝐼 → (∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘) → (𝑆‘𝑥) ⊆ (𝑇‘𝑥)))
16	11, 15	mpan9 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ⊆ (𝑇‘𝑥))
17	16	3ad2antr1 1226	. . . 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 1067	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐼)
21		simpr2 1068	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐼)
22		simpr3 1069	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
23	18, 19, 20, 21, 22, 1	dprdcntz 18407	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑇‘𝑦)))
24	4, 7	dprdf2 18406	. . . . . . . . 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 6360	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑦) ∈ (SubGrp‘𝐺))
27		eqid 2622	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
28	27	subgss 17595	. . . . . . 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	11	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → ∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘))
31		fveq2 6191	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → (𝑆‘𝑘) = (𝑆‘𝑦))
32		fveq2 6191	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → (𝑇‘𝑘) = (𝑇‘𝑦))
33	31, 32	sseq12d 3634	. . . . . . . 8 ⊢ (𝑘 = 𝑦 → ((𝑆‘𝑘) ⊆ (𝑇‘𝑘) ↔ (𝑆‘𝑦) ⊆ (𝑇‘𝑦)))
34	33	rspcv 3305	. . . . . . 7 ⊢ (𝑦 ∈ 𝐼 → (∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘) → (𝑆‘𝑦) ⊆ (𝑇‘𝑦)))
35	21, 30, 34	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘𝑦) ⊆ (𝑇‘𝑦))
36	27, 1	cntz2ss 17765	. . . . . 6 ⊢ (((𝑇‘𝑦) ⊆ (Base‘𝐺) ∧ (𝑆‘𝑦) ⊆ (𝑇‘𝑦)) → ((Cntz‘𝐺)‘(𝑇‘𝑦)) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)))
37	29, 35, 36	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → ((Cntz‘𝐺)‘(𝑇‘𝑦)) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)))
38	23, 37	sstrd 3613	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)))
39	17, 38	sstrd 3613	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)))
40	6	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Grp)
41	27	subgacs 17629	. . . . . . 7 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
42		acsmre 16313	. . . . . . 7 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
43	40, 41, 42	3syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
44		difss 3737	. . . . . . . . 9 ⊢ (𝐼 ∖ {𝑥}) ⊆ 𝐼
45	11	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘))
46		ssralv 3666	. . . . . . . . 9 ⊢ ((𝐼 ∖ {𝑥}) ⊆ 𝐼 → (∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘) → ∀𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) ⊆ (𝑇‘𝑘)))
47	44, 45, 46	mpsyl 68	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∀𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) ⊆ (𝑇‘𝑘))
48		ss2iun 4536	. . . . . . . 8 ⊢ (∀𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) ⊆ (𝑇‘𝑘) → ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) ⊆ ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑇‘𝑘))
49	47, 48	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) ⊆ ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑇‘𝑘))
50	9	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺))
51		ffun 6048	. . . . . . . 8 ⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → Fun 𝑆)
52		funiunfv 6506	. . . . . . . 8 ⊢ (Fun 𝑆 → ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) = ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
53	50, 51, 52	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) = ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
54	24	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑇:𝐼⟶(SubGrp‘𝐺))
55		ffun 6048	. . . . . . . 8 ⊢ (𝑇:𝐼⟶(SubGrp‘𝐺) → Fun 𝑇)
56		funiunfv 6506	. . . . . . . 8 ⊢ (Fun 𝑇 → ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑇‘𝑘) = ∪ (𝑇 “ (𝐼 ∖ {𝑥})))
57	54, 55, 56	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑇‘𝑘) = ∪ (𝑇 “ (𝐼 ∖ {𝑥})))
58	49, 53, 57	3sstr3d 3647	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ∪ (𝑇 “ (𝐼 ∖ {𝑥})))
59		imassrn 5477	. . . . . . . 8 ⊢ (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑇
60		frn 6053	. . . . . . . . . 10 ⊢ (𝑇:𝐼⟶(SubGrp‘𝐺) → ran 𝑇 ⊆ (SubGrp‘𝐺))
61	54, 60	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ran 𝑇 ⊆ (SubGrp‘𝐺))
62		mresspw 16252	. . . . . . . . . 10 ⊢ ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
63	43, 62	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
64	61, 63	sstrd 3613	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ran 𝑇 ⊆ 𝒫 (Base‘𝐺))
65	59, 64	syl5ss 3614	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
66		sspwuni 4611	. . . . . . 7 ⊢ ((𝑇 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
67	65, 66	sylib 208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
68	43, 3, 58, 67	mrcssd 16284	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇 “ (𝐼 ∖ {𝑥}))))
69		ss2in 3840	. . . . 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})))))
70	16, 68, 69	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ ((𝑇‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇 “ (𝐼 ∖ {𝑥})))))
71	4	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd 𝑇)
72	7	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → dom 𝑇 = 𝐼)
73		simpr 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
74	71, 72, 73, 2, 3	dprddisj 18408	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑇‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)})
75	70, 74	sseqtrd 3641	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ {(0g‘𝐺)})
76	1, 2, 3, 6, 8, 9, 39, 75	dmdprdd 18398	. 2 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
77	4	a1d 25	. . . . 5 ⊢ (𝜑 → (𝐺dom DProd 𝑆 → 𝐺dom DProd 𝑇))
78		ss2ixp 7921	. . . . . . 7 ⊢ (∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘) → X𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ X𝑘 ∈ 𝐼 (𝑇‘𝑘))
79	11, 78	syl 17	. . . . . 6 ⊢ (𝜑 → X𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ X𝑘 ∈ 𝐼 (𝑇‘𝑘))
80		rabss2 3685	. . . . . 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)})
81		ssrexv 3667	. . . . . 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)))
82	79, 80, 81	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)))
83	77, 82	anim12d 586	. . . 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))))
84		fdm 6051	. . . . 5 ⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → dom 𝑆 = 𝐼)
85		eqid 2622	. . . . . 6 ⊢ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)} = {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}
86	2, 85	eldprd 18403	. . . . 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))))
87	9, 84, 86	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))))
88		eqid 2622	. . . . . 6 ⊢ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)} = {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}
89	2, 88	eldprd 18403	. . . . 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))))
90	7, 89	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))))
91	83, 87, 90	3imtr4d 283	. . 3 ⊢ (𝜑 → (𝑎 ∈ (𝐺 DProd 𝑆) → 𝑎 ∈ (𝐺 DProd 𝑇)))
92	91	ssrdv 3609	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ (𝐺 DProd 𝑇))
93	76, 92	jca 554	1 ⊢ (𝜑 → (𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) ⊆ (𝐺 DProd 𝑇)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436  ∪ ciun 4520   class class class wbr 4653  dom cdm 5114  ran crn 5115   “ cima 5117  Fun wfun 5882  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Xcixp 7908   finSupp cfsupp 8275  Basecbs 15857  0_gc0g 16100   Σ_g cgsu 16101  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245  Grpcgrp 17422  SubGrpcsubg 17588  Cntzccntz 17748   DProd cdprd 18392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-subg 17591  df-cntz 17750  df-dprd 18394

This theorem is referenced by: (None)