Theorem dmdprdsplit2lem 20066

Description: Lemma for dmdprdsplit 20068. (Contributed by Mario Carneiro, 26-Apr-2016.)

Hypotheses

Ref	Expression
dprdsplit.2	⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
dprdsplit.i	⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)
dprdsplit.u	⊢ (𝜑 → 𝐼 = (𝐶 ∪ 𝐷))
dmdprdsplit.z	⊢ 𝑍 = (Cntz‘𝐺)
dmdprdsplit.0	⊢ 0 = (0g‘𝐺)
dmdprdsplit2.1	⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐶))
dmdprdsplit2.2	⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐷))
dmdprdsplit2.3	⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))))
dmdprdsplit2.4	⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })
dmdprdsplit2lem.k	⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))

Assertion

Ref	Expression
dmdprdsplit2lem	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑌 ∈ 𝐼 → (𝑋 ≠ 𝑌 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))) ∧ ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) ⊆ { 0 }))

Proof of Theorem dmdprdsplit2lem

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dprdsplit.u	. . . . . 6 ⊢ (𝜑 → 𝐼 = (𝐶 ∪ 𝐷))
2	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐼 = (𝐶 ∪ 𝐷))
3	2	eleq2d 2826	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑌 ∈ 𝐼 ↔ 𝑌 ∈ (𝐶 ∪ 𝐷)))
4		elun 4152	. . . 4 ⊢ (𝑌 ∈ (𝐶 ∪ 𝐷) ↔ (𝑌 ∈ 𝐶 ∨ 𝑌 ∈ 𝐷))
5	3, 4	bitrdi 287	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑌 ∈ 𝐼 ↔ (𝑌 ∈ 𝐶 ∨ 𝑌 ∈ 𝐷)))
6		dmdprdsplit2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐶))
7	6	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → 𝐺dom DProd (𝑆 ↾ 𝐶))
8		dprdsplit.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
9		ssun1 4177	. . . . . . . . . . 11 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷)
10	9, 1	sseqtrrid 4026	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ⊆ 𝐼)
11	8, 10	fssresd 6774	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ↾ 𝐶):𝐶⟶(SubGrp‘𝐺))
12	11	fdmd 6745	. . . . . . . 8 ⊢ (𝜑 → dom (𝑆 ↾ 𝐶) = 𝐶)
13	12	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → dom (𝑆 ↾ 𝐶) = 𝐶)
14		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ∈ 𝐶)
15		simprl 770	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → 𝑌 ∈ 𝐶)
16		simprr 772	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ≠ 𝑌)
17		dmdprdsplit.z	. . . . . . 7 ⊢ 𝑍 = (Cntz‘𝐺)
18	7, 13, 14, 15, 16, 17	dprdcntz 20029	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐶)‘𝑋) ⊆ (𝑍‘((𝑆 ↾ 𝐶)‘𝑌)))
19		fvres 6924	. . . . . . 7 ⊢ (𝑋 ∈ 𝐶 → ((𝑆 ↾ 𝐶)‘𝑋) = (𝑆‘𝑋))
20	19	ad2antlr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐶)‘𝑋) = (𝑆‘𝑋))
21		fvres 6924	. . . . . . . 8 ⊢ (𝑌 ∈ 𝐶 → ((𝑆 ↾ 𝐶)‘𝑌) = (𝑆‘𝑌))
22	21	ad2antrl 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐶)‘𝑌) = (𝑆‘𝑌))
23	22	fveq2d 6909	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → (𝑍‘((𝑆 ↾ 𝐶)‘𝑌)) = (𝑍‘(𝑆‘𝑌)))
24	18, 20, 23	3sstr3d 4037	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))
25	24	exp32 420	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑌 ∈ 𝐶 → (𝑋 ≠ 𝑌 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))))
26	19	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐶)‘𝑋) = (𝑆‘𝑋))
27	6	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → 𝐺dom DProd (𝑆 ↾ 𝐶))
28	12	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → dom (𝑆 ↾ 𝐶) = 𝐶)
29		simplr 768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ∈ 𝐶)
30	27, 28, 29	dprdub 20046	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐶)‘𝑋) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
31	26, 30	eqsstrrd 4018	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝑆‘𝑋) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
32		dmdprdsplit2.3	. . . . . . . 8 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))))
33	32	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))))
34		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
35	34	dprdssv 20037	. . . . . . . 8 ⊢ (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (Base‘𝐺)
36		fvres 6924	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝐷 → ((𝑆 ↾ 𝐷)‘𝑌) = (𝑆‘𝑌))
37	36	ad2antrl 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐷)‘𝑌) = (𝑆‘𝑌))
38		dmdprdsplit2.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐷))
39	38	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → 𝐺dom DProd (𝑆 ↾ 𝐷))
40		ssun2 4178	. . . . . . . . . . . . . 14 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷)
41	40, 1	sseqtrrid 4026	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ⊆ 𝐼)
42	8, 41	fssresd 6774	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
43	42	fdmd 6745	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝑆 ↾ 𝐷) = 𝐷)
44	43	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → dom (𝑆 ↾ 𝐷) = 𝐷)
45		simprl 770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → 𝑌 ∈ 𝐷)
46	39, 44, 45	dprdub 20046	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐷)‘𝑌) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷)))
47	37, 46	eqsstrrd 4018	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝑆‘𝑌) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷)))
48	34, 17	cntz2ss 19354	. . . . . . . 8 ⊢ (((𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (Base‘𝐺) ∧ (𝑆‘𝑌) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷))) → (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ (𝑍‘(𝑆‘𝑌)))
49	35, 47, 48	sylancr 587	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ (𝑍‘(𝑆‘𝑌)))
50	33, 49	sstrd 3993	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝑆‘𝑌)))
51	31, 50	sstrd 3993	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))
52	51	exp32 420	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑌 ∈ 𝐷 → (𝑋 ≠ 𝑌 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))))
53	25, 52	jaod 859	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑌 ∈ 𝐶 ∨ 𝑌 ∈ 𝐷) → (𝑋 ≠ 𝑌 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))))
54	5, 53	sylbid 240	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑌 ∈ 𝐼 → (𝑋 ≠ 𝑌 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))))
55		dprdgrp 20026	. . . . . . . 8 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐶) → 𝐺 ∈ Grp)
56	6, 55	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Grp)
57	56	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐺 ∈ Grp)
58	34	subgacs 19180	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
59		acsmre 17696	. . . . . 6 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
60	57, 58, 59	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
61		difundir 4290	. . . . . . . . . . 11 ⊢ ((𝐶 ∪ 𝐷) ∖ {𝑋}) = ((𝐶 ∖ {𝑋}) ∪ (𝐷 ∖ {𝑋}))
62	2	difeq1d 4124	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐼 ∖ {𝑋}) = ((𝐶 ∪ 𝐷) ∖ {𝑋}))
63		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝐶)
64	63	snssd 4808	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {𝑋} ⊆ 𝐶)
65		sslin 4242	. . . . . . . . . . . . . . 15 ⊢ ({𝑋} ⊆ 𝐶 → (𝐷 ∩ {𝑋}) ⊆ (𝐷 ∩ 𝐶))
66	64, 65	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐷 ∩ {𝑋}) ⊆ (𝐷 ∩ 𝐶))
67		incom 4208	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∩ 𝐷) = (𝐷 ∩ 𝐶)
68		dprdsplit.i	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)
69	68	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐶 ∩ 𝐷) = ∅)
70	67, 69	eqtr3id 2790	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐷 ∩ 𝐶) = ∅)
71		sseq0 4402	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∩ {𝑋}) ⊆ (𝐷 ∩ 𝐶) ∧ (𝐷 ∩ 𝐶) = ∅) → (𝐷 ∩ {𝑋}) = ∅)
72	66, 70, 71	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐷 ∩ {𝑋}) = ∅)
73		disj3 4453	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∩ {𝑋}) = ∅ ↔ 𝐷 = (𝐷 ∖ {𝑋}))
74	72, 73	sylib 218	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐷 = (𝐷 ∖ {𝑋}))
75	74	uneq2d 4167	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐶 ∖ {𝑋}) ∪ 𝐷) = ((𝐶 ∖ {𝑋}) ∪ (𝐷 ∖ {𝑋})))
76	61, 62, 75	3eqtr4a 2802	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐼 ∖ {𝑋}) = ((𝐶 ∖ {𝑋}) ∪ 𝐷))
77	76	imaeq2d 6077	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆 “ (𝐼 ∖ {𝑋})) = (𝑆 “ ((𝐶 ∖ {𝑋}) ∪ 𝐷)))
78		imaundi 6168	. . . . . . . . 9 ⊢ (𝑆 “ ((𝐶 ∖ {𝑋}) ∪ 𝐷)) = ((𝑆 “ (𝐶 ∖ {𝑋})) ∪ (𝑆 “ 𝐷))
79	77, 78	eqtrdi 2792	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆 “ (𝐼 ∖ {𝑋})) = ((𝑆 “ (𝐶 ∖ {𝑋})) ∪ (𝑆 “ 𝐷)))
80	79	unieqd 4919	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐼 ∖ {𝑋})) = ∪ ((𝑆 “ (𝐶 ∖ {𝑋})) ∪ (𝑆 “ 𝐷)))
81		uniun 4929	. . . . . . 7 ⊢ ∪ ((𝑆 “ (𝐶 ∖ {𝑋})) ∪ (𝑆 “ 𝐷)) = (∪ (𝑆 “ (𝐶 ∖ {𝑋})) ∪ ∪ (𝑆 “ 𝐷))
82	80, 81	eqtrdi 2792	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐼 ∖ {𝑋})) = (∪ (𝑆 “ (𝐶 ∖ {𝑋})) ∪ ∪ (𝑆 “ 𝐷)))
83		dmdprdsplit2lem.k	. . . . . . . . 9 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
84		difss 4135	. . . . . . . . . . 11 ⊢ (𝐶 ∖ {𝑋}) ⊆ 𝐶
85		imass2 6119	. . . . . . . . . . 11 ⊢ ((𝐶 ∖ {𝑋}) ⊆ 𝐶 → (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (𝑆 “ 𝐶))
86		uniss 4914	. . . . . . . . . . 11 ⊢ ((𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (𝑆 “ 𝐶) → ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ ∪ (𝑆 “ 𝐶))
87	84, 85, 86	mp2b 10	. . . . . . . . . 10 ⊢ ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ ∪ (𝑆 “ 𝐶)
88		imassrn 6088	. . . . . . . . . . . 12 ⊢ (𝑆 “ 𝐶) ⊆ ran 𝑆
89	8	frnd 6743	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
90	89	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ran 𝑆 ⊆ (SubGrp‘𝐺))
91		mresspw 17636	. . . . . . . . . . . . . 14 ⊢ ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
92	60, 91	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
93	90, 92	sstrd 3993	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
94	88, 93	sstrid 3994	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆 “ 𝐶) ⊆ 𝒫 (Base‘𝐺))
95		sspwuni 5099	. . . . . . . . . . 11 ⊢ ((𝑆 “ 𝐶) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆 “ 𝐶) ⊆ (Base‘𝐺))
96	94, 95	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ 𝐶) ⊆ (Base‘𝐺))
97	87, 96	sstrid 3994	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (Base‘𝐺))
98	60, 83, 97	mrcssidd 17669	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))))
99		imassrn 6088	. . . . . . . . . . . 12 ⊢ (𝑆 “ 𝐷) ⊆ ran 𝑆
100	99, 93	sstrid 3994	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆 “ 𝐷) ⊆ 𝒫 (Base‘𝐺))
101		sspwuni 5099	. . . . . . . . . . 11 ⊢ ((𝑆 “ 𝐷) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆 “ 𝐷) ⊆ (Base‘𝐺))
102	100, 101	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ 𝐷) ⊆ (Base‘𝐺))
103	60, 83, 102	mrcssidd 17669	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ 𝐷) ⊆ (𝐾‘∪ (𝑆 “ 𝐷)))
104	83	dprdspan 20048	. . . . . . . . . . . 12 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐷) → (𝐺 DProd (𝑆 ↾ 𝐷)) = (𝐾‘∪ ran (𝑆 ↾ 𝐷)))
105	38, 104	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) = (𝐾‘∪ ran (𝑆 ↾ 𝐷)))
106		df-ima 5697	. . . . . . . . . . . . 13 ⊢ (𝑆 “ 𝐷) = ran (𝑆 ↾ 𝐷)
107	106	unieqi 4918	. . . . . . . . . . . 12 ⊢ ∪ (𝑆 “ 𝐷) = ∪ ran (𝑆 ↾ 𝐷)
108	107	fveq2i 6908	. . . . . . . . . . 11 ⊢ (𝐾‘∪ (𝑆 “ 𝐷)) = (𝐾‘∪ ran (𝑆 ↾ 𝐷))
109	105, 108	eqtr4di 2794	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) = (𝐾‘∪ (𝑆 “ 𝐷)))
110	109	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐷)) = (𝐾‘∪ (𝑆 “ 𝐷)))
111	103, 110	sseqtrrd 4020	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ 𝐷) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷)))
112		unss12 4187	. . . . . . . 8 ⊢ ((∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∧ ∪ (𝑆 “ 𝐷) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷))) → (∪ (𝑆 “ (𝐶 ∖ {𝑋})) ∪ ∪ (𝑆 “ 𝐷)) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∪ (𝐺 DProd (𝑆 ↾ 𝐷))))
113	98, 111, 112	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (∪ (𝑆 “ (𝐶 ∖ {𝑋})) ∪ ∪ (𝑆 “ 𝐷)) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∪ (𝐺 DProd (𝑆 ↾ 𝐷))))
114	83	mrccl 17655	. . . . . . . . 9 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (Base‘𝐺)) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∈ (SubGrp‘𝐺))
115	60, 97, 114	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∈ (SubGrp‘𝐺))
116		dprdsubg 20045	. . . . . . . . . 10 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐷) → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
117	38, 116	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
118	117	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
119		eqid 2736	. . . . . . . . 9 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
120	119	lsmunss 19678	. . . . . . . 8 ⊢ (((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∈ (SubGrp‘𝐺) ∧ (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺)) → ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∪ (𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))))
121	115, 118, 120	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∪ (𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))))
122	113, 121	sstrd 3993	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (∪ (𝑆 “ (𝐶 ∖ {𝑋})) ∪ ∪ (𝑆 “ 𝐷)) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))))
123	82, 122	eqsstrd 4017	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐼 ∖ {𝑋})) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))))
124	87	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ ∪ (𝑆 “ 𝐶))
125	60, 83, 124, 96	mrcssd 17668	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝐾‘∪ (𝑆 “ 𝐶)))
126	83	dprdspan 20048	. . . . . . . . . . 11 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) = (𝐾‘∪ ran (𝑆 ↾ 𝐶)))
127	6, 126	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) = (𝐾‘∪ ran (𝑆 ↾ 𝐶)))
128		df-ima 5697	. . . . . . . . . . . 12 ⊢ (𝑆 “ 𝐶) = ran (𝑆 ↾ 𝐶)
129	128	unieqi 4918	. . . . . . . . . . 11 ⊢ ∪ (𝑆 “ 𝐶) = ∪ ran (𝑆 ↾ 𝐶)
130	129	fveq2i 6908	. . . . . . . . . 10 ⊢ (𝐾‘∪ (𝑆 “ 𝐶)) = (𝐾‘∪ ran (𝑆 ↾ 𝐶))
131	127, 130	eqtr4di 2794	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) = (𝐾‘∪ (𝑆 “ 𝐶)))
132	131	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) = (𝐾‘∪ (𝑆 “ 𝐶)))
133	125, 132	sseqtrrd 4020	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
134	32	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))))
135	133, 134	sstrd 3993	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))))
136	119, 17	lsmsubg 19673	. . . . . 6 ⊢ (((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∈ (SubGrp‘𝐺) ∧ (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺) ∧ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷)))) → ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))) ∈ (SubGrp‘𝐺))
137	115, 118, 135, 136	syl3anc 1372	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))) ∈ (SubGrp‘𝐺))
138	83	mrcsscl 17664	. . . . 5 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐼 ∖ {𝑋})) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))) ∧ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))) ∈ (SubGrp‘𝐺)) → (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋}))) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))))
139	60, 123, 137, 138	syl3anc 1372	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋}))) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))))
140		sslin 4242	. . . 4 ⊢ ((𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋}))) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))) → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) ⊆ ((𝑆‘𝑋) ∩ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷)))))
141	139, 140	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) ⊆ ((𝑆‘𝑋) ∩ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷)))))
142	10	sselda 3982	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝐼)
143	8	ffvelcdmda 7103	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐼) → (𝑆‘𝑋) ∈ (SubGrp‘𝐺))
144	142, 143	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆‘𝑋) ∈ (SubGrp‘𝐺))
145		dmdprdsplit.0	. . . 4 ⊢ 0 = (0g‘𝐺)
146	19	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆 ↾ 𝐶)‘𝑋) = (𝑆‘𝑋))
147	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐺dom DProd (𝑆 ↾ 𝐶))
148	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → dom (𝑆 ↾ 𝐶) = 𝐶)
149	147, 148, 63	dprdub 20046	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆 ↾ 𝐶)‘𝑋) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
150	146, 149	eqsstrrd 4018	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆‘𝑋) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
151		dprdsubg 20045	. . . . . . . . . . 11 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺))
152	6, 151	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺))
153	152	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺))
154	119	lsmlub 19683	. . . . . . . . 9 ⊢ (((𝑆‘𝑋) ∈ (SubGrp‘𝐺) ∧ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∈ (SubGrp‘𝐺) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺)) → (((𝑆‘𝑋) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)) ∧ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶))) ↔ ((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶))))
155	144, 115, 153, 154	syl3anc 1372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝑆‘𝑋) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)) ∧ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶))) ↔ ((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶))))
156	150, 133, 155	mpbi2and 712	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
157	156	ssrind 4243	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))))
158		dmdprdsplit2.4	. . . . . . 7 ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })
159	158	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })
160	157, 159	sseqtrd 4019	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ { 0 })
161	119	lsmub1 19676	. . . . . . . . 9 ⊢ (((𝑆‘𝑋) ∈ (SubGrp‘𝐺) ∧ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∈ (SubGrp‘𝐺)) → (𝑆‘𝑋) ⊆ ((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))))
162	144, 115, 161	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆‘𝑋) ⊆ ((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))))
163	145	subg0cl 19153	. . . . . . . . 9 ⊢ ((𝑆‘𝑋) ∈ (SubGrp‘𝐺) → 0 ∈ (𝑆‘𝑋))
164	144, 163	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 0 ∈ (𝑆‘𝑋))
165	162, 164	sseldd 3983	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 0 ∈ ((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))))
166	145	subg0cl 19153	. . . . . . . 8 ⊢ ((𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺) → 0 ∈ (𝐺 DProd (𝑆 ↾ 𝐷)))
167	118, 166	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 0 ∈ (𝐺 DProd (𝑆 ↾ 𝐷)))
168	165, 167	elind 4199	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 0 ∈ (((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))))
169	168	snssd 4808	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → { 0 } ⊆ (((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))))
170	160, 169	eqssd 4000	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })
171		resima2 6033	. . . . . . . . 9 ⊢ ((𝐶 ∖ {𝑋}) ⊆ 𝐶 → ((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋})) = (𝑆 “ (𝐶 ∖ {𝑋})))
172	84, 171	mp1i 13	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋})) = (𝑆 “ (𝐶 ∖ {𝑋})))
173	172	unieqd 4919	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ ((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋})) = ∪ (𝑆 “ (𝐶 ∖ {𝑋})))
174	173	fveq2d 6909	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ ((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋}))) = (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))))
175	146, 174	ineq12d 4220	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝑆 ↾ 𝐶)‘𝑋) ∩ (𝐾‘∪ ((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋})))) = ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))))
176	147, 148, 63, 145, 83	dprddisj 20030	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝑆 ↾ 𝐶)‘𝑋) ∩ (𝐾‘∪ ((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋})))) = { 0 })
177	175, 176	eqtr3d 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) = { 0 })
178	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑆:𝐼⟶(SubGrp‘𝐺))
179		ffun 6738	. . . . . . . 8 ⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → Fun 𝑆)
180		funiunfv 7269	. . . . . . . 8 ⊢ (Fun 𝑆 → ∪ 𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) = ∪ (𝑆 “ (𝐶 ∖ {𝑋})))
181	178, 179, 180	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ 𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) = ∪ (𝑆 “ (𝐶 ∖ {𝑋})))
182	6	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → 𝐺dom DProd (𝑆 ↾ 𝐶))
183	12	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → dom (𝑆 ↾ 𝐶) = 𝐶)
184		eldifi 4130	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐶 ∖ {𝑋}) → 𝑦 ∈ 𝐶)
185	184	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → 𝑦 ∈ 𝐶)
186		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → 𝑋 ∈ 𝐶)
187		eldifsni 4789	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐶 ∖ {𝑋}) → 𝑦 ≠ 𝑋)
188	187	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → 𝑦 ≠ 𝑋)
189	182, 183, 185, 186, 188, 17	dprdcntz 20029	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → ((𝑆 ↾ 𝐶)‘𝑦) ⊆ (𝑍‘((𝑆 ↾ 𝐶)‘𝑋)))
190	185	fvresd 6925	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → ((𝑆 ↾ 𝐶)‘𝑦) = (𝑆‘𝑦))
191	19	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → ((𝑆 ↾ 𝐶)‘𝑋) = (𝑆‘𝑋))
192	191	fveq2d 6909	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → (𝑍‘((𝑆 ↾ 𝐶)‘𝑋)) = (𝑍‘(𝑆‘𝑋)))
193	189, 190, 192	3sstr3d 4037	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → (𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑋)))
194	193	ralrimiva 3145	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∀𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑋)))
195		iunss 5044	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑋)) ↔ ∀𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑋)))
196	194, 195	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ 𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑋)))
197	181, 196	eqsstrrd 4018	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (𝑍‘(𝑆‘𝑋)))
198	34	subgss 19146	. . . . . . . 8 ⊢ ((𝑆‘𝑋) ∈ (SubGrp‘𝐺) → (𝑆‘𝑋) ⊆ (Base‘𝐺))
199	144, 198	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆‘𝑋) ⊆ (Base‘𝐺))
200	34, 17	cntzsubg 19358	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑆‘𝑋) ⊆ (Base‘𝐺)) → (𝑍‘(𝑆‘𝑋)) ∈ (SubGrp‘𝐺))
201	57, 199, 200	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑍‘(𝑆‘𝑋)) ∈ (SubGrp‘𝐺))
202	83	mrcsscl 17664	. . . . . 6 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (𝑍‘(𝑆‘𝑋)) ∧ (𝑍‘(𝑆‘𝑋)) ∈ (SubGrp‘𝐺)) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝑍‘(𝑆‘𝑋)))
203	60, 197, 201, 202	syl3anc 1372	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝑍‘(𝑆‘𝑋)))
204	17, 115, 144, 203	cntzrecd 19697	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆‘𝑋) ⊆ (𝑍‘(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))))
205	119, 144, 115, 118, 145, 170, 177, 17, 204	lsmdisj3 19702	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆‘𝑋) ∩ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷)))) = { 0 })
206	141, 205	sseqtrd 4019	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) ⊆ { 0 })
207	54, 206	jca 511	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑌 ∈ 𝐼 → (𝑋 ≠ 𝑌 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))) ∧ ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) ⊆ { 0 }))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060   ∖ cdif 3947   ∪ cun 3948   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  𝒫 cpw 4599  {csn 4625  ∪ cuni 4906  ∪ ciun 4990   class class class wbr 5142  dom cdm 5684  ran crn 5685   ↾ cres 5686   “ cima 5687  Fun wfun 6554  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432  Basecbs 17248  0_gc0g 17485  Moorecmre 17626  mrClscmrc 17627  ACScacs 17629  Grpcgrp 18952  SubGrpcsubg 19139  Cntzccntz 19334  LSSumclsm 19653   DProd cdprd 20014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-0g 17487  df-gsum 17488  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-gim 19278  df-cntz 19336  df-oppg 19365  df-lsm 19655  df-cmn 19801  df-dprd 20016

This theorem is referenced by:  dmdprdsplit2  20067