Theorem dmdprdsplit2lem 20112

Description: Lemma for dmdprdsplit 20114. (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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐼 = (𝐶 ∪ 𝐷))
3	2	eleq2d 2849	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑌 ∈ 𝐼 ↔ 𝑌 ∈ (𝐶 ∪ 𝐷)))
4		elun 4107	. . . 4 ⊢ (𝑌 ∈ (𝐶 ∪ 𝐷) ↔ (𝑌 ∈ 𝐶 ∨ 𝑌 ∈ 𝐷))
5	3, 4	bitrdi 290	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑌 ∈ 𝐼 ↔ (𝑌 ∈ 𝐶 ∨ 𝑌 ∈ 𝐷)))
6		dmdprdsplit2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐶))
7	6	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → 𝐺dom DProd (𝑆 ↾ 𝐶))
8		dprdsplit.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
9		ssun1 4131	. . . . . . . . . . 11 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷)
10	9, 1	sseqtrrid 3980	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ⊆ 𝐼)
11	8, 10	fssresd 6745	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ↾ 𝐶):𝐶⟶(SubGrp‘𝐺))
12	11	fdmd 6716	. . . . . . . 8 ⊢ (𝜑 → dom (𝑆 ↾ 𝐶) = 𝐶)
13	12	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → dom (𝑆 ↾ 𝐶) = 𝐶)
14		simplr 780	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ∈ 𝐶)
15		simprl 782	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → 𝑌 ∈ 𝐶)
16		simprr 784	. . . . . . 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 20075	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐶)‘𝑋) ⊆ (𝑍‘((𝑆 ↾ 𝐶)‘𝑌)))
19		fvres 6900	. . . . . . 7 ⊢ (𝑋 ∈ 𝐶 → ((𝑆 ↾ 𝐶)‘𝑋) = (𝑆‘𝑋))
20	19	ad2antlr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐶)‘𝑋) = (𝑆‘𝑋))
21		fvres 6900	. . . . . . . 8 ⊢ (𝑌 ∈ 𝐶 → ((𝑆 ↾ 𝐶)‘𝑌) = (𝑆‘𝑌))
22	21	ad2antrl 740	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐶)‘𝑌) = (𝑆‘𝑌))
23	22	fveq2d 6885	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → (𝑍‘((𝑆 ↾ 𝐶)‘𝑌)) = (𝑍‘(𝑆‘𝑌)))
24	18, 20, 23	3sstr3d 3991	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))
25	24	exp32 425	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑌 ∈ 𝐶 → (𝑋 ≠ 𝑌 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))))
26	19	ad2antlr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐶)‘𝑋) = (𝑆‘𝑋))
27	6	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → 𝐺dom DProd (𝑆 ↾ 𝐶))
28	12	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → dom (𝑆 ↾ 𝐶) = 𝐶)
29		simplr 780	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ∈ 𝐶)
30	27, 28, 29	dprdub 20092	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐶)‘𝑋) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
31	26, 30	eqsstrrd 3972	. . . . . 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 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))))
34		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
35	34	dprdssv 20083	. . . . . . . 8 ⊢ (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (Base‘𝐺)
36		fvres 6900	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝐷 → ((𝑆 ↾ 𝐷)‘𝑌) = (𝑆‘𝑌))
37	36	ad2antrl 740	. . . . . . . . 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 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → 𝐺dom DProd (𝑆 ↾ 𝐷))
40		ssun2 4132	. . . . . . . . . . . . . 14 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷)
41	40, 1	sseqtrrid 3980	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ⊆ 𝐼)
42	8, 41	fssresd 6745	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
43	42	fdmd 6716	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝑆 ↾ 𝐷) = 𝐷)
44	43	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → dom (𝑆 ↾ 𝐷) = 𝐷)
45		simprl 782	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → 𝑌 ∈ 𝐷)
46	39, 44, 45	dprdub 20092	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐷)‘𝑌) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷)))
47	37, 46	eqsstrrd 3972	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝑆‘𝑌) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷)))
48	34, 17	cntz2ss 19400	. . . . . . . 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 598	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ (𝑍‘(𝑆‘𝑌)))
50	33, 49	sstrd 3947	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝑆‘𝑌)))
51	31, 50	sstrd 3947	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))
52	51	exp32 425	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑌 ∈ 𝐷 → (𝑋 ≠ 𝑌 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))))
53	25, 52	jaod 872	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑌 ∈ 𝐶 ∨ 𝑌 ∈ 𝐷) → (𝑋 ≠ 𝑌 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))))
54	5, 53	sylbid 243	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑌 ∈ 𝐼 → (𝑋 ≠ 𝑌 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))))
55		dprdgrp 20072	. . . . . . . 8 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐶) → 𝐺 ∈ Grp)
56	6, 55	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Grp)
57	56	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐺 ∈ Grp)
58	34	subgacs 19222	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
59		acsmre 17703	. . . . . 6 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
60	57, 58, 59	3syl 19	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
61		difundir 4244	. . . . . . . . . . 11 ⊢ ((𝐶 ∪ 𝐷) ∖ {𝑋}) = ((𝐶 ∖ {𝑋}) ∪ (𝐷 ∖ {𝑋}))
62	2	difeq1d 4080	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐼 ∖ {𝑋}) = ((𝐶 ∪ 𝐷) ∖ {𝑋}))
63		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝐶)
64	63	snssd 4752	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {𝑋} ⊆ 𝐶)
65		sslin 4195	. . . . . . . . . . . . . . 15 ⊢ ({𝑋} ⊆ 𝐶 → (𝐷 ∩ {𝑋}) ⊆ (𝐷 ∩ 𝐶))
66	64, 65	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐷 ∩ {𝑋}) ⊆ (𝐷 ∩ 𝐶))
67		incom 4162	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∩ 𝐷) = (𝐷 ∩ 𝐶)
68		dprdsplit.i	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)
69	68	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐶 ∩ 𝐷) = ∅)
70	67, 69	eqtr3id 2812	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐷 ∩ 𝐶) = ∅)
71		sseq0 4361	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∩ {𝑋}) ⊆ (𝐷 ∩ 𝐶) ∧ (𝐷 ∩ 𝐶) = ∅) → (𝐷 ∩ {𝑋}) = ∅)
72	66, 70, 71	syl2anc 595	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐷 ∩ {𝑋}) = ∅)
73		disj3 4414	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∩ {𝑋}) = ∅ ↔ 𝐷 = (𝐷 ∖ {𝑋}))
74	72, 73	sylib 221	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐷 = (𝐷 ∖ {𝑋}))
75	74	uneq2d 4122	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐶 ∖ {𝑋}) ∪ 𝐷) = ((𝐶 ∖ {𝑋}) ∪ (𝐷 ∖ {𝑋})))
76	61, 62, 75	3eqtr4a 2824	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐼 ∖ {𝑋}) = ((𝐶 ∖ {𝑋}) ∪ 𝐷))
77	76	imaeq2d 6062	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆 “ (𝐼 ∖ {𝑋})) = (𝑆 “ ((𝐶 ∖ {𝑋}) ∪ 𝐷)))
78		imaundi 6147	. . . . . . . . 9 ⊢ (𝑆 “ ((𝐶 ∖ {𝑋}) ∪ 𝐷)) = ((𝑆 “ (𝐶 ∖ {𝑋})) ∪ (𝑆 “ 𝐷))
79	77, 78	eqtrdi 2814	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆 “ (𝐼 ∖ {𝑋})) = ((𝑆 “ (𝐶 ∖ {𝑋})) ∪ (𝑆 “ 𝐷)))
80	79	unieqd 4885	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐼 ∖ {𝑋})) = ∪ ((𝑆 “ (𝐶 ∖ {𝑋})) ∪ (𝑆 “ 𝐷)))
81		uniun 4895	. . . . . . 7 ⊢ ∪ ((𝑆 “ (𝐶 ∖ {𝑋})) ∪ (𝑆 “ 𝐷)) = (∪ (𝑆 “ (𝐶 ∖ {𝑋})) ∪ ∪ (𝑆 “ 𝐷))
82	80, 81	eqtrdi 2814	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐼 ∖ {𝑋})) = (∪ (𝑆 “ (𝐶 ∖ {𝑋})) ∪ ∪ (𝑆 “ 𝐷)))
83		dmdprdsplit2lem.k	. . . . . . . . 9 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
84		difss 4090	. . . . . . . . . . 11 ⊢ (𝐶 ∖ {𝑋}) ⊆ 𝐶
85		imass2 6104	. . . . . . . . . . 11 ⊢ ((𝐶 ∖ {𝑋}) ⊆ 𝐶 → (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (𝑆 “ 𝐶))
86		uniss 4880	. . . . . . . . . . 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 6073	. . . . . . . . . . . 12 ⊢ (𝑆 “ 𝐶) ⊆ ran 𝑆
89	8	frnd 6714	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
90	89	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ran 𝑆 ⊆ (SubGrp‘𝐺))
91		mresspw 17639	. . . . . . . . . . . . . 14 ⊢ ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
92	60, 91	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
93	90, 92	sstrd 3947	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
94	88, 93	sstrid 3948	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆 “ 𝐶) ⊆ 𝒫 (Base‘𝐺))
95		sspwuni 5066	. . . . . . . . . . 11 ⊢ ((𝑆 “ 𝐶) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆 “ 𝐶) ⊆ (Base‘𝐺))
96	94, 95	sylib 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ 𝐶) ⊆ (Base‘𝐺))
97	87, 96	sstrid 3948	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (Base‘𝐺))
98	60, 83, 97	mrcssidd 17676	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))))
99		imassrn 6073	. . . . . . . . . . . 12 ⊢ (𝑆 “ 𝐷) ⊆ ran 𝑆
100	99, 93	sstrid 3948	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆 “ 𝐷) ⊆ 𝒫 (Base‘𝐺))
101		sspwuni 5066	. . . . . . . . . . 11 ⊢ ((𝑆 “ 𝐷) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆 “ 𝐷) ⊆ (Base‘𝐺))
102	100, 101	sylib 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ 𝐷) ⊆ (Base‘𝐺))
103	60, 83, 102	mrcssidd 17676	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ 𝐷) ⊆ (𝐾‘∪ (𝑆 “ 𝐷)))
104	83	dprdspan 20094	. . . . . . . . . . . 12 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐷) → (𝐺 DProd (𝑆 ↾ 𝐷)) = (𝐾‘∪ ran (𝑆 ↾ 𝐷)))
105	38, 104	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) = (𝐾‘∪ ran (𝑆 ↾ 𝐷)))
106		df-ima 5674	. . . . . . . . . . . . 13 ⊢ (𝑆 “ 𝐷) = ran (𝑆 ↾ 𝐷)
107	106	unieqi 4884	. . . . . . . . . . . 12 ⊢ ∪ (𝑆 “ 𝐷) = ∪ ran (𝑆 ↾ 𝐷)
108	107	fveq2i 6884	. . . . . . . . . . 11 ⊢ (𝐾‘∪ (𝑆 “ 𝐷)) = (𝐾‘∪ ran (𝑆 ↾ 𝐷))
109	105, 108	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) = (𝐾‘∪ (𝑆 “ 𝐷)))
110	109	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐷)) = (𝐾‘∪ (𝑆 “ 𝐷)))
111	103, 110	sseqtrrd 3974	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ 𝐷) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷)))
112		unss12 4141	. . . . . . . 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 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (∪ (𝑆 “ (𝐶 ∖ {𝑋})) ∪ ∪ (𝑆 “ 𝐷)) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∪ (𝐺 DProd (𝑆 ↾ 𝐷))))
114	83	mrccl 17662	. . . . . . . . 9 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (Base‘𝐺)) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∈ (SubGrp‘𝐺))
115	60, 97, 114	syl2anc 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∈ (SubGrp‘𝐺))
116		dprdsubg 20091	. . . . . . . . . 10 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐷) → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
117	38, 116	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
118	117	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
119		eqid 2763	. . . . . . . . 9 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
120	119	lsmunss 19724	. . . . . . . 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 595	. . . . . . 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 3947	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (∪ (𝑆 “ (𝐶 ∖ {𝑋})) ∪ ∪ (𝑆 “ 𝐷)) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))))
123	82, 122	eqsstrd 3971	. . . . 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 17675	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝐾‘∪ (𝑆 “ 𝐶)))
126	83	dprdspan 20094	. . . . . . . . . . 11 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) = (𝐾‘∪ ran (𝑆 ↾ 𝐶)))
127	6, 126	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) = (𝐾‘∪ ran (𝑆 ↾ 𝐶)))
128		df-ima 5674	. . . . . . . . . . . 12 ⊢ (𝑆 “ 𝐶) = ran (𝑆 ↾ 𝐶)
129	128	unieqi 4884	. . . . . . . . . . 11 ⊢ ∪ (𝑆 “ 𝐶) = ∪ ran (𝑆 ↾ 𝐶)
130	129	fveq2i 6884	. . . . . . . . . 10 ⊢ (𝐾‘∪ (𝑆 “ 𝐶)) = (𝐾‘∪ ran (𝑆 ↾ 𝐶))
131	127, 130	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) = (𝐾‘∪ (𝑆 “ 𝐶)))
132	131	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) = (𝐾‘∪ (𝑆 “ 𝐶)))
133	125, 132	sseqtrrd 3974	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
134	32	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))))
135	133, 134	sstrd 3947	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))))
136	119, 17	lsmsubg 19719	. . . . . 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 1398	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))) ∈ (SubGrp‘𝐺))
138	83	mrcsscl 17671	. . . . 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 1398	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋}))) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))))
140		sslin 4195	. . . 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 18	. . 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 3937	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝐼)
143	8	ffvelcdmda 7079	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐼) → (𝑆‘𝑋) ∈ (SubGrp‘𝐺))
144	142, 143	syldan 602	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆‘𝑋) ∈ (SubGrp‘𝐺))
145		dmdprdsplit.0	. . . 4 ⊢ 0 = (0g‘𝐺)
146	19	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆 ↾ 𝐶)‘𝑋) = (𝑆‘𝑋))
147	6	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐺dom DProd (𝑆 ↾ 𝐶))
148	12	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → dom (𝑆 ↾ 𝐶) = 𝐶)
149	147, 148, 63	dprdub 20092	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆 ↾ 𝐶)‘𝑋) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
150	146, 149	eqsstrrd 3972	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆‘𝑋) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
151		dprdsubg 20091	. . . . . . . . . . 11 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺))
152	6, 151	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺))
153	152	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺))
154	119	lsmlub 19729	. . . . . . . . 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 1398	. . . . . . . 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 724	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
157	156	ssrind 4196	. . . . . 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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })
160	157, 159	sseqtrd 3973	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ { 0 })
161	119	lsmub1 19722	. . . . . . . . 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 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆‘𝑋) ⊆ ((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))))
163	145	subg0cl 19195	. . . . . . . . 9 ⊢ ((𝑆‘𝑋) ∈ (SubGrp‘𝐺) → 0 ∈ (𝑆‘𝑋))
164	144, 163	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 0 ∈ (𝑆‘𝑋))
165	162, 164	sseldd 3938	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 0 ∈ ((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))))
166	145	subg0cl 19195	. . . . . . . 8 ⊢ ((𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺) → 0 ∈ (𝐺 DProd (𝑆 ↾ 𝐷)))
167	118, 166	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 0 ∈ (𝐺 DProd (𝑆 ↾ 𝐷)))
168	165, 167	elind 4153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 0 ∈ (((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))))
169	168	snssd 4752	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → { 0 } ⊆ (((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))))
170	160, 169	eqssd 3954	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })
171		resima2 6015	. . . . . . . . 9 ⊢ ((𝐶 ∖ {𝑋}) ⊆ 𝐶 → ((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋})) = (𝑆 “ (𝐶 ∖ {𝑋})))
172	84, 171	mp1i 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋})) = (𝑆 “ (𝐶 ∖ {𝑋})))
173	172	unieqd 4885	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ ((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋})) = ∪ (𝑆 “ (𝐶 ∖ {𝑋})))
174	173	fveq2d 6885	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ ((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋}))) = (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))))
175	146, 174	ineq12d 4174	. . . . 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 20076	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝑆 ↾ 𝐶)‘𝑋) ∩ (𝐾‘∪ ((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋})))) = { 0 })
177	175, 176	eqtr3d 2800	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) = { 0 })
178	8	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑆:𝐼⟶(SubGrp‘𝐺))
179		ffun 6708	. . . . . . . 8 ⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → Fun 𝑆)
180		funiunfv 7246	. . . . . . . 8 ⊢ (Fun 𝑆 → ∪ 𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) = ∪ (𝑆 “ (𝐶 ∖ {𝑋})))
181	178, 179, 180	3syl 19	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ 𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) = ∪ (𝑆 “ (𝐶 ∖ {𝑋})))
182	6	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → 𝐺dom DProd (𝑆 ↾ 𝐶))
183	12	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → dom (𝑆 ↾ 𝐶) = 𝐶)
184		eldifi 4085	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐶 ∖ {𝑋}) → 𝑦 ∈ 𝐶)
185	184	adantl 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → 𝑦 ∈ 𝐶)
186		simplr 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → 𝑋 ∈ 𝐶)
187		eldifsni 4758	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐶 ∖ {𝑋}) → 𝑦 ≠ 𝑋)
188	187	adantl 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → 𝑦 ≠ 𝑋)
189	182, 183, 185, 186, 188, 17	dprdcntz 20075	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → ((𝑆 ↾ 𝐶)‘𝑦) ⊆ (𝑍‘((𝑆 ↾ 𝐶)‘𝑋)))
190	185	fvresd 6901	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → ((𝑆 ↾ 𝐶)‘𝑦) = (𝑆‘𝑦))
191	19	ad2antlr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → ((𝑆 ↾ 𝐶)‘𝑋) = (𝑆‘𝑋))
192	191	fveq2d 6885	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → (𝑍‘((𝑆 ↾ 𝐶)‘𝑋)) = (𝑍‘(𝑆‘𝑋)))
193	189, 190, 192	3sstr3d 3991	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → (𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑋)))
194	193	ralrimiva 3157	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∀𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑋)))
195		iunss 5009	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑋)) ↔ ∀𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑋)))
196	194, 195	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ 𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑋)))
197	181, 196	eqsstrrd 3972	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (𝑍‘(𝑆‘𝑋)))
198	34	subgss 19188	. . . . . . . 8 ⊢ ((𝑆‘𝑋) ∈ (SubGrp‘𝐺) → (𝑆‘𝑋) ⊆ (Base‘𝐺))
199	144, 198	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆‘𝑋) ⊆ (Base‘𝐺))
200	34, 17	cntzsubg 19404	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑆‘𝑋) ⊆ (Base‘𝐺)) → (𝑍‘(𝑆‘𝑋)) ∈ (SubGrp‘𝐺))
201	57, 199, 200	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑍‘(𝑆‘𝑋)) ∈ (SubGrp‘𝐺))
202	83	mrcsscl 17671	. . . . . 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 1398	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝑍‘(𝑆‘𝑋)))
204	17, 115, 144, 203	cntzrecd 19743	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆‘𝑋) ⊆ (𝑍‘(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))))
205	119, 144, 115, 118, 145, 170, 177, 17, 204	lsmdisj3 19748	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆‘𝑋) ∩ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷)))) = { 0 })
206	141, 205	sseqtrd 3973	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) ⊆ { 0 })
207	54, 206	jca 520	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 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4562  {csn 4589  ∪ cuni 4872  ∪ ciun 4956   class class class wbr 5109  dom cdm 5661  ran crn 5662   ↾ cres 5663   “ cima 5664  Fun wfun 6530  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  0_gc0g 17487  Moorecmre 17629  mrClscmrc 17630  ACScacs 17632  Grpcgrp 18995  SubGrpcsubg 19181  Cntzccntz 19380  LSSumclsm 19699   DProd cdprd 20060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-oi 9468  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-n0 12500  df-z 12587  df-uz 12858  df-fz 13531  df-fzo 13679  df-seq 14034  df-hash 14363  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-0g 17489  df-gsum 17490  df-mre 17633  df-mrc 17634  df-acs 17636  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-mhm 18836  df-submnd 18837  df-grp 18998  df-minusg 18999  df-sbg 19000  df-mulg 19129  df-subg 19184  df-ghm 19279  df-gim 19324  df-cntz 19382  df-oppg 19411  df-lsm 19701  df-cmn 19847  df-dprd 20062

This theorem is referenced by:  dmdprdsplit2  20113