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Theorem dprdsplit 19172

Description: The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 25-Apr-2016.)

**Hypotheses**
Ref	Expression
dprdsplit.2	⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
dprdsplit.i	⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)
dprdsplit.u	⊢ (𝜑 → 𝐼 = (𝐶 ∪ 𝐷))
dprdsplit.s	⊢ ⊕ = (LSSum‘𝐺)
dprdsplit.1	⊢ (𝜑 → 𝐺dom DProd 𝑆)

**Assertion**
Ref	Expression
dprdsplit	⊢ (𝜑 → (𝐺 DProd 𝑆) = ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))

Proof of Theorem dprdsplit Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dprdsplit.1	. . 3 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
2		dprdsplit.2	. . . 4 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
3	2	fdmd 6525	. . 3 ⊢ (𝜑 → dom 𝑆 = 𝐼)
4		ssun1 4150	. . . . . . . 8 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷)
5		dprdsplit.u	. . . . . . . 8 ⊢ (𝜑 → 𝐼 = (𝐶 ∪ 𝐷))
6	4, 5	sseqtrrid 4022	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐼)
7	1, 3, 6	dprdres 19152	. . . . . 6 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ 𝐶) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝐺 DProd 𝑆)))
8	7	simpld 497	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐶))
9		dprdsubg 19148	. . . . 5 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺))
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺))
11		ssun2 4151	. . . . . . . 8 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷)
12	11, 5	sseqtrrid 4022	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ 𝐼)
13	1, 3, 12	dprdres 19152	. . . . . 6 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ 𝐷) ∧ (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (𝐺 DProd 𝑆)))
14	13	simpld 497	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐷))
15		dprdsubg 19148	. . . . 5 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐷) → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
16	14, 15	syl 17	. . . 4 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
17		dprdsplit.i	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)
18		eqid 2823	. . . . . . 7 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
19		eqid 2823	. . . . . . 7 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
20	2, 17, 5, 18, 19	dmdprdsplit 19171	. . . . . 6 ⊢ (𝜑 → (𝐺dom DProd 𝑆 ↔ ((𝐺dom DProd (𝑆 ↾ 𝐶) ∧ 𝐺dom DProd (𝑆 ↾ 𝐷)) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ 𝐷))) ∧ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = {(0_g‘𝐺)})))
21	1, 20	mpbid 234	. . . . 5 ⊢ (𝜑 → ((𝐺dom DProd (𝑆 ↾ 𝐶) ∧ 𝐺dom DProd (𝑆 ↾ 𝐷)) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ 𝐷))) ∧ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = {(0_g‘𝐺)}))
22	21	simp2d 1139	. . . 4 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ 𝐷))))
23		dprdsplit.s	. . . . 5 ⊢ ⊕ = (LSSum‘𝐺)
24	23, 18	lsmsubg 18781	. . . 4 ⊢ (((𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺) ∧ (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ 𝐷)))) → ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))) ∈ (SubGrp‘𝐺))
25	10, 16, 22, 24	syl3anc 1367	. . 3 ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))) ∈ (SubGrp‘𝐺))
26	5	eleq2d 2900	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↔ 𝑥 ∈ (𝐶 ∪ 𝐷)))
27		elun 4127	. . . . . 6 ⊢ (𝑥 ∈ (𝐶 ∪ 𝐷) ↔ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷))
28	26, 27	syl6bb 289	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↔ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷)))
29	28	biimpa 479	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷))
30		fvres 6691	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐶 → ((𝑆 ↾ 𝐶)‘𝑥) = (𝑆‘𝑥))
31	30	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑆 ↾ 𝐶)‘𝑥) = (𝑆‘𝑥))
32	8	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐺dom DProd (𝑆 ↾ 𝐶))
33	2, 6	fssresd 6547	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ↾ 𝐶):𝐶⟶(SubGrp‘𝐺))
34	33	fdmd 6525	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑆 ↾ 𝐶) = 𝐶)
35	34	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → dom (𝑆 ↾ 𝐶) = 𝐶)
36		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
37	32, 35, 36	dprdub 19149	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑆 ↾ 𝐶)‘𝑥) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
38	31, 37	eqsstrrd 4008	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑆‘𝑥) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
39	23	lsmub1 18784	. . . . . . . 8 ⊢ (((𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺) ∧ (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺)) → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
40	10, 16, 39	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
41	40	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
42	38, 41	sstrd 3979	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑆‘𝑥) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
43		fvres 6691	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((𝑆 ↾ 𝐷)‘𝑥) = (𝑆‘𝑥))
44	43	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑆 ↾ 𝐷)‘𝑥) = (𝑆‘𝑥))
45	14	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐺dom DProd (𝑆 ↾ 𝐷))
46	2, 12	fssresd 6547	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
47	46	fdmd 6525	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑆 ↾ 𝐷) = 𝐷)
48	47	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → dom (𝑆 ↾ 𝐷) = 𝐷)
49		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
50	45, 48, 49	dprdub 19149	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑆 ↾ 𝐷)‘𝑥) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷)))
51	44, 50	eqsstrrd 4008	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑆‘𝑥) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷)))
52	23	lsmub2 18785	. . . . . . . 8 ⊢ (((𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺) ∧ (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺)) → (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
53	10, 16, 52	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
54	53	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
55	51, 54	sstrd 3979	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑆‘𝑥) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
56	42, 55	jaodan 954	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷)) → (𝑆‘𝑥) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
57	29, 56	syldan 593	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
58	1, 3, 25, 57	dprdlub 19150	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
59	7	simprd 498	. . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝐺 DProd 𝑆))
60	13	simprd 498	. . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (𝐺 DProd 𝑆))
61		dprdsubg 19148	. . . . 5 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
62	1, 61	syl 17	. . . 4 ⊢ (𝜑 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
63	23	lsmlub 18792	. . . 4 ⊢ (((𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺) ∧ (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺) ∧ (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺)) → (((𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝐺 DProd 𝑆) ∧ (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (𝐺 DProd 𝑆)) ↔ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ (𝐺 DProd 𝑆)))
64	10, 16, 62, 63	syl3anc 1367	. . 3 ⊢ (𝜑 → (((𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝐺 DProd 𝑆) ∧ (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (𝐺 DProd 𝑆)) ↔ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ (𝐺 DProd 𝑆)))
65	59, 60, 64	mpbi2and 710	. 2 ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ (𝐺 DProd 𝑆))
66	58, 65	eqssd 3986	1 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∪ cun 3936 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 {csn 4569 class class class wbr 5068 dom cdm 5557 ↾ cres 5559 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 0_gc0g 16715 SubGrpcsubg 18275 Cntzccntz 18447 LSSumclsm 18761 DProd cdprd 19117

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-gim 18401 df-cntz 18449 df-oppg 18476 df-lsm 18763 df-cmn 18910 df-dprd 19119

This theorem is referenced by: dprdpr 19174 dpjlsm 19178 ablfac1eulem 19196 ablfac1eu 19197 pgpfaclem1 19205

W3C validator