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Theorem dprdsplit 20151

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 6723	. . 3 ⊢ (𝜑 → dom 𝑆 = 𝐼)
4		ssun1 4134	. . . . . . . 8 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷)
5		dprdsplit.u	. . . . . . . 8 ⊢ (𝜑 → 𝐼 = (𝐶 ∪ 𝐷))
6	4, 5	sseqtrrid 3983	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐼)
7	1, 3, 6	dprdres 20131	. . . . . 6 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ 𝐶) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝐺 DProd 𝑆)))
8	7	simpld 500	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐶))
9		dprdsubg 20127	. . . . 5 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺))
10	8, 9	syl 18	. . . 4 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺))
11		ssun2 4135	. . . . . . . 8 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷)
12	11, 5	sseqtrrid 3983	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ 𝐼)
13	1, 3, 12	dprdres 20131	. . . . . 6 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ 𝐷) ∧ (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (𝐺 DProd 𝑆)))
14	13	simpld 500	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐷))
15		dprdsubg 20127	. . . . 5 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐷) → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
16	14, 15	syl 18	. . . 4 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
17		dprdsplit.i	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)
18		eqid 2766	. . . . . . 7 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
19		eqid 2766	. . . . . . 7 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
20	2, 17, 5, 18, 19	dmdprdsplit 20150	. . . . . 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 235	. . . . 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 1161	. . . 4 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ 𝐷))))
23		dprdsplit.s	. . . . 5 ⊢ ⊕ = (LSSum‘𝐺)
24	23, 18	lsmsubg 19755	. . . 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 1398	. . 3 ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))) ∈ (SubGrp‘𝐺))
26	5	eleq2d 2852	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↔ 𝑥 ∈ (𝐶 ∪ 𝐷)))
27		elun 4110	. . . . . 6 ⊢ (𝑥 ∈ (𝐶 ∪ 𝐷) ↔ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷))
28	26, 27	bitrdi 290	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↔ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷)))
29	28	biimpa 482	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷))
30		fvres 6907	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐶 → ((𝑆 ↾ 𝐶)‘𝑥) = (𝑆‘𝑥))
31	30	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑆 ↾ 𝐶)‘𝑥) = (𝑆‘𝑥))
32	8	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐺dom DProd (𝑆 ↾ 𝐶))
33	2, 6	fssresd 6752	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ↾ 𝐶):𝐶⟶(SubGrp‘𝐺))
34	33	fdmd 6723	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑆 ↾ 𝐶) = 𝐶)
35	34	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → dom (𝑆 ↾ 𝐶) = 𝐶)
36		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
37	32, 35, 36	dprdub 20128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑆 ↾ 𝐶)‘𝑥) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
38	31, 37	eqsstrrd 3975	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑆‘𝑥) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
39	23	lsmub1 19758	. . . . . . . 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 596	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
41	40	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
42	38, 41	sstrd 3950	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑆‘𝑥) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
43		fvres 6907	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((𝑆 ↾ 𝐷)‘𝑥) = (𝑆‘𝑥))
44	43	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑆 ↾ 𝐷)‘𝑥) = (𝑆‘𝑥))
45	14	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐺dom DProd (𝑆 ↾ 𝐷))
46	2, 12	fssresd 6752	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
47	46	fdmd 6723	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑆 ↾ 𝐷) = 𝐷)
48	47	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → dom (𝑆 ↾ 𝐷) = 𝐷)
49		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
50	45, 48, 49	dprdub 20128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑆 ↾ 𝐷)‘𝑥) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷)))
51	44, 50	eqsstrrd 3975	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑆‘𝑥) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷)))
52	23	lsmub2 19759	. . . . . . . 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 596	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
54	53	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
55	51, 54	sstrd 3950	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑆‘𝑥) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
56	42, 55	jaodan 972	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷)) → (𝑆‘𝑥) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
57	29, 56	syldan 603	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
58	1, 3, 25, 57	dprdlub 20129	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
59	7	simprd 501	. . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝐺 DProd 𝑆))
60	13	simprd 501	. . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (𝐺 DProd 𝑆))
61		dprdsubg 20127	. . . . 5 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
62	1, 61	syl 18	. . . 4 ⊢ (𝜑 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
63	23	lsmlub 19765	. . . 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 1398	. . 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 725	. 2 ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ (𝐺 DProd 𝑆))
66	58, 65	eqssd 3957	1 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∪ cun 3906 ∩ cin 3907 ⊆ wss 3908 ∅c0 4289 {csn 4594 class class class wbr 5114 dom cdm 5666 ↾ cres 5668 ⟶wf 6539 ‘cfv 6543 (class class class)co 7423 0_gc0g 17517 SubGrpcsubg 19217 Cntzccntz 19416 LSSumclsm 19735 DProd cdprd 20096

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-tpos 8231 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-oi 9482 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-n0 12523 df-z 12610 df-uz 12881 df-fz 13554 df-fzo 13702 df-seq 14058 df-hash 14387 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-0g 17519 df-gsum 17520 df-mre 17663 df-mrc 17664 df-acs 17666 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-mhm 18872 df-submnd 18873 df-grp 19034 df-minusg 19035 df-sbg 19036 df-mulg 19165 df-subg 19220 df-ghm 19315 df-gim 19360 df-cntz 19418 df-oppg 19447 df-lsm 19737 df-cmn 19883 df-dprd 20098

This theorem is used by: dprdpr 20153 dpjlsm 20157 ablfac1eulem 20175 ablfac1eu 20176 pgpfaclem1 20184

W3C validator