dprdsplit - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dprdsplit		Structured version Visualization version GIF version

Theorem dprdsplit 19964

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 6666	. . 3 ⊢ (𝜑 → dom 𝑆 = 𝐼)
4		ssun1 4127	. . . . . . . 8 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷)
5		dprdsplit.u	. . . . . . . 8 ⊢ (𝜑 → 𝐼 = (𝐶 ∪ 𝐷))
6	4, 5	sseqtrrid 3974	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐼)
7	1, 3, 6	dprdres 19944	. . . . . 6 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ 𝐶) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝐺 DProd 𝑆)))
8	7	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐶))
9		dprdsubg 19940	. . . . 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 4128	. . . . . . . 8 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷)
12	11, 5	sseqtrrid 3974	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ 𝐼)
13	1, 3, 12	dprdres 19944	. . . . . 6 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ 𝐷) ∧ (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (𝐺 DProd 𝑆)))
14	13	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐷))
15		dprdsubg 19940	. . . . 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 2733	. . . . . . 7 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
19		eqid 2733	. . . . . . 7 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
20	2, 17, 5, 18, 19	dmdprdsplit 19963	. . . . . 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 232	. . . . 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 1143	. . . 4 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ 𝐷))))
23		dprdsplit.s	. . . . 5 ⊢ ⊕ = (LSSum‘𝐺)
24	23, 18	lsmsubg 19568	. . . 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 1373	. . 3 ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))) ∈ (SubGrp‘𝐺))
26	5	eleq2d 2819	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↔ 𝑥 ∈ (𝐶 ∪ 𝐷)))
27		elun 4102	. . . . . 6 ⊢ (𝑥 ∈ (𝐶 ∪ 𝐷) ↔ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷))
28	26, 27	bitrdi 287	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↔ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷)))
29	28	biimpa 476	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷))
30		fvres 6847	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐶 → ((𝑆 ↾ 𝐶)‘𝑥) = (𝑆‘𝑥))
31	30	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑆 ↾ 𝐶)‘𝑥) = (𝑆‘𝑥))
32	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐺dom DProd (𝑆 ↾ 𝐶))
33	2, 6	fssresd 6695	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ↾ 𝐶):𝐶⟶(SubGrp‘𝐺))
34	33	fdmd 6666	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑆 ↾ 𝐶) = 𝐶)
35	34	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → dom (𝑆 ↾ 𝐶) = 𝐶)
36		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
37	32, 35, 36	dprdub 19941	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑆 ↾ 𝐶)‘𝑥) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
38	31, 37	eqsstrrd 3966	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑆‘𝑥) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))
39	23	lsmub1 19571	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
41	40	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
42	38, 41	sstrd 3941	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑆‘𝑥) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
43		fvres 6847	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((𝑆 ↾ 𝐷)‘𝑥) = (𝑆‘𝑥))
44	43	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑆 ↾ 𝐷)‘𝑥) = (𝑆‘𝑥))
45	14	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐺dom DProd (𝑆 ↾ 𝐷))
46	2, 12	fssresd 6695	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
47	46	fdmd 6666	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑆 ↾ 𝐷) = 𝐷)
48	47	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → dom (𝑆 ↾ 𝐷) = 𝐷)
49		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
50	45, 48, 49	dprdub 19941	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑆 ↾ 𝐷)‘𝑥) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷)))
51	44, 50	eqsstrrd 3966	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑆‘𝑥) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷)))
52	23	lsmub2 19572	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
54	53	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
55	51, 54	sstrd 3941	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑆‘𝑥) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
56	42, 55	jaodan 959	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷)) → (𝑆‘𝑥) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
57	29, 56	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
58	1, 3, 25, 57	dprdlub 19942	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))
59	7	simprd 495	. . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝐺 DProd 𝑆))
60	13	simprd 495	. . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (𝐺 DProd 𝑆))
61		dprdsubg 19940	. . . . 5 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
62	1, 61	syl 17	. . . 4 ⊢ (𝜑 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
63	23	lsmlub 19578	. . . 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 1373	. . 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 712	. 2 ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ (𝐺 DProd 𝑆))
66	58, 65	eqssd 3948	1 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∪ cun 3896 ∩ cin 3897 ⊆ wss 3898 ∅c0 4282 {csn 4575 class class class wbr 5093 dom cdm 5619 ↾ cres 5621 ⟶wf 6482 ‘cfv 6486 (class class class)co 7352 0_gc0g 17345 SubGrpcsubg 19035 Cntzccntz 19229 LSSumclsm 19548 DProd cdprd 19909

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-nn 12133 df-2 12195 df-n0 12389 df-z 12476 df-uz 12739 df-fz 13410 df-fzo 13557 df-seq 13911 df-hash 14240 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-0g 17347 df-gsum 17348 df-mre 17490 df-mrc 17491 df-acs 17493 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-mhm 18693 df-submnd 18694 df-grp 18851 df-minusg 18852 df-sbg 18853 df-mulg 18983 df-subg 19038 df-ghm 19127 df-gim 19173 df-cntz 19231 df-oppg 19260 df-lsm 19550 df-cmn 19696 df-dprd 19911

This theorem is referenced by: dprdpr 19966 dpjlsm 19970 ablfac1eulem 19988 ablfac1eu 19989 pgpfaclem1 19997

W3C validator