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Theorem dmdprdsplit2 19977

Description: The direct product splits into the direct product of any partition of the index set. (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))
dmdprdsplit.z	⊢ 𝑍 = (Cntz‘𝐺)
dmdprdsplit.0	⊢ 0 = (0_g‘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 })

**Assertion**
Ref	Expression
dmdprdsplit2	⊢ (𝜑 → 𝐺dom DProd 𝑆)

Proof of Theorem dmdprdsplit2 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmdprdsplit.z	. 2 ⊢ 𝑍 = (Cntz‘𝐺)
2		dmdprdsplit.0	. 2 ⊢ 0 = (0_g‘𝐺)
3		eqid 2736	. 2 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		dmdprdsplit2.1	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐶))
5		dprdgrp 19936	. . 3 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐶) → 𝐺 ∈ Grp)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		dprdsplit.u	. . 3 ⊢ (𝜑 → 𝐼 = (𝐶 ∪ 𝐷))
8		dprdsplit.2	. . . . . . 7 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
9		ssun1 4130	. . . . . . . 8 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷)
10	9, 7	sseqtrrid 3977	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐼)
11	8, 10	fssresd 6701	. . . . . 6 ⊢ (𝜑 → (𝑆 ↾ 𝐶):𝐶⟶(SubGrp‘𝐺))
12	11	fdmd 6672	. . . . 5 ⊢ (𝜑 → dom (𝑆 ↾ 𝐶) = 𝐶)
13	4, 12	dprddomcld 19932	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
14		dmdprdsplit2.2	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐷))
15		ssun2 4131	. . . . . . . 8 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷)
16	15, 7	sseqtrrid 3977	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ 𝐼)
17	8, 16	fssresd 6701	. . . . . 6 ⊢ (𝜑 → (𝑆 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
18	17	fdmd 6672	. . . . 5 ⊢ (𝜑 → dom (𝑆 ↾ 𝐷) = 𝐷)
19	14, 18	dprddomcld 19932	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
20		unexg 7688	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ∪ 𝐷) ∈ V)
21	13, 19, 20	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐶 ∪ 𝐷) ∈ V)
22	7, 21	eqeltrd 2836	. 2 ⊢ (𝜑 → 𝐼 ∈ V)
23	7	eleq2d 2822	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↔ 𝑥 ∈ (𝐶 ∪ 𝐷)))
24		elun 4105	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ∪ 𝐷) ↔ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷))
25	23, 24	bitrdi 287	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↔ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷)))
26		dprdsplit.i	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)
27		dmdprdsplit2.3	. . . . . . . 8 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))))
28		dmdprdsplit2.4	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })
29	8, 26, 7, 1, 2, 4, 14, 27, 28, 3	dmdprdsplit2lem 19976	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑦 ∈ 𝐼 → (𝑥 ≠ 𝑦 → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)))) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 }))
30		incom 4161	. . . . . . . . 9 ⊢ (𝐶 ∩ 𝐷) = (𝐷 ∩ 𝐶)
31	30, 26	eqtr3id 2785	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ∩ 𝐶) = ∅)
32		uncom 4110	. . . . . . . . 9 ⊢ (𝐶 ∪ 𝐷) = (𝐷 ∪ 𝐶)
33	7, 32	eqtrdi 2787	. . . . . . . 8 ⊢ (𝜑 → 𝐼 = (𝐷 ∪ 𝐶))
34		dprdsubg 19955	. . . . . . . . . 10 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺))
35	4, 34	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺))
36		dprdsubg 19955	. . . . . . . . . 10 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐷) → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
37	14, 36	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
38	1, 35, 37, 27	cntzrecd 19607	. . . . . . . 8 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐶))))
39		incom 4161	. . . . . . . . 9 ⊢ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = ((𝐺 DProd (𝑆 ↾ 𝐷)) ∩ (𝐺 DProd (𝑆 ↾ 𝐶)))
40	39, 28	eqtr3id 2785	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐷)) ∩ (𝐺 DProd (𝑆 ↾ 𝐶))) = { 0 })
41	8, 31, 33, 1, 2, 14, 4, 38, 40, 3	dmdprdsplit2lem 19976	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑦 ∈ 𝐼 → (𝑥 ≠ 𝑦 → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)))) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 }))
42	29, 41	jaodan 959	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷)) → ((𝑦 ∈ 𝐼 → (𝑥 ≠ 𝑦 → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)))) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 }))
43	42	simpld 494	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷)) → (𝑦 ∈ 𝐼 → (𝑥 ≠ 𝑦 → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)))))
44	43	ex 412	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷) → (𝑦 ∈ 𝐼 → (𝑥 ≠ 𝑦 → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦))))))
45	25, 44	sylbid 240	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 → (𝑦 ∈ 𝐼 → (𝑥 ≠ 𝑦 → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦))))))
46	45	3imp2 1350	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)))
47	25	biimpa 476	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷))
48	29	simprd 495	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 })
49	41	simprd 495	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 })
50	48, 49	jaodan 959	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷)) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 })
51	47, 50	syldan 591	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 })
52	1, 2, 3, 6, 22, 8, 46, 51	dmdprdd 19930	1 ⊢ (𝜑 → 𝐺dom DProd 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 Vcvv 3440 ∖ cdif 3898 ∪ cun 3899 ∩ cin 3900 ⊆ wss 3901 ∅c0 4285 {csn 4580 ∪ cuni 4863 class class class wbr 5098 dom cdm 5624 ↾ cres 5626 “ cima 5627 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 0_gc0g 17359 mrClscmrc 17502 Grpcgrp 18863 SubGrpcsubg 19050 Cntzccntz 19244 DProd cdprd 19924

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 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-n0 12402 df-z 12489 df-uz 12752 df-fz 13424 df-fzo 13571 df-seq 13925 df-hash 14254 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-0g 17361 df-gsum 17362 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18998 df-subg 19053 df-ghm 19142 df-gim 19188 df-cntz 19246 df-oppg 19275 df-lsm 19565 df-cmn 19711 df-dprd 19926

This theorem is referenced by: dmdprdsplit 19978 pgpfaclem1 20012

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