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Theorem dmdprdsplit2 19978

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 2729	. 2 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		dmdprdsplit2.1	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐶))
5		dprdgrp 19937	. . 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 4141	. . . . . . . 8 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷)
10	9, 7	sseqtrrid 3990	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐼)
11	8, 10	fssresd 6727	. . . . . 6 ⊢ (𝜑 → (𝑆 ↾ 𝐶):𝐶⟶(SubGrp‘𝐺))
12	11	fdmd 6698	. . . . 5 ⊢ (𝜑 → dom (𝑆 ↾ 𝐶) = 𝐶)
13	4, 12	dprddomcld 19933	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
14		dmdprdsplit2.2	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐷))
15		ssun2 4142	. . . . . . . 8 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷)
16	15, 7	sseqtrrid 3990	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ 𝐼)
17	8, 16	fssresd 6727	. . . . . 6 ⊢ (𝜑 → (𝑆 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
18	17	fdmd 6698	. . . . 5 ⊢ (𝜑 → dom (𝑆 ↾ 𝐷) = 𝐷)
19	14, 18	dprddomcld 19933	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
20		unexg 7719	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ∪ 𝐷) ∈ V)
21	13, 19, 20	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐶 ∪ 𝐷) ∈ V)
22	7, 21	eqeltrd 2828	. 2 ⊢ (𝜑 → 𝐼 ∈ V)
23	7	eleq2d 2814	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↔ 𝑥 ∈ (𝐶 ∪ 𝐷)))
24		elun 4116	. . . . 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 19977	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑦 ∈ 𝐼 → (𝑥 ≠ 𝑦 → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)))) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 }))
30		incom 4172	. . . . . . . . 9 ⊢ (𝐶 ∩ 𝐷) = (𝐷 ∩ 𝐶)
31	30, 26	eqtr3id 2778	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ∩ 𝐶) = ∅)
32		uncom 4121	. . . . . . . . 9 ⊢ (𝐶 ∪ 𝐷) = (𝐷 ∪ 𝐶)
33	7, 32	eqtrdi 2780	. . . . . . . 8 ⊢ (𝜑 → 𝐼 = (𝐷 ∪ 𝐶))
34		dprdsubg 19956	. . . . . . . . . 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 19956	. . . . . . . . . 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 19608	. . . . . . . 8 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐶))))
39		incom 4172	. . . . . . . . 9 ⊢ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = ((𝐺 DProd (𝑆 ↾ 𝐷)) ∩ (𝐺 DProd (𝑆 ↾ 𝐶)))
40	39, 28	eqtr3id 2778	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐷)) ∩ (𝐺 DProd (𝑆 ↾ 𝐶))) = { 0 })
41	8, 31, 33, 1, 2, 14, 4, 38, 40, 3	dmdprdsplit2lem 19977	. . . . . . 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 19931	1 ⊢ (𝜑 → 𝐺dom DProd 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 Vcvv 3447 ∖ cdif 3911 ∪ cun 3912 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 {csn 4589 ∪ cuni 4871 class class class wbr 5107 dom cdm 5638 ↾ cres 5640 “ cima 5641 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 0_gc0g 17402 mrClscmrc 17544 Grpcgrp 18865 SubGrpcsubg 19052 Cntzccntz 19247 DProd cdprd 19925

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-fzo 13616 df-seq 13967 df-hash 14296 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-0g 17404 df-gsum 17405 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-ghm 19145 df-gim 19191 df-cntz 19249 df-oppg 19278 df-lsm 19566 df-cmn 19712 df-dprd 19927

This theorem is referenced by: dmdprdsplit 19979 pgpfaclem1 20013

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