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Theorem dmdprdsplit2 19989

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 2737	. 2 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		dmdprdsplit2.1	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐶))
5		dprdgrp 19948	. . 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 4132	. . . . . . . 8 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷)
10	9, 7	sseqtrrid 3979	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐼)
11	8, 10	fssresd 6709	. . . . . 6 ⊢ (𝜑 → (𝑆 ↾ 𝐶):𝐶⟶(SubGrp‘𝐺))
12	11	fdmd 6680	. . . . 5 ⊢ (𝜑 → dom (𝑆 ↾ 𝐶) = 𝐶)
13	4, 12	dprddomcld 19944	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
14		dmdprdsplit2.2	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐷))
15		ssun2 4133	. . . . . . . 8 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷)
16	15, 7	sseqtrrid 3979	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ 𝐼)
17	8, 16	fssresd 6709	. . . . . 6 ⊢ (𝜑 → (𝑆 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
18	17	fdmd 6680	. . . . 5 ⊢ (𝜑 → dom (𝑆 ↾ 𝐷) = 𝐷)
19	14, 18	dprddomcld 19944	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
20		unexg 7698	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ∪ 𝐷) ∈ V)
21	13, 19, 20	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐶 ∪ 𝐷) ∈ V)
22	7, 21	eqeltrd 2837	. 2 ⊢ (𝜑 → 𝐼 ∈ V)
23	7	eleq2d 2823	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↔ 𝑥 ∈ (𝐶 ∪ 𝐷)))
24		elun 4107	. . . . 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 19988	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑦 ∈ 𝐼 → (𝑥 ≠ 𝑦 → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)))) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 }))
30		incom 4163	. . . . . . . . 9 ⊢ (𝐶 ∩ 𝐷) = (𝐷 ∩ 𝐶)
31	30, 26	eqtr3id 2786	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ∩ 𝐶) = ∅)
32		uncom 4112	. . . . . . . . 9 ⊢ (𝐶 ∪ 𝐷) = (𝐷 ∪ 𝐶)
33	7, 32	eqtrdi 2788	. . . . . . . 8 ⊢ (𝜑 → 𝐼 = (𝐷 ∪ 𝐶))
34		dprdsubg 19967	. . . . . . . . . 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 19967	. . . . . . . . . 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 19619	. . . . . . . 8 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐶))))
39		incom 4163	. . . . . . . . 9 ⊢ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = ((𝐺 DProd (𝑆 ↾ 𝐷)) ∩ (𝐺 DProd (𝑆 ↾ 𝐶)))
40	39, 28	eqtr3id 2786	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐷)) ∩ (𝐺 DProd (𝑆 ↾ 𝐶))) = { 0 })
41	8, 31, 33, 1, 2, 14, 4, 38, 40, 3	dmdprdsplit2lem 19988	. . . . . . 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 960	. . . . . 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 1351	. 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 960	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷)) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 })
51	47, 50	syldan 592	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 })
52	1, 2, 3, 6, 22, 8, 46, 51	dmdprdd 19942	1 ⊢ (𝜑 → 𝐺dom DProd 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3442 ∖ cdif 3900 ∪ cun 3901 ∩ cin 3902 ⊆ wss 3903 ∅c0 4287 {csn 4582 ∪ cuni 4865 class class class wbr 5100 dom cdm 5632 ↾ cres 5634 “ cima 5635 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 0_gc0g 17371 mrClscmrc 17514 Grpcgrp 18875 SubGrpcsubg 19062 Cntzccntz 19256 DProd cdprd 19936

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-seq 13937 df-hash 14266 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-0g 17373 df-gsum 17374 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-ghm 19154 df-gim 19200 df-cntz 19258 df-oppg 19287 df-lsm 19577 df-cmn 19723 df-dprd 19938

This theorem is referenced by: dmdprdsplit 19990 pgpfaclem1 20024

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