dmdprdsplit2 - Metamath Proof Explorer

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Theorem dmdprdsplit2 20313

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 2763	. 2 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		dmdprdsplit2.1	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐶))
5		dprdgrp 20272	. . 3 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐶) → 𝐺 ∈ Grp)
6	4, 5	syl 18	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		dprdsplit.u	. . 3 ⊢ (𝜑 → 𝐼 = (𝐶 ∪ 𝐷))
8		dprdsplit.2	. . . . . . 7 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
9		ssun1 4179	. . . . . . . 8 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷)
10	9, 7	sseqtrrid 3956	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐼)
11	8, 10	fssresd 6779	. . . . . 6 ⊢ (𝜑 → (𝑆 ↾ 𝐶):𝐶⟶(SubGrp‘𝐺))
12	11	fdmd 6750	. . . . 5 ⊢ (𝜑 → dom (𝑆 ↾ 𝐶) = 𝐶)
13	4, 12	dprddomcld 20268	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
14		dmdprdsplit2.2	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐷))
15		ssun2 4180	. . . . . . . 8 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷)
16	15, 7	sseqtrrid 3956	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ 𝐼)
17	8, 16	fssresd 6779	. . . . . 6 ⊢ (𝜑 → (𝑆 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
18	17	fdmd 6750	. . . . 5 ⊢ (𝜑 → dom (𝑆 ↾ 𝐷) = 𝐷)
19	14, 18	dprddomcld 20268	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
20		unexg 7792	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ∪ 𝐷) ∈ V)
21	13, 19, 20	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐶 ∪ 𝐷) ∈ V)
22	7, 21	eqeltrd 2864	. 2 ⊢ (𝜑 → 𝐼 ∈ V)
23	7	eleq2d 2850	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↔ 𝑥 ∈ (𝐶 ∪ 𝐷)))
24		elun 4155	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ∪ 𝐷) ↔ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷))
25	23, 24	bitrdi 290	. . . 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 20312	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑦 ∈ 𝐼 → (𝑥 ≠ 𝑦 → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)))) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 }))
30		incom 4109	. . . . . . . . 9 ⊢ (𝐶 ∩ 𝐷) = (𝐷 ∩ 𝐶)
31	30, 26	eqtr3id 2812	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ∩ 𝐶) = ∅)
32		uncom 4160	. . . . . . . . 9 ⊢ (𝐶 ∪ 𝐷) = (𝐷 ∪ 𝐶)
33	7, 32	eqtrdi 2814	. . . . . . . 8 ⊢ (𝜑 → 𝐼 = (𝐷 ∪ 𝐶))
34		dprdsubg 20291	. . . . . . . . . 10 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺))
35	4, 34	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺))
36		dprdsubg 20291	. . . . . . . . . 10 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐷) → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
37	14, 36	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
38	1, 35, 37, 27	cntzrecd 19943	. . . . . . . 8 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐶))))
39		incom 4109	. . . . . . . . 9 ⊢ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = ((𝐺 DProd (𝑆 ↾ 𝐷)) ∩ (𝐺 DProd (𝑆 ↾ 𝐶)))
40	39, 28	eqtr3id 2812	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐷)) ∩ (𝐺 DProd (𝑆 ↾ 𝐶))) = { 0 })
41	8, 31, 33, 1, 2, 14, 4, 38, 40, 3	dmdprdsplit2lem 20312	. . . . . . 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 972	. . . . . 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 500	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷)) → (𝑦 ∈ 𝐼 → (𝑥 ≠ 𝑦 → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)))))
44	43	ex 418	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷) → (𝑦 ∈ 𝐼 → (𝑥 ≠ 𝑦 → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦))))))
45	25, 44	sylbid 243	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 → (𝑦 ∈ 𝐼 → (𝑥 ≠ 𝑦 → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦))))))
46	45	3imp2 1368	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)))
47	25	biimpa 482	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷))
48	29	simprd 501	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 })
49	41	simprd 501	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 })
50	48, 49	jaodan 972	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷)) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 })
51	47, 50	syldan 603	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 })
52	1, 2, 3, 6, 22, 8, 46, 51	dmdprdd 20266	1 ⊢ (𝜑 → 𝐺dom DProd 𝑆)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 Vcvv 3454 ⊆ wss 3900 ∖ cdif 4048 ∩ cin 4092 ∪ cun 4152 ∅c0 4281 {csn 4588 ∪ cuni 4872 class class class wbr 5108 dom cdm 5658 ↾ cres 5660 “ cima 5661 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 0_gc0g 17660 mrClscmrc 17803 Grpcgrp 19194 SubGrpcsubg 19380 Cntzccntz 19580 DProd cdprd 20260

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-tpos 8281 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-n0 12647 df-z 12734 df-uz 13006 df-fz 13680 df-fzo 13829 df-seq 14185 df-hash 14516 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-0g 17662 df-gsum 17663 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-mhm 19028 df-submnd 19029 df-grp 19197 df-minusg 19198 df-sbg 19199 df-mulg 19328 df-subg 19383 df-ghm 19478 df-gim 19523 df-cntz 19582 df-oppg 19611 df-lsm 19901 df-cmn 20047 df-dprd 20262

This theorem is used by: dmdprdsplit 20314 pgpfaclem1 20348

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