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Mirrors > Home > MPE Home > Th. List > dmdprdsplit2 | Structured version Visualization version GIF version |
Description: The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.) |
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 = (0g‘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 }) |
Ref | Expression |
---|---|
dmdprdsplit2 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmdprdsplit.z | . 2 ⊢ 𝑍 = (Cntz‘𝐺) | |
2 | dmdprdsplit.0 | . 2 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2818 | . 2 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺)) | |
4 | dmdprdsplit2.1 | . . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐶)) | |
5 | dprdgrp 19056 | . . 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 4145 | . . . . . . . 8 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷) | |
10 | 9, 7 | sseqtrrid 4017 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐼) |
11 | 8, 10 | fssresd 6538 | . . . . . 6 ⊢ (𝜑 → (𝑆 ↾ 𝐶):𝐶⟶(SubGrp‘𝐺)) |
12 | 11 | fdmd 6516 | . . . . 5 ⊢ (𝜑 → dom (𝑆 ↾ 𝐶) = 𝐶) |
13 | 4, 12 | dprddomcld 19052 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ V) |
14 | dmdprdsplit2.2 | . . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐷)) | |
15 | ssun2 4146 | . . . . . . . 8 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷) | |
16 | 15, 7 | sseqtrrid 4017 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ 𝐼) |
17 | 8, 16 | fssresd 6538 | . . . . . 6 ⊢ (𝜑 → (𝑆 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺)) |
18 | 17 | fdmd 6516 | . . . . 5 ⊢ (𝜑 → dom (𝑆 ↾ 𝐷) = 𝐷) |
19 | 14, 18 | dprddomcld 19052 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) |
20 | unexg 7461 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ∪ 𝐷) ∈ V) | |
21 | 13, 19, 20 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐶 ∪ 𝐷) ∈ V) |
22 | 7, 21 | eqeltrd 2910 | . 2 ⊢ (𝜑 → 𝐼 ∈ V) |
23 | 7 | eleq2d 2895 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↔ 𝑥 ∈ (𝐶 ∪ 𝐷))) |
24 | elun 4122 | . . . . 5 ⊢ (𝑥 ∈ (𝐶 ∪ 𝐷) ↔ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷)) | |
25 | 23, 24 | syl6bb 288 | . . . 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 19096 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑦 ∈ 𝐼 → (𝑥 ≠ 𝑦 → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)))) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 })) |
30 | incom 4175 | . . . . . . . . 9 ⊢ (𝐶 ∩ 𝐷) = (𝐷 ∩ 𝐶) | |
31 | 30, 26 | syl5eqr 2867 | . . . . . . . 8 ⊢ (𝜑 → (𝐷 ∩ 𝐶) = ∅) |
32 | uncom 4126 | . . . . . . . . 9 ⊢ (𝐶 ∪ 𝐷) = (𝐷 ∪ 𝐶) | |
33 | 7, 32 | syl6eq 2869 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 = (𝐷 ∪ 𝐶)) |
34 | dprdsubg 19075 | . . . . . . . . . 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 19075 | . . . . . . . . . 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 18733 | . . . . . . . 8 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐶)))) |
39 | incom 4175 | . . . . . . . . 9 ⊢ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = ((𝐺 DProd (𝑆 ↾ 𝐷)) ∩ (𝐺 DProd (𝑆 ↾ 𝐶))) | |
40 | 39, 28 | syl5eqr 2867 | . . . . . . . 8 ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐷)) ∩ (𝐺 DProd (𝑆 ↾ 𝐶))) = { 0 }) |
41 | 8, 31, 33, 1, 2, 14, 4, 38, 40, 3 | dmdprdsplit2lem 19096 | . . . . . . 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 951 | . . . . . 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 495 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷)) → (𝑦 ∈ 𝐼 → (𝑥 ≠ 𝑦 → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦))))) |
44 | 43 | ex 413 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷) → (𝑦 ∈ 𝐼 → (𝑥 ≠ 𝑦 → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)))))) |
45 | 25, 44 | sylbid 241 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 → (𝑦 ∈ 𝐼 → (𝑥 ≠ 𝑦 → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)))))) |
46 | 45 | 3imp2 1341 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦))) |
47 | 25 | biimpa 477 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐷)) |
48 | 29 | simprd 496 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 }) |
49 | 41 | simprd 496 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 }) |
50 | 48, 49 | jaodan 951 | . . 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 19050 | 1 ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 ∖ cdif 3930 ∪ cun 3931 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 {csn 4557 ∪ cuni 4830 class class class wbr 5057 dom cdm 5548 ↾ cres 5550 “ cima 5551 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 0gc0g 16701 mrClscmrc 16842 Grpcgrp 18041 SubGrpcsubg 18211 Cntzccntz 18383 DProd cdprd 19044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-0g 16703 df-gsum 16704 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-ghm 18294 df-gim 18337 df-cntz 18385 df-oppg 18412 df-lsm 18690 df-cmn 18837 df-dprd 19046 |
This theorem is referenced by: dmdprdsplit 19098 pgpfaclem1 19132 |
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