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Mirrors > Home > MPE Home > Th. List > dmdprdsplit | 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) |
Ref | Expression |
---|---|
dmdprdsplit | ⊢ (𝜑 → (𝐺dom DProd 𝑆 ↔ ((𝐺dom DProd (𝑆 ↾ 𝐶) ∧ 𝐺dom DProd (𝑆 ↾ 𝐷)) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ∧ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 }))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐺dom DProd 𝑆) → 𝐺dom DProd 𝑆) | |
2 | dprdsplit.2 | . . . . . . . 8 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) | |
3 | 2 | fdmd 6613 | . . . . . . 7 ⊢ (𝜑 → dom 𝑆 = 𝐼) |
4 | 3 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐺dom DProd 𝑆) → dom 𝑆 = 𝐼) |
5 | ssun1 4107 | . . . . . . 7 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷) | |
6 | dprdsplit.u | . . . . . . . 8 ⊢ (𝜑 → 𝐼 = (𝐶 ∪ 𝐷)) | |
7 | 6 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐺dom DProd 𝑆) → 𝐼 = (𝐶 ∪ 𝐷)) |
8 | 5, 7 | sseqtrrid 3975 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐺dom DProd 𝑆) → 𝐶 ⊆ 𝐼) |
9 | 1, 4, 8 | dprdres 19629 | . . . . 5 ⊢ ((𝜑 ∧ 𝐺dom DProd 𝑆) → (𝐺dom DProd (𝑆 ↾ 𝐶) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝐺 DProd 𝑆))) |
10 | 9 | simpld 495 | . . . 4 ⊢ ((𝜑 ∧ 𝐺dom DProd 𝑆) → 𝐺dom DProd (𝑆 ↾ 𝐶)) |
11 | ssun2 4108 | . . . . . . 7 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷) | |
12 | 11, 7 | sseqtrrid 3975 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐺dom DProd 𝑆) → 𝐷 ⊆ 𝐼) |
13 | 1, 4, 12 | dprdres 19629 | . . . . 5 ⊢ ((𝜑 ∧ 𝐺dom DProd 𝑆) → (𝐺dom DProd (𝑆 ↾ 𝐷) ∧ (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (𝐺 DProd 𝑆))) |
14 | 13 | simpld 495 | . . . 4 ⊢ ((𝜑 ∧ 𝐺dom DProd 𝑆) → 𝐺dom DProd (𝑆 ↾ 𝐷)) |
15 | 10, 14 | jca 512 | . . 3 ⊢ ((𝜑 ∧ 𝐺dom DProd 𝑆) → (𝐺dom DProd (𝑆 ↾ 𝐶) ∧ 𝐺dom DProd (𝑆 ↾ 𝐷))) |
16 | dprdsplit.i | . . . . 5 ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) | |
17 | 16 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐺dom DProd 𝑆) → (𝐶 ∩ 𝐷) = ∅) |
18 | dmdprdsplit.z | . . . 4 ⊢ 𝑍 = (Cntz‘𝐺) | |
19 | 1, 4, 8, 12, 17, 18 | dprdcntz2 19639 | . . 3 ⊢ ((𝜑 ∧ 𝐺dom DProd 𝑆) → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷)))) |
20 | dmdprdsplit.0 | . . . 4 ⊢ 0 = (0g‘𝐺) | |
21 | 1, 4, 8, 12, 17, 20 | dprddisj2 19640 | . . 3 ⊢ ((𝜑 ∧ 𝐺dom DProd 𝑆) → ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 }) |
22 | 15, 19, 21 | 3jca 1127 | . 2 ⊢ ((𝜑 ∧ 𝐺dom DProd 𝑆) → ((𝐺dom DProd (𝑆 ↾ 𝐶) ∧ 𝐺dom DProd (𝑆 ↾ 𝐷)) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ∧ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })) |
23 | 2 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ((𝐺dom DProd (𝑆 ↾ 𝐶) ∧ 𝐺dom DProd (𝑆 ↾ 𝐷)) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ∧ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })) → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
24 | 16 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ((𝐺dom DProd (𝑆 ↾ 𝐶) ∧ 𝐺dom DProd (𝑆 ↾ 𝐷)) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ∧ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })) → (𝐶 ∩ 𝐷) = ∅) |
25 | 6 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ((𝐺dom DProd (𝑆 ↾ 𝐶) ∧ 𝐺dom DProd (𝑆 ↾ 𝐷)) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ∧ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })) → 𝐼 = (𝐶 ∪ 𝐷)) |
26 | simpr1l 1229 | . . 3 ⊢ ((𝜑 ∧ ((𝐺dom DProd (𝑆 ↾ 𝐶) ∧ 𝐺dom DProd (𝑆 ↾ 𝐷)) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ∧ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })) → 𝐺dom DProd (𝑆 ↾ 𝐶)) | |
27 | simpr1r 1230 | . . 3 ⊢ ((𝜑 ∧ ((𝐺dom DProd (𝑆 ↾ 𝐶) ∧ 𝐺dom DProd (𝑆 ↾ 𝐷)) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ∧ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })) → 𝐺dom DProd (𝑆 ↾ 𝐷)) | |
28 | simpr2 1194 | . . 3 ⊢ ((𝜑 ∧ ((𝐺dom DProd (𝑆 ↾ 𝐶) ∧ 𝐺dom DProd (𝑆 ↾ 𝐷)) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ∧ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })) → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷)))) | |
29 | simpr3 1195 | . . 3 ⊢ ((𝜑 ∧ ((𝐺dom DProd (𝑆 ↾ 𝐶) ∧ 𝐺dom DProd (𝑆 ↾ 𝐷)) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ∧ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })) → ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 }) | |
30 | 23, 24, 25, 18, 20, 26, 27, 28, 29 | dmdprdsplit2 19647 | . 2 ⊢ ((𝜑 ∧ ((𝐺dom DProd (𝑆 ↾ 𝐶) ∧ 𝐺dom DProd (𝑆 ↾ 𝐷)) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ∧ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })) → 𝐺dom DProd 𝑆) |
31 | 22, 30 | impbida 798 | 1 ⊢ (𝜑 → (𝐺dom DProd 𝑆 ↔ ((𝐺dom DProd (𝑆 ↾ 𝐶) ∧ 𝐺dom DProd (𝑆 ↾ 𝐷)) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ∧ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 }))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∪ cun 3886 ∩ cin 3887 ⊆ wss 3888 ∅c0 4258 {csn 4563 class class class wbr 5076 dom cdm 5591 ↾ cres 5593 ⟶wf 6431 ‘cfv 6435 (class class class)co 7277 0gc0g 17148 SubGrpcsubg 18747 Cntzccntz 18919 DProd cdprd 19594 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-isom 6444 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7976 df-tpos 8040 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-er 8496 df-map 8615 df-ixp 8684 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-fsupp 9127 df-oi 9267 df-card 9695 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-n0 12232 df-z 12318 df-uz 12581 df-fz 13238 df-fzo 13381 df-seq 13720 df-hash 14043 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-0g 17150 df-gsum 17151 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-mulg 18699 df-subg 18750 df-ghm 18830 df-gim 18873 df-cntz 18921 df-oppg 18948 df-lsm 19239 df-cmn 19386 df-dprd 19596 |
This theorem is referenced by: dprdsplit 19649 dmdprdpr 19650 dpjcntz 19653 dpjdisj 19654 |
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