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Mirrors > Home > MPE Home > Th. List > dpjghm | Structured version Visualization version GIF version |
Description: The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.) |
Ref | Expression |
---|---|
dpjfval.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
dpjfval.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
dpjfval.p | ⊢ 𝑃 = (𝐺dProj𝑆) |
dpjlid.3 | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
Ref | Expression |
---|---|
dpjghm | ⊢ (𝜑 → (𝑃‘𝑋) ∈ ((𝐺 ↾s (𝐺 DProd 𝑆)) GrpHom 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
2 | eqid 2738 | . . 3 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺) | |
3 | eqid 2738 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | eqid 2738 | . . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
5 | dpjfval.1 | . . . . 5 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
6 | dpjfval.2 | . . . . 5 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
7 | 5, 6 | dprdf2 19241 | . . . 4 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
8 | dpjlid.3 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
9 | 7, 8 | ffvelrnd 6856 | . . 3 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (SubGrp‘𝐺)) |
10 | difssd 4021 | . . . . . 6 ⊢ (𝜑 → (𝐼 ∖ {𝑋}) ⊆ 𝐼) | |
11 | 5, 6, 10 | dprdres 19262 | . . . . 5 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) ∧ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ⊆ (𝐺 DProd 𝑆))) |
12 | 11 | simpld 498 | . . . 4 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) |
13 | dprdsubg 19258 | . . . 4 ⊢ (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ∈ (SubGrp‘𝐺)) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ∈ (SubGrp‘𝐺)) |
15 | 5, 6, 8, 3 | dpjdisj 19287 | . . 3 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = {(0g‘𝐺)}) |
16 | 5, 6, 8, 4 | dpjcntz 19286 | . . 3 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
17 | eqid 2738 | . . 3 ⊢ (proj1‘𝐺) = (proj1‘𝐺) | |
18 | 1, 2, 3, 4, 9, 14, 15, 16, 17 | pj1ghm 18940 | . 2 ⊢ (𝜑 → ((𝑆‘𝑋)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) ∈ ((𝐺 ↾s ((𝑆‘𝑋)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) GrpHom 𝐺)) |
19 | dpjfval.p | . . 3 ⊢ 𝑃 = (𝐺dProj𝑆) | |
20 | 5, 6, 19, 17, 8 | dpjval 19290 | . 2 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
21 | 5, 6, 8, 2 | dpjlsm 19288 | . . . 4 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((𝑆‘𝑋)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
22 | 21 | oveq2d 7180 | . . 3 ⊢ (𝜑 → (𝐺 ↾s (𝐺 DProd 𝑆)) = (𝐺 ↾s ((𝑆‘𝑋)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))) |
23 | 22 | oveq1d 7179 | . 2 ⊢ (𝜑 → ((𝐺 ↾s (𝐺 DProd 𝑆)) GrpHom 𝐺) = ((𝐺 ↾s ((𝑆‘𝑋)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) GrpHom 𝐺)) |
24 | 18, 20, 23 | 3eltr4d 2848 | 1 ⊢ (𝜑 → (𝑃‘𝑋) ∈ ((𝐺 ↾s (𝐺 DProd 𝑆)) GrpHom 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ∖ cdif 3838 ⊆ wss 3841 {csn 4513 class class class wbr 5027 dom cdm 5519 ↾ cres 5521 ‘cfv 6333 (class class class)co 7164 ↾s cress 16580 +gcplusg 16661 0gc0g 16809 SubGrpcsubg 18384 GrpHom cghm 18466 Cntzccntz 18556 LSSumclsm 18870 proj1cpj1 18871 DProd cdprd 19227 dProjcdpj 19228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-supp 7850 df-tpos 7914 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-fsupp 8900 df-oi 9040 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-n0 11970 df-z 12056 df-uz 12318 df-fz 12975 df-fzo 13118 df-seq 13454 df-hash 13776 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-0g 16811 df-gsum 16812 df-mre 16953 df-mrc 16954 df-acs 16956 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-mhm 18065 df-submnd 18066 df-grp 18215 df-minusg 18216 df-sbg 18217 df-mulg 18336 df-subg 18387 df-ghm 18467 df-gim 18510 df-cntz 18558 df-oppg 18585 df-lsm 18872 df-pj1 18873 df-cmn 19019 df-dprd 19229 df-dpj 19230 |
This theorem is referenced by: dpjghm2 19298 dchrptlem2 25993 |
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