![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2724 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
2 | eqid 2724 | . . 3 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺) | |
3 | eqid 2724 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | eqid 2724 | . . 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 19921 | . . . 4 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
8 | dpjlid.3 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
9 | 7, 8 | ffvelcdmd 7078 | . . 3 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (SubGrp‘𝐺)) |
10 | difssd 4125 | . . . . . 6 ⊢ (𝜑 → (𝐼 ∖ {𝑋}) ⊆ 𝐼) | |
11 | 5, 6, 10 | dprdres 19942 | . . . . 5 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) ∧ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ⊆ (𝐺 DProd 𝑆))) |
12 | 11 | simpld 494 | . . . 4 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) |
13 | dprdsubg 19938 | . . . 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 19967 | . . 3 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = {(0g‘𝐺)}) |
16 | 5, 6, 8, 4 | dpjcntz 19966 | . . 3 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
17 | eqid 2724 | . . 3 ⊢ (proj1‘𝐺) = (proj1‘𝐺) | |
18 | 1, 2, 3, 4, 9, 14, 15, 16, 17 | pj1ghm 19615 | . 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 19970 | . 2 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
21 | 5, 6, 8, 2 | dpjlsm 19968 | . . . 4 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((𝑆‘𝑋)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
22 | 21 | oveq2d 7418 | . . 3 ⊢ (𝜑 → (𝐺 ↾s (𝐺 DProd 𝑆)) = (𝐺 ↾s ((𝑆‘𝑋)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))) |
23 | 22 | oveq1d 7417 | . 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 2840 | 1 ⊢ (𝜑 → (𝑃‘𝑋) ∈ ((𝐺 ↾s (𝐺 DProd 𝑆)) GrpHom 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∖ cdif 3938 ⊆ wss 3941 {csn 4621 class class class wbr 5139 dom cdm 5667 ↾ cres 5669 ‘cfv 6534 (class class class)co 7402 ↾s cress 17174 +gcplusg 17198 0gc0g 17386 SubGrpcsubg 19039 GrpHom cghm 19130 Cntzccntz 19223 LSSumclsm 19546 proj1cpj1 19547 DProd cdprd 19907 dProjcdpj 19908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-oi 9502 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-n0 12471 df-z 12557 df-uz 12821 df-fz 13483 df-fzo 13626 df-seq 13965 df-hash 14289 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-0g 17388 df-gsum 17389 df-mre 17531 df-mrc 17532 df-acs 17534 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18988 df-subg 19042 df-ghm 19131 df-gim 19176 df-cntz 19225 df-oppg 19254 df-lsm 19548 df-pj1 19549 df-cmn 19694 df-dprd 19909 df-dpj 19910 |
This theorem is referenced by: dpjghm2 19978 dchrptlem2 27117 |
Copyright terms: Public domain | W3C validator |