Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dprdfinv | Structured version Visualization version GIF version |
Description: Take the inverse of a group sum over a family of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.) |
Ref | Expression |
---|---|
eldprdi.0 | ⊢ 0 = (0g‘𝐺) |
eldprdi.w | ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } |
eldprdi.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
eldprdi.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
eldprdi.3 | ⊢ (𝜑 → 𝐹 ∈ 𝑊) |
dprdfinv.b | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
dprdfinv | ⊢ (𝜑 → ((𝑁 ∘ 𝐹) ∈ 𝑊 ∧ (𝐺 Σg (𝑁 ∘ 𝐹)) = (𝑁‘(𝐺 Σg 𝐹)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldprdi.1 | . . . . . 6 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
2 | dprdgrp 19056 | . . . . . 6 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp) |
4 | eqid 2818 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | dprdfinv.b | . . . . . 6 ⊢ 𝑁 = (invg‘𝐺) | |
6 | 4, 5 | grpinvf 18088 | . . . . 5 ⊢ (𝐺 ∈ Grp → 𝑁:(Base‘𝐺)⟶(Base‘𝐺)) |
7 | 3, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁:(Base‘𝐺)⟶(Base‘𝐺)) |
8 | eldprdi.w | . . . . 5 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } | |
9 | eldprdi.2 | . . . . 5 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
10 | eldprdi.3 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑊) | |
11 | 8, 1, 9, 10, 4 | dprdff 19063 | . . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺)) |
12 | fcompt 6887 | . . . 4 ⊢ ((𝑁:(Base‘𝐺)⟶(Base‘𝐺) ∧ 𝐹:𝐼⟶(Base‘𝐺)) → (𝑁 ∘ 𝐹) = (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝐹‘𝑥)))) | |
13 | 7, 11, 12 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑁 ∘ 𝐹) = (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝐹‘𝑥)))) |
14 | 1, 9 | dprdf2 19058 | . . . . . 6 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
15 | 14 | ffvelrnda 6843 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ∈ (SubGrp‘𝐺)) |
16 | 8, 1, 9, 10 | dprdfcl 19064 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (𝑆‘𝑥)) |
17 | 5 | subginvcl 18226 | . . . . 5 ⊢ (((𝑆‘𝑥) ∈ (SubGrp‘𝐺) ∧ (𝐹‘𝑥) ∈ (𝑆‘𝑥)) → (𝑁‘(𝐹‘𝑥)) ∈ (𝑆‘𝑥)) |
18 | 15, 16, 17 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑁‘(𝐹‘𝑥)) ∈ (𝑆‘𝑥)) |
19 | 1, 9 | dprddomcld 19052 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) |
20 | 19 | mptexd 6978 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝐹‘𝑥))) ∈ V) |
21 | funmpt 6386 | . . . . . 6 ⊢ Fun (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝐹‘𝑥))) | |
22 | 21 | a1i 11 | . . . . 5 ⊢ (𝜑 → Fun (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝐹‘𝑥)))) |
23 | 8, 1, 9, 10 | dprdffsupp 19065 | . . . . 5 ⊢ (𝜑 → 𝐹 finSupp 0 ) |
24 | ssidd 3987 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )) | |
25 | eldprdi.0 | . . . . . . . . . . 11 ⊢ 0 = (0g‘𝐺) | |
26 | 25 | fvexi 6677 | . . . . . . . . . 10 ⊢ 0 ∈ V |
27 | 26 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 0 ∈ V) |
28 | 11, 24, 19, 27 | suppssr 7850 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐹 supp 0 ))) → (𝐹‘𝑥) = 0 ) |
29 | 28 | fveq2d 6667 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐹 supp 0 ))) → (𝑁‘(𝐹‘𝑥)) = (𝑁‘ 0 )) |
30 | 25, 5 | grpinvid 18098 | . . . . . . . . 9 ⊢ (𝐺 ∈ Grp → (𝑁‘ 0 ) = 0 ) |
31 | 3, 30 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘ 0 ) = 0 ) |
32 | 31 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐹 supp 0 ))) → (𝑁‘ 0 ) = 0 ) |
33 | 29, 32 | eqtrd 2853 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐹 supp 0 ))) → (𝑁‘(𝐹‘𝑥)) = 0 ) |
34 | 33, 19 | suppss2 7853 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ (𝑁‘(𝐹‘𝑥))) supp 0 ) ⊆ (𝐹 supp 0 )) |
35 | fsuppsssupp 8837 | . . . . 5 ⊢ ((((𝑥 ∈ 𝐼 ↦ (𝑁‘(𝐹‘𝑥))) ∈ V ∧ Fun (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝐹‘𝑥)))) ∧ (𝐹 finSupp 0 ∧ ((𝑥 ∈ 𝐼 ↦ (𝑁‘(𝐹‘𝑥))) supp 0 ) ⊆ (𝐹 supp 0 ))) → (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝐹‘𝑥))) finSupp 0 ) | |
36 | 20, 22, 23, 34, 35 | syl22anc 834 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝐹‘𝑥))) finSupp 0 ) |
37 | 8, 1, 9, 18, 36 | dprdwd 19062 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝐹‘𝑥))) ∈ 𝑊) |
38 | 13, 37 | eqeltrd 2910 | . 2 ⊢ (𝜑 → (𝑁 ∘ 𝐹) ∈ 𝑊) |
39 | eqid 2818 | . . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
40 | 8, 1, 9, 10, 39 | dprdfcntz 19066 | . . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ((Cntz‘𝐺)‘ran 𝐹)) |
41 | 4, 25, 39, 5, 3, 19, 11, 40, 23 | gsumzinv 18994 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑁 ∘ 𝐹)) = (𝑁‘(𝐺 Σg 𝐹))) |
42 | 38, 41 | jca 512 | 1 ⊢ (𝜑 → ((𝑁 ∘ 𝐹) ∈ 𝑊 ∧ (𝐺 Σg (𝑁 ∘ 𝐹)) = (𝑁‘(𝐺 Σg 𝐹)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 ∖ cdif 3930 ⊆ wss 3933 class class class wbr 5057 ↦ cmpt 5137 dom cdm 5548 ∘ ccom 5552 Fun wfun 6342 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 supp csupp 7819 Xcixp 8449 finSupp cfsupp 8821 Basecbs 16471 0gc0g 16701 Σg cgsu 16702 Grpcgrp 18041 invgcminusg 18042 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-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-subg 18214 df-ghm 18294 df-gim 18337 df-cntz 18385 df-oppg 18412 df-cmn 18837 df-dprd 19046 |
This theorem is referenced by: dprdfsub 19072 |
Copyright terms: Public domain | W3C validator |