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Mirrors > Home > MPE Home > Th. List > gsumzinv | Structured version Visualization version GIF version |
Description: Inverse of a group sum. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.) |
Ref | Expression |
---|---|
gsumzinv.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumzinv.0 | ⊢ 0 = (0g‘𝐺) |
gsumzinv.z | ⊢ 𝑍 = (Cntz‘𝐺) |
gsumzinv.i | ⊢ 𝐼 = (invg‘𝐺) |
gsumzinv.g | ⊢ (𝜑 → 𝐺 ∈ Grp) |
gsumzinv.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumzinv.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
gsumzinv.c | ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
gsumzinv.n | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Ref | Expression |
---|---|
gsumzinv | ⊢ (𝜑 → (𝐺 Σg (𝐼 ∘ 𝐹)) = (𝐼‘(𝐺 Σg 𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumzinv.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumzinv.0 | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | gsumzinv.z | . . 3 ⊢ 𝑍 = (Cntz‘𝐺) | |
4 | eqid 2739 | . . 3 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
5 | gsumzinv.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
6 | 5 | grpmndd 18244 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
7 | gsumzinv.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | gsumzinv.i | . . . . . 6 ⊢ 𝐼 = (invg‘𝐺) | |
9 | 1, 8 | grpinvf 18281 | . . . . 5 ⊢ (𝐺 ∈ Grp → 𝐼:𝐵⟶𝐵) |
10 | 5, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐼:𝐵⟶𝐵) |
11 | gsumzinv.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
12 | fco 6539 | . . . 4 ⊢ ((𝐼:𝐵⟶𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐼 ∘ 𝐹):𝐴⟶𝐵) | |
13 | 10, 11, 12 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝐼 ∘ 𝐹):𝐴⟶𝐵) |
14 | 4, 8 | invoppggim 18619 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpIso (oppg‘𝐺))) |
15 | gimghm 18535 | . . . . . 6 ⊢ (𝐼 ∈ (𝐺 GrpIso (oppg‘𝐺)) → 𝐼 ∈ (𝐺 GrpHom (oppg‘𝐺))) | |
16 | ghmmhm 18499 | . . . . . 6 ⊢ (𝐼 ∈ (𝐺 GrpHom (oppg‘𝐺)) → 𝐼 ∈ (𝐺 MndHom (oppg‘𝐺))) | |
17 | 5, 14, 15, 16 | 4syl 19 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝐺 MndHom (oppg‘𝐺))) |
18 | gsumzinv.c | . . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) | |
19 | eqid 2739 | . . . . . 6 ⊢ (Cntz‘(oppg‘𝐺)) = (Cntz‘(oppg‘𝐺)) | |
20 | 3, 19 | cntzmhm2 18601 | . . . . 5 ⊢ ((𝐼 ∈ (𝐺 MndHom (oppg‘𝐺)) ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐼 “ ran 𝐹) ⊆ ((Cntz‘(oppg‘𝐺))‘(𝐼 “ ran 𝐹))) |
21 | 17, 18, 20 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐼 “ ran 𝐹) ⊆ ((Cntz‘(oppg‘𝐺))‘(𝐼 “ ran 𝐹))) |
22 | rnco2 6096 | . . . 4 ⊢ ran (𝐼 ∘ 𝐹) = (𝐼 “ ran 𝐹) | |
23 | 22 | fveq2i 6690 | . . . . 5 ⊢ (𝑍‘ran (𝐼 ∘ 𝐹)) = (𝑍‘(𝐼 “ ran 𝐹)) |
24 | 4, 3 | oppgcntz 18623 | . . . . 5 ⊢ (𝑍‘(𝐼 “ ran 𝐹)) = ((Cntz‘(oppg‘𝐺))‘(𝐼 “ ran 𝐹)) |
25 | 23, 24 | eqtri 2762 | . . . 4 ⊢ (𝑍‘ran (𝐼 ∘ 𝐹)) = ((Cntz‘(oppg‘𝐺))‘(𝐼 “ ran 𝐹)) |
26 | 21, 22, 25 | 3sstr4g 3932 | . . 3 ⊢ (𝜑 → ran (𝐼 ∘ 𝐹) ⊆ (𝑍‘ran (𝐼 ∘ 𝐹))) |
27 | 2 | fvexi 6701 | . . . . 5 ⊢ 0 ∈ V |
28 | 27 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
29 | 1 | fvexi 6701 | . . . . 5 ⊢ 𝐵 ∈ V |
30 | 29 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
31 | gsumzinv.n | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
32 | 2, 8 | grpinvid 18291 | . . . . 5 ⊢ (𝐺 ∈ Grp → (𝐼‘ 0 ) = 0 ) |
33 | 5, 32 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼‘ 0 ) = 0 ) |
34 | 28, 11, 10, 7, 30, 31, 33 | fsuppco2 8953 | . . 3 ⊢ (𝜑 → (𝐼 ∘ 𝐹) finSupp 0 ) |
35 | 1, 2, 3, 4, 6, 7, 13, 26, 34 | gsumzoppg 19196 | . 2 ⊢ (𝜑 → ((oppg‘𝐺) Σg (𝐼 ∘ 𝐹)) = (𝐺 Σg (𝐼 ∘ 𝐹))) |
36 | 4 | oppgmnd 18613 | . . . 4 ⊢ (𝐺 ∈ Mnd → (oppg‘𝐺) ∈ Mnd) |
37 | 6, 36 | syl 17 | . . 3 ⊢ (𝜑 → (oppg‘𝐺) ∈ Mnd) |
38 | 1, 3, 6, 37, 7, 17, 11, 18, 2, 31 | gsumzmhm 19189 | . 2 ⊢ (𝜑 → ((oppg‘𝐺) Σg (𝐼 ∘ 𝐹)) = (𝐼‘(𝐺 Σg 𝐹))) |
39 | 35, 38 | eqtr3d 2776 | 1 ⊢ (𝜑 → (𝐺 Σg (𝐼 ∘ 𝐹)) = (𝐼‘(𝐺 Σg 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3400 ⊆ wss 3853 class class class wbr 5040 ran crn 5536 “ cima 5538 ∘ ccom 5539 ⟶wf 6346 ‘cfv 6350 (class class class)co 7183 finSupp cfsupp 8919 Basecbs 16599 0gc0g 16829 Σg cgsu 16830 Mndcmnd 18040 MndHom cmhm 18083 Grpcgrp 18232 invgcminusg 18233 GrpHom cghm 18486 GrpIso cgim 18528 Cntzccntz 18576 oppgcoppg 18604 |
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 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
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 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-supp 7870 df-tpos 7934 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-er 8333 df-map 8452 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-fsupp 8920 df-oi 9060 df-card 9454 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-2 11792 df-n0 11990 df-z 12076 df-uz 12338 df-fz 12995 df-fzo 13138 df-seq 13474 df-hash 13796 df-ndx 16602 df-slot 16603 df-base 16605 df-sets 16606 df-ress 16607 df-plusg 16694 df-0g 16831 df-gsum 16832 df-mre 16973 df-mrc 16974 df-acs 16976 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-mhm 18085 df-submnd 18086 df-grp 18235 df-minusg 18236 df-ghm 18487 df-gim 18530 df-cntz 18578 df-oppg 18605 df-cmn 19039 |
This theorem is referenced by: dprdfinv 19273 |
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