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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 2798 | . . 3 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
5 | gsumzinv.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
6 | grpmnd 18102 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
8 | gsumzinv.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
9 | gsumzinv.i | . . . . . 6 ⊢ 𝐼 = (invg‘𝐺) | |
10 | 1, 9 | grpinvf 18142 | . . . . 5 ⊢ (𝐺 ∈ Grp → 𝐼:𝐵⟶𝐵) |
11 | 5, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐼:𝐵⟶𝐵) |
12 | gsumzinv.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
13 | fco 6505 | . . . 4 ⊢ ((𝐼:𝐵⟶𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐼 ∘ 𝐹):𝐴⟶𝐵) | |
14 | 11, 12, 13 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝐼 ∘ 𝐹):𝐴⟶𝐵) |
15 | 4, 9 | invoppggim 18480 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpIso (oppg‘𝐺))) |
16 | gimghm 18396 | . . . . . 6 ⊢ (𝐼 ∈ (𝐺 GrpIso (oppg‘𝐺)) → 𝐼 ∈ (𝐺 GrpHom (oppg‘𝐺))) | |
17 | ghmmhm 18360 | . . . . . 6 ⊢ (𝐼 ∈ (𝐺 GrpHom (oppg‘𝐺)) → 𝐼 ∈ (𝐺 MndHom (oppg‘𝐺))) | |
18 | 5, 15, 16, 17 | 4syl 19 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝐺 MndHom (oppg‘𝐺))) |
19 | gsumzinv.c | . . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) | |
20 | eqid 2798 | . . . . . 6 ⊢ (Cntz‘(oppg‘𝐺)) = (Cntz‘(oppg‘𝐺)) | |
21 | 3, 20 | cntzmhm2 18462 | . . . . 5 ⊢ ((𝐼 ∈ (𝐺 MndHom (oppg‘𝐺)) ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐼 “ ran 𝐹) ⊆ ((Cntz‘(oppg‘𝐺))‘(𝐼 “ ran 𝐹))) |
22 | 18, 19, 21 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐼 “ ran 𝐹) ⊆ ((Cntz‘(oppg‘𝐺))‘(𝐼 “ ran 𝐹))) |
23 | rnco2 6073 | . . . 4 ⊢ ran (𝐼 ∘ 𝐹) = (𝐼 “ ran 𝐹) | |
24 | 23 | fveq2i 6648 | . . . . 5 ⊢ (𝑍‘ran (𝐼 ∘ 𝐹)) = (𝑍‘(𝐼 “ ran 𝐹)) |
25 | 4, 3 | oppgcntz 18484 | . . . . 5 ⊢ (𝑍‘(𝐼 “ ran 𝐹)) = ((Cntz‘(oppg‘𝐺))‘(𝐼 “ ran 𝐹)) |
26 | 24, 25 | eqtri 2821 | . . . 4 ⊢ (𝑍‘ran (𝐼 ∘ 𝐹)) = ((Cntz‘(oppg‘𝐺))‘(𝐼 “ ran 𝐹)) |
27 | 22, 23, 26 | 3sstr4g 3960 | . . 3 ⊢ (𝜑 → ran (𝐼 ∘ 𝐹) ⊆ (𝑍‘ran (𝐼 ∘ 𝐹))) |
28 | 2 | fvexi 6659 | . . . . 5 ⊢ 0 ∈ V |
29 | 28 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
30 | 1 | fvexi 6659 | . . . . 5 ⊢ 𝐵 ∈ V |
31 | 30 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
32 | gsumzinv.n | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
33 | 2, 9 | grpinvid 18152 | . . . . 5 ⊢ (𝐺 ∈ Grp → (𝐼‘ 0 ) = 0 ) |
34 | 5, 33 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼‘ 0 ) = 0 ) |
35 | 29, 12, 11, 8, 31, 32, 34 | fsuppco2 8850 | . . 3 ⊢ (𝜑 → (𝐼 ∘ 𝐹) finSupp 0 ) |
36 | 1, 2, 3, 4, 7, 8, 14, 27, 35 | gsumzoppg 19057 | . 2 ⊢ (𝜑 → ((oppg‘𝐺) Σg (𝐼 ∘ 𝐹)) = (𝐺 Σg (𝐼 ∘ 𝐹))) |
37 | 4 | oppgmnd 18474 | . . . 4 ⊢ (𝐺 ∈ Mnd → (oppg‘𝐺) ∈ Mnd) |
38 | 7, 37 | syl 17 | . . 3 ⊢ (𝜑 → (oppg‘𝐺) ∈ Mnd) |
39 | 1, 3, 7, 38, 8, 18, 12, 19, 2, 32 | gsumzmhm 19050 | . 2 ⊢ (𝜑 → ((oppg‘𝐺) Σg (𝐼 ∘ 𝐹)) = (𝐼‘(𝐺 Σg 𝐹))) |
40 | 36, 39 | eqtr3d 2835 | 1 ⊢ (𝜑 → (𝐺 Σg (𝐼 ∘ 𝐹)) = (𝐼‘(𝐺 Σg 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 ran crn 5520 “ cima 5522 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 finSupp cfsupp 8817 Basecbs 16475 0gc0g 16705 Σg cgsu 16706 Mndcmnd 17903 MndHom cmhm 17946 Grpcgrp 18095 invgcminusg 18096 GrpHom cghm 18347 GrpIso cgim 18389 Cntzccntz 18437 oppgcoppg 18465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 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 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-ghm 18348 df-gim 18391 df-cntz 18439 df-oppg 18466 df-cmn 18900 |
This theorem is referenced by: dprdfinv 19134 |
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