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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 2728 | . . 3 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
5 | gsumzinv.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
6 | 5 | grpmndd 18910 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
7 | gsumzinv.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | gsumzinv.i | . . . . . 6 ⊢ 𝐼 = (invg‘𝐺) | |
9 | 1, 8 | grpinvf 18950 | . . . . 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 6752 | . . . 4 ⊢ ((𝐼:𝐵⟶𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐼 ∘ 𝐹):𝐴⟶𝐵) | |
13 | 10, 11, 12 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝐼 ∘ 𝐹):𝐴⟶𝐵) |
14 | 4, 8 | invoppggim 19321 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpIso (oppg‘𝐺))) |
15 | gimghm 19225 | . . . . . 6 ⊢ (𝐼 ∈ (𝐺 GrpIso (oppg‘𝐺)) → 𝐼 ∈ (𝐺 GrpHom (oppg‘𝐺))) | |
16 | ghmmhm 19187 | . . . . . 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 2728 | . . . . . 6 ⊢ (Cntz‘(oppg‘𝐺)) = (Cntz‘(oppg‘𝐺)) | |
20 | 3, 19 | cntzmhm2 19300 | . . . . 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 582 | . . . 4 ⊢ (𝜑 → (𝐼 “ ran 𝐹) ⊆ ((Cntz‘(oppg‘𝐺))‘(𝐼 “ ran 𝐹))) |
22 | rnco2 6262 | . . . 4 ⊢ ran (𝐼 ∘ 𝐹) = (𝐼 “ ran 𝐹) | |
23 | 22 | fveq2i 6905 | . . . . 5 ⊢ (𝑍‘ran (𝐼 ∘ 𝐹)) = (𝑍‘(𝐼 “ ran 𝐹)) |
24 | 4, 3 | oppgcntz 19325 | . . . . 5 ⊢ (𝑍‘(𝐼 “ ran 𝐹)) = ((Cntz‘(oppg‘𝐺))‘(𝐼 “ ran 𝐹)) |
25 | 23, 24 | eqtri 2756 | . . . 4 ⊢ (𝑍‘ran (𝐼 ∘ 𝐹)) = ((Cntz‘(oppg‘𝐺))‘(𝐼 “ ran 𝐹)) |
26 | 21, 22, 25 | 3sstr4g 4027 | . . 3 ⊢ (𝜑 → ran (𝐼 ∘ 𝐹) ⊆ (𝑍‘ran (𝐼 ∘ 𝐹))) |
27 | 2 | fvexi 6916 | . . . . 5 ⊢ 0 ∈ V |
28 | 27 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
29 | 1 | fvexi 6916 | . . . . 5 ⊢ 𝐵 ∈ V |
30 | 29 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
31 | gsumzinv.n | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
32 | 2, 8 | grpinvid 18963 | . . . . 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 9434 | . . 3 ⊢ (𝜑 → (𝐼 ∘ 𝐹) finSupp 0 ) |
35 | 1, 2, 3, 4, 6, 7, 13, 26, 34 | gsumzoppg 19906 | . 2 ⊢ (𝜑 → ((oppg‘𝐺) Σg (𝐼 ∘ 𝐹)) = (𝐺 Σg (𝐼 ∘ 𝐹))) |
36 | 4 | oppgmnd 19315 | . . . 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 19899 | . 2 ⊢ (𝜑 → ((oppg‘𝐺) Σg (𝐼 ∘ 𝐹)) = (𝐼‘(𝐺 Σg 𝐹))) |
39 | 35, 38 | eqtr3d 2770 | 1 ⊢ (𝜑 → (𝐺 Σg (𝐼 ∘ 𝐹)) = (𝐼‘(𝐺 Σg 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ⊆ wss 3949 class class class wbr 5152 ran crn 5683 “ cima 5685 ∘ ccom 5686 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 finSupp cfsupp 9393 Basecbs 17187 0gc0g 17428 Σg cgsu 17429 Mndcmnd 18701 MndHom cmhm 18745 Grpcgrp 18897 invgcminusg 18898 GrpHom cghm 19174 GrpIso cgim 19218 Cntzccntz 19273 oppgcoppg 19303 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-0g 17430 df-gsum 17431 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-ghm 19175 df-gim 19220 df-cntz 19275 df-oppg 19304 df-cmn 19744 |
This theorem is referenced by: dprdfinv 19983 |
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