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Mirrors > Home > MPE Home > Th. List > gsumsub | Structured version Visualization version GIF version |
Description: The difference of two group sums. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 6-Jun-2019.) |
Ref | Expression |
---|---|
gsumsub.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumsub.z | ⊢ 0 = (0g‘𝐺) |
gsumsub.m | ⊢ − = (-g‘𝐺) |
gsumsub.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
gsumsub.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumsub.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
gsumsub.h | ⊢ (𝜑 → 𝐻:𝐴⟶𝐵) |
gsumsub.fn | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
gsumsub.hn | ⊢ (𝜑 → 𝐻 finSupp 0 ) |
Ref | Expression |
---|---|
gsumsub | ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f − 𝐻)) = ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumsub.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumsub.z | . . . 4 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2737 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | gsumsub.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
5 | ablcmn 19528 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
7 | gsumsub.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | gsumsub.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
9 | eqid 2737 | . . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
10 | ablgrp 19526 | . . . . . . . 8 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
11 | 4, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Grp) |
12 | 1, 9, 11 | grpinvf1o 18776 | . . . . . 6 ⊢ (𝜑 → (invg‘𝐺):𝐵–1-1-onto→𝐵) |
13 | f1of 6781 | . . . . . 6 ⊢ ((invg‘𝐺):𝐵–1-1-onto→𝐵 → (invg‘𝐺):𝐵⟶𝐵) | |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → (invg‘𝐺):𝐵⟶𝐵) |
15 | gsumsub.h | . . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵) | |
16 | fco 6689 | . . . . 5 ⊢ (((invg‘𝐺):𝐵⟶𝐵 ∧ 𝐻:𝐴⟶𝐵) → ((invg‘𝐺) ∘ 𝐻):𝐴⟶𝐵) | |
17 | 14, 15, 16 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((invg‘𝐺) ∘ 𝐻):𝐴⟶𝐵) |
18 | gsumsub.fn | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
19 | 2 | fvexi 6853 | . . . . . 6 ⊢ 0 ∈ V |
20 | 19 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 ∈ V) |
21 | 1 | fvexi 6853 | . . . . . 6 ⊢ 𝐵 ∈ V |
22 | 21 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
23 | gsumsub.hn | . . . . 5 ⊢ (𝜑 → 𝐻 finSupp 0 ) | |
24 | 2, 9 | grpinvid 18767 | . . . . . 6 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) = 0 ) |
25 | 11, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → ((invg‘𝐺)‘ 0 ) = 0 ) |
26 | 20, 15, 14, 7, 22, 23, 25 | fsuppco2 9297 | . . . 4 ⊢ (𝜑 → ((invg‘𝐺) ∘ 𝐻) finSupp 0 ) |
27 | 1, 2, 3, 6, 7, 8, 17, 18, 26 | gsumadd 19659 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻))) = ((𝐺 Σg 𝐹)(+g‘𝐺)(𝐺 Σg ((invg‘𝐺) ∘ 𝐻)))) |
28 | 1, 2, 9, 4, 7, 15, 23 | gsuminv 19682 | . . . 4 ⊢ (𝜑 → (𝐺 Σg ((invg‘𝐺) ∘ 𝐻)) = ((invg‘𝐺)‘(𝐺 Σg 𝐻))) |
29 | 28 | oveq2d 7367 | . . 3 ⊢ (𝜑 → ((𝐺 Σg 𝐹)(+g‘𝐺)(𝐺 Σg ((invg‘𝐺) ∘ 𝐻))) = ((𝐺 Σg 𝐹)(+g‘𝐺)((invg‘𝐺)‘(𝐺 Σg 𝐻)))) |
30 | 27, 29 | eqtrd 2777 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻))) = ((𝐺 Σg 𝐹)(+g‘𝐺)((invg‘𝐺)‘(𝐺 Σg 𝐻)))) |
31 | 8 | ffvelcdmda 7031 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵) |
32 | 15 | ffvelcdmda 7031 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐻‘𝑘) ∈ 𝐵) |
33 | gsumsub.m | . . . . . . 7 ⊢ − = (-g‘𝐺) | |
34 | 1, 3, 9, 33 | grpsubval 18756 | . . . . . 6 ⊢ (((𝐹‘𝑘) ∈ 𝐵 ∧ (𝐻‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘) − (𝐻‘𝑘)) = ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘)))) |
35 | 31, 32, 34 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘) − (𝐻‘𝑘)) = ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘)))) |
36 | 35 | mpteq2dva 5203 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) − (𝐻‘𝑘))) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘))))) |
37 | 8 | feqmptd 6907 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
38 | 15 | feqmptd 6907 | . . . . 5 ⊢ (𝜑 → 𝐻 = (𝑘 ∈ 𝐴 ↦ (𝐻‘𝑘))) |
39 | 7, 31, 32, 37, 38 | offval2 7629 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f − 𝐻) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) − (𝐻‘𝑘)))) |
40 | fvexd 6854 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((invg‘𝐺)‘(𝐻‘𝑘)) ∈ V) | |
41 | 14 | feqmptd 6907 | . . . . . 6 ⊢ (𝜑 → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ ((invg‘𝐺)‘𝑥))) |
42 | fveq2 6839 | . . . . . 6 ⊢ (𝑥 = (𝐻‘𝑘) → ((invg‘𝐺)‘𝑥) = ((invg‘𝐺)‘(𝐻‘𝑘))) | |
43 | 32, 38, 41, 42 | fmptco 7071 | . . . . 5 ⊢ (𝜑 → ((invg‘𝐺) ∘ 𝐻) = (𝑘 ∈ 𝐴 ↦ ((invg‘𝐺)‘(𝐻‘𝑘)))) |
44 | 7, 31, 40, 37, 43 | offval2 7629 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘))))) |
45 | 36, 39, 44 | 3eqtr4d 2787 | . . 3 ⊢ (𝜑 → (𝐹 ∘f − 𝐻) = (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻))) |
46 | 45 | oveq2d 7367 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f − 𝐻)) = (𝐺 Σg (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)))) |
47 | 1, 2, 6, 7, 8, 18 | gsumcl 19651 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) |
48 | 1, 2, 6, 7, 15, 23 | gsumcl 19651 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐻) ∈ 𝐵) |
49 | 1, 3, 9, 33 | grpsubval 18756 | . . 3 ⊢ (((𝐺 Σg 𝐹) ∈ 𝐵 ∧ (𝐺 Σg 𝐻) ∈ 𝐵) → ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻)) = ((𝐺 Σg 𝐹)(+g‘𝐺)((invg‘𝐺)‘(𝐺 Σg 𝐻)))) |
50 | 47, 48, 49 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻)) = ((𝐺 Σg 𝐹)(+g‘𝐺)((invg‘𝐺)‘(𝐺 Σg 𝐻)))) |
51 | 30, 46, 50 | 3eqtr4d 2787 | 1 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f − 𝐻)) = ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 class class class wbr 5103 ↦ cmpt 5186 ∘ ccom 5635 ⟶wf 6489 –1-1-onto→wf1o 6492 ‘cfv 6493 (class class class)co 7351 ∘f cof 7607 finSupp cfsupp 9263 Basecbs 17043 +gcplusg 17093 0gc0g 17281 Σg cgsu 17282 Grpcgrp 18708 invgcminusg 18709 -gcsg 18710 CMndccmn 19521 Abelcabl 19522 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-fzo 13522 df-seq 13861 df-hash 14185 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-0g 17283 df-gsum 17284 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-mhm 18561 df-submnd 18562 df-grp 18711 df-minusg 18712 df-sbg 18713 df-ghm 18965 df-cntz 19056 df-cmn 19523 df-abl 19524 |
This theorem is referenced by: gsummptfssub 19685 tsmsxplem2 23457 |
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