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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 2736 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | gsumsub.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
5 | ablcmn 19131 | . . . . 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 2736 | . . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
10 | ablgrp 19129 | . . . . . . . 8 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
11 | 4, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Grp) |
12 | 1, 9, 11 | grpinvf1o 18387 | . . . . . 6 ⊢ (𝜑 → (invg‘𝐺):𝐵–1-1-onto→𝐵) |
13 | f1of 6639 | . . . . . 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 6547 | . . . . 5 ⊢ (((invg‘𝐺):𝐵⟶𝐵 ∧ 𝐻:𝐴⟶𝐵) → ((invg‘𝐺) ∘ 𝐻):𝐴⟶𝐵) | |
17 | 14, 15, 16 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ((invg‘𝐺) ∘ 𝐻):𝐴⟶𝐵) |
18 | gsumsub.fn | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
19 | 2 | fvexi 6709 | . . . . . 6 ⊢ 0 ∈ V |
20 | 19 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 ∈ V) |
21 | 1 | fvexi 6709 | . . . . . 6 ⊢ 𝐵 ∈ V |
22 | 21 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
23 | gsumsub.hn | . . . . 5 ⊢ (𝜑 → 𝐻 finSupp 0 ) | |
24 | 2, 9 | grpinvid 18378 | . . . . . 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 8997 | . . . 4 ⊢ (𝜑 → ((invg‘𝐺) ∘ 𝐻) finSupp 0 ) |
27 | 1, 2, 3, 6, 7, 8, 17, 18, 26 | gsumadd 19262 | . . 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 19285 | . . . 4 ⊢ (𝜑 → (𝐺 Σg ((invg‘𝐺) ∘ 𝐻)) = ((invg‘𝐺)‘(𝐺 Σg 𝐻))) |
29 | 28 | oveq2d 7207 | . . 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 2771 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻))) = ((𝐺 Σg 𝐹)(+g‘𝐺)((invg‘𝐺)‘(𝐺 Σg 𝐻)))) |
31 | 8 | ffvelrnda 6882 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵) |
32 | 15 | ffvelrnda 6882 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐻‘𝑘) ∈ 𝐵) |
33 | gsumsub.m | . . . . . . 7 ⊢ − = (-g‘𝐺) | |
34 | 1, 3, 9, 33 | grpsubval 18367 | . . . . . 6 ⊢ (((𝐹‘𝑘) ∈ 𝐵 ∧ (𝐻‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘) − (𝐻‘𝑘)) = ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘)))) |
35 | 31, 32, 34 | syl2anc 587 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘) − (𝐻‘𝑘)) = ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘)))) |
36 | 35 | mpteq2dva 5135 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) − (𝐻‘𝑘))) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘))))) |
37 | 8 | feqmptd 6758 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
38 | 15 | feqmptd 6758 | . . . . 5 ⊢ (𝜑 → 𝐻 = (𝑘 ∈ 𝐴 ↦ (𝐻‘𝑘))) |
39 | 7, 31, 32, 37, 38 | offval2 7466 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f − 𝐻) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) − (𝐻‘𝑘)))) |
40 | fvexd 6710 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((invg‘𝐺)‘(𝐻‘𝑘)) ∈ V) | |
41 | 14 | feqmptd 6758 | . . . . . 6 ⊢ (𝜑 → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ ((invg‘𝐺)‘𝑥))) |
42 | fveq2 6695 | . . . . . 6 ⊢ (𝑥 = (𝐻‘𝑘) → ((invg‘𝐺)‘𝑥) = ((invg‘𝐺)‘(𝐻‘𝑘))) | |
43 | 32, 38, 41, 42 | fmptco 6922 | . . . . 5 ⊢ (𝜑 → ((invg‘𝐺) ∘ 𝐻) = (𝑘 ∈ 𝐴 ↦ ((invg‘𝐺)‘(𝐻‘𝑘)))) |
44 | 7, 31, 40, 37, 43 | offval2 7466 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘))))) |
45 | 36, 39, 44 | 3eqtr4d 2781 | . . 3 ⊢ (𝜑 → (𝐹 ∘f − 𝐻) = (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻))) |
46 | 45 | oveq2d 7207 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f − 𝐻)) = (𝐺 Σg (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)))) |
47 | 1, 2, 6, 7, 8, 18 | gsumcl 19254 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) |
48 | 1, 2, 6, 7, 15, 23 | gsumcl 19254 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐻) ∈ 𝐵) |
49 | 1, 3, 9, 33 | grpsubval 18367 | . . 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 587 | . 2 ⊢ (𝜑 → ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻)) = ((𝐺 Σg 𝐹)(+g‘𝐺)((invg‘𝐺)‘(𝐺 Σg 𝐻)))) |
51 | 30, 46, 50 | 3eqtr4d 2781 | 1 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f − 𝐻)) = ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 class class class wbr 5039 ↦ cmpt 5120 ∘ ccom 5540 ⟶wf 6354 –1-1-onto→wf1o 6357 ‘cfv 6358 (class class class)co 7191 ∘f cof 7445 finSupp cfsupp 8963 Basecbs 16666 +gcplusg 16749 0gc0g 16898 Σg cgsu 16899 Grpcgrp 18319 invgcminusg 18320 -gcsg 18321 CMndccmn 19124 Abelcabl 19125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-0g 16900 df-gsum 16901 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-ghm 18574 df-cntz 18665 df-cmn 19126 df-abl 19127 |
This theorem is referenced by: gsummptfssub 19288 tsmsxplem2 23005 |
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