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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 2734 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | gsumsub.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
5 | ablcmn 19819 | . . . . 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 2734 | . . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
10 | ablgrp 19817 | . . . . . . . 8 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
11 | 4, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Grp) |
12 | 1, 9, 11 | grpinvf1o 19039 | . . . . . 6 ⊢ (𝜑 → (invg‘𝐺):𝐵–1-1-onto→𝐵) |
13 | f1of 6848 | . . . . . 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 6760 | . . . . 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 6920 | . . . . . 6 ⊢ 0 ∈ V |
20 | 19 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 ∈ V) |
21 | 1 | fvexi 6920 | . . . . . 6 ⊢ 𝐵 ∈ V |
22 | 21 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
23 | gsumsub.hn | . . . . 5 ⊢ (𝜑 → 𝐻 finSupp 0 ) | |
24 | 2, 9 | grpinvid 19029 | . . . . . 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 9440 | . . . 4 ⊢ (𝜑 → ((invg‘𝐺) ∘ 𝐻) finSupp 0 ) |
27 | 1, 2, 3, 6, 7, 8, 17, 18, 26 | gsumadd 19955 | . . 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 19978 | . . . 4 ⊢ (𝜑 → (𝐺 Σg ((invg‘𝐺) ∘ 𝐻)) = ((invg‘𝐺)‘(𝐺 Σg 𝐻))) |
29 | 28 | oveq2d 7446 | . . 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 2774 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻))) = ((𝐺 Σg 𝐹)(+g‘𝐺)((invg‘𝐺)‘(𝐺 Σg 𝐻)))) |
31 | 8 | ffvelcdmda 7103 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵) |
32 | 15 | ffvelcdmda 7103 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐻‘𝑘) ∈ 𝐵) |
33 | gsumsub.m | . . . . . . 7 ⊢ − = (-g‘𝐺) | |
34 | 1, 3, 9, 33 | grpsubval 19015 | . . . . . 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 5247 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) − (𝐻‘𝑘))) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘))))) |
37 | 8 | feqmptd 6976 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
38 | 15 | feqmptd 6976 | . . . . 5 ⊢ (𝜑 → 𝐻 = (𝑘 ∈ 𝐴 ↦ (𝐻‘𝑘))) |
39 | 7, 31, 32, 37, 38 | offval2 7716 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f − 𝐻) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) − (𝐻‘𝑘)))) |
40 | fvexd 6921 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((invg‘𝐺)‘(𝐻‘𝑘)) ∈ V) | |
41 | 14 | feqmptd 6976 | . . . . . 6 ⊢ (𝜑 → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ ((invg‘𝐺)‘𝑥))) |
42 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = (𝐻‘𝑘) → ((invg‘𝐺)‘𝑥) = ((invg‘𝐺)‘(𝐻‘𝑘))) | |
43 | 32, 38, 41, 42 | fmptco 7148 | . . . . 5 ⊢ (𝜑 → ((invg‘𝐺) ∘ 𝐻) = (𝑘 ∈ 𝐴 ↦ ((invg‘𝐺)‘(𝐻‘𝑘)))) |
44 | 7, 31, 40, 37, 43 | offval2 7716 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘))))) |
45 | 36, 39, 44 | 3eqtr4d 2784 | . . 3 ⊢ (𝜑 → (𝐹 ∘f − 𝐻) = (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻))) |
46 | 45 | oveq2d 7446 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f − 𝐻)) = (𝐺 Σg (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)))) |
47 | 1, 2, 6, 7, 8, 18 | gsumcl 19947 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) |
48 | 1, 2, 6, 7, 15, 23 | gsumcl 19947 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐻) ∈ 𝐵) |
49 | 1, 3, 9, 33 | grpsubval 19015 | . . 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 2784 | 1 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f − 𝐻)) = ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 class class class wbr 5147 ↦ cmpt 5230 ∘ ccom 5692 ⟶wf 6558 –1-1-onto→wf1o 6561 ‘cfv 6562 (class class class)co 7430 ∘f cof 7694 finSupp cfsupp 9398 Basecbs 17244 +gcplusg 17297 0gc0g 17485 Σg cgsu 17486 Grpcgrp 18963 invgcminusg 18964 -gcsg 18965 CMndccmn 19812 Abelcabl 19813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-0g 17487 df-gsum 17488 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-ghm 19243 df-cntz 19347 df-cmn 19814 df-abl 19815 |
This theorem is referenced by: gsummptfssub 19981 tsmsxplem2 24177 |
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