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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 19805 | . . . . 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 19803 | . . . . . . . 8 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
| 11 | 4, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Grp) |
| 12 | 1, 9, 11 | grpinvf1o 19027 | . . . . . 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 19017 | . . . . . 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 9443 | . . . 4 ⊢ (𝜑 → ((invg‘𝐺) ∘ 𝐻) finSupp 0 ) |
| 27 | 1, 2, 3, 6, 7, 8, 17, 18, 26 | gsumadd 19941 | . . 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 19964 | . . . 4 ⊢ (𝜑 → (𝐺 Σg ((invg‘𝐺) ∘ 𝐻)) = ((invg‘𝐺)‘(𝐺 Σg 𝐻))) |
| 29 | 28 | oveq2d 7447 | . . 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 7104 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵) |
| 32 | 15 | ffvelcdmda 7104 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐻‘𝑘) ∈ 𝐵) |
| 33 | gsumsub.m | . . . . . . 7 ⊢ − = (-g‘𝐺) | |
| 34 | 1, 3, 9, 33 | grpsubval 19003 | . . . . . 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 5242 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) − (𝐻‘𝑘))) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(+g‘𝐺)((invg‘𝐺)‘(𝐻‘𝑘))))) |
| 37 | 8 | feqmptd 6977 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
| 38 | 15 | feqmptd 6977 | . . . . 5 ⊢ (𝜑 → 𝐻 = (𝑘 ∈ 𝐴 ↦ (𝐻‘𝑘))) |
| 39 | 7, 31, 32, 37, 38 | offval2 7717 | . . . 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 6977 | . . . . . 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 7149 | . . . . 5 ⊢ (𝜑 → ((invg‘𝐺) ∘ 𝐻) = (𝑘 ∈ 𝐴 ↦ ((invg‘𝐺)‘(𝐻‘𝑘)))) |
| 44 | 7, 31, 40, 37, 43 | offval2 7717 | . . . 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 7447 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f − 𝐻)) = (𝐺 Σg (𝐹 ∘f (+g‘𝐺)((invg‘𝐺) ∘ 𝐻)))) |
| 47 | 1, 2, 6, 7, 8, 18 | gsumcl 19933 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) |
| 48 | 1, 2, 6, 7, 15, 23 | gsumcl 19933 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐻) ∈ 𝐵) |
| 49 | 1, 3, 9, 33 | grpsubval 19003 | . . 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 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 ↦ cmpt 5225 ∘ ccom 5689 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 finSupp cfsupp 9401 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Σg cgsu 17485 Grpcgrp 18951 invgcminusg 18952 -gcsg 18953 CMndccmn 19798 Abelcabl 19799 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-0g 17486 df-gsum 17487 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 |
| This theorem is referenced by: gsummptfssub 19967 tsmsxplem2 24162 |
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