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Mirrors > Home > MPE Home > Th. List > gsummptfssub | Structured version Visualization version GIF version |
Description: The difference of two group sums expressed as mappings. (Contributed by AV, 7-Nov-2019.) |
Ref | Expression |
---|---|
gsummptfssub.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptfssub.z | ⊢ 0 = (0g‘𝐺) |
gsummptfssub.s | ⊢ − = (-g‘𝐺) |
gsummptfssub.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
gsummptfssub.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsummptfssub.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
gsummptfssub.d | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵) |
gsummptfssub.f | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
gsummptfssub.h | ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷)) |
gsummptfssub.w | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
gsummptfssub.v | ⊢ (𝜑 → 𝐻 finSupp 0 ) |
Ref | Expression |
---|---|
gsummptfssub | ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 − 𝐷))) = ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptfssub.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | gsummptfssub.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) | |
3 | gsummptfssub.d | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵) | |
4 | gsummptfssub.f | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
5 | gsummptfssub.h | . . . . 5 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷)) | |
6 | 1, 2, 3, 4, 5 | offval2 7684 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f − 𝐻) = (𝑥 ∈ 𝐴 ↦ (𝐶 − 𝐷))) |
7 | 6 | eqcomd 2739 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 − 𝐷)) = (𝐹 ∘f − 𝐻)) |
8 | 7 | oveq2d 7419 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 − 𝐷))) = (𝐺 Σg (𝐹 ∘f − 𝐻))) |
9 | gsummptfssub.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
10 | gsummptfssub.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
11 | gsummptfssub.s | . . 3 ⊢ − = (-g‘𝐺) | |
12 | gsummptfssub.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
13 | 4, 2 | fmpt3d 7110 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
14 | 5, 3 | fmpt3d 7110 | . . 3 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵) |
15 | gsummptfssub.w | . . 3 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
16 | gsummptfssub.v | . . 3 ⊢ (𝜑 → 𝐻 finSupp 0 ) | |
17 | 9, 10, 11, 12, 1, 13, 14, 15, 16 | gsumsub 19807 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f − 𝐻)) = ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻))) |
18 | 8, 17 | eqtrd 2773 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 − 𝐷))) = ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5146 ↦ cmpt 5229 ‘cfv 6539 (class class class)co 7403 ∘f cof 7662 finSupp cfsupp 9356 Basecbs 17139 0gc0g 17380 Σg cgsu 17381 -gcsg 18816 Abelcabl 19641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8141 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-oi 9500 df-card 9929 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-n0 12468 df-z 12554 df-uz 12818 df-fz 13480 df-fzo 13623 df-seq 13962 df-hash 14286 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17140 df-ress 17169 df-plusg 17205 df-0g 17382 df-gsum 17383 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-mhm 18666 df-submnd 18667 df-grp 18817 df-minusg 18818 df-sbg 18819 df-ghm 19083 df-cntz 19174 df-cmn 19642 df-abl 19643 |
This theorem is referenced by: gsummptfidmsub 19809 |
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