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Mirrors > Home > MPE Home > Th. List > gsummptfidmsub | Structured version Visualization version GIF version |
Description: The difference of two group sums expressed as mappings with finite domain. (Contributed by AV, 7-Nov-2019.) |
Ref | Expression |
---|---|
gsummptfidmsub.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptfidmsub.s | ⊢ − = (-g‘𝐺) |
gsummptfidmsub.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
gsummptfidmsub.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
gsummptfidmsub.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
gsummptfidmsub.d | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵) |
gsummptfidmsub.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) |
gsummptfidmsub.h | ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷) |
Ref | Expression |
---|---|
gsummptfidmsub | ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 − 𝐷))) = ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptfidmsub.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2736 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
3 | gsummptfidmsub.s | . 2 ⊢ − = (-g‘𝐺) | |
4 | gsummptfidmsub.g | . 2 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
5 | gsummptfidmsub.a | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
6 | gsummptfidmsub.c | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) | |
7 | gsummptfidmsub.d | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵) | |
8 | gsummptfidmsub.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
10 | gsummptfidmsub.h | . . 3 ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷) | |
11 | 10 | a1i 11 | . 2 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷)) |
12 | fvexd 6854 | . . 3 ⊢ (𝜑 → (0g‘𝐺) ∈ V) | |
13 | 8, 5, 6, 12 | fsuppmptdm 9312 | . 2 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝐺)) |
14 | 10, 5, 7, 12 | fsuppmptdm 9312 | . 2 ⊢ (𝜑 → 𝐻 finSupp (0g‘𝐺)) |
15 | 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 14 | gsummptfssub 19717 | 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 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ↦ cmpt 5186 ‘cfv 6493 (class class class)co 7353 Fincfn 8879 Basecbs 17075 0gc0g 17313 Σg cgsu 17314 -gcsg 18742 Abelcabl 19554 |
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 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 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 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7613 df-om 7799 df-1st 7917 df-2nd 7918 df-supp 8089 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-map 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9302 df-oi 9442 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-n0 12410 df-z 12496 df-uz 12760 df-fz 13417 df-fzo 13560 df-seq 13899 df-hash 14223 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-ress 17105 df-plusg 17138 df-0g 17315 df-gsum 17316 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-mhm 18593 df-submnd 18594 df-grp 18743 df-minusg 18744 df-sbg 18745 df-ghm 18997 df-cntz 19088 df-cmn 19555 df-abl 19556 |
This theorem is referenced by: cpmadugsumlemF 22209 |
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