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Mirrors > Home > MPE Home > Th. List > gsummptfidmsplit | Structured version Visualization version GIF version |
Description: Split a group sum expressed as mapping with a finite domain into two parts. (Contributed by AV, 23-Jul-2019.) |
Ref | Expression |
---|---|
gsummptfidmsplit.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptfidmsplit.p | ⊢ + = (+g‘𝐺) |
gsummptfidmsplit.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptfidmsplit.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
gsummptfidmsplit.y | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑌 ∈ 𝐵) |
gsummptfidmsplit.i | ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) |
gsummptfidmsplit.u | ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) |
Ref | Expression |
---|---|
gsummptfidmsplit | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑌)) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑌)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptfidmsplit.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2740 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
3 | gsummptfidmsplit.p | . 2 ⊢ + = (+g‘𝐺) | |
4 | gsummptfidmsplit.g | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
5 | gsummptfidmsplit.a | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
6 | gsummptfidmsplit.y | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑌 ∈ 𝐵) | |
7 | eqid 2740 | . . 3 ⊢ (𝑘 ∈ 𝐴 ↦ 𝑌) = (𝑘 ∈ 𝐴 ↦ 𝑌) | |
8 | fvexd 6786 | . . 3 ⊢ (𝜑 → (0g‘𝐺) ∈ V) | |
9 | 7, 5, 6, 8 | fsuppmptdm 9117 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑌) finSupp (0g‘𝐺)) |
10 | gsummptfidmsplit.i | . 2 ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) | |
11 | gsummptfidmsplit.u | . 2 ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) | |
12 | 1, 2, 3, 4, 5, 6, 9, 10, 11 | gsumsplit2 19528 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑌)) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑌)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ∪ cun 3890 ∩ cin 3891 ∅c0 4262 ↦ cmpt 5162 ‘cfv 6432 (class class class)co 7271 Fincfn 8716 Basecbs 16910 +gcplusg 16960 0gc0g 17148 Σg cgsu 17149 CMndccmn 19384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-fzo 13382 df-seq 13720 df-hash 14043 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-0g 17150 df-gsum 17151 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-cntz 18921 df-cmn 19386 |
This theorem is referenced by: gsummptfzsplit 19531 gsummptfzsplitl 19532 gsumunsnfd 19556 gsummptun 19563 telgsumfzslem 19587 mdetdiaglem 21745 mdetrlin 21749 mdetrsca 21750 m2detleib 21778 smadiadet 21817 |
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