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Mirrors > Home > MPE Home > Th. List > gsumsn | Structured version Visualization version GIF version |
Description: Group sum of a singleton. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Proof shortened by AV, 11-Dec-2019.) |
Ref | Expression |
---|---|
gsumsn.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumsn.s | ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐶) |
Ref | Expression |
---|---|
gsumsn | ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ 𝑉 ∧ 𝐶 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2895 | . 2 ⊢ Ⅎ𝑘𝐶 | |
2 | gsumsn.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
3 | gsumsn.s | . 2 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐶) | |
4 | 1, 2, 3 | gsumsnf 19865 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ 𝑉 ∧ 𝐶 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {csn 4621 ↦ cmpt 5222 ‘cfv 6534 (class class class)co 7402 Basecbs 17145 Σg cgsu 17387 Mndcmnd 18659 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-oi 9502 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-n0 12471 df-z 12557 df-uz 12821 df-fz 13483 df-fzo 13626 df-seq 13965 df-hash 14289 df-0g 17388 df-gsum 17389 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mulg 18988 df-cntz 19225 |
This theorem is referenced by: gsumpr 19867 gsumpt 19874 dpjidcl 19972 srgbinomlem3 20125 srgbinomlem4 20126 srgbinom 20128 islindf4 21703 psrlidm 21835 psrridm 21836 mplmonmul 21903 ply1coe 22141 mat1dimmul 22302 mdet0pr 22418 m1detdiag 22423 mdetdiaglem 22424 mdetrlin 22428 mdetrsca 22429 m2detleib 22457 chfacfscmulgsum 22686 chfacfpmmulgsum 22690 tmdgsum 23923 tsmsxplem1 23981 tsmsxplem2 23982 imasdsf1olem 24203 tdeglem4 25919 tdeglem4OLD 25920 tdeglem2 25921 amgm 26842 wilthlem2 26920 gsumle 32713 signstf0 34071 evlsbagval 41631 selvvvval 41650 sge0tsms 45606 lincvalsng 47310 snlindsntor 47365 |
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