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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 2899 | . 2 ⊢ Ⅎ𝑘𝐶 | |
2 | gsumsn.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
3 | gsumsn.s | . 2 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐶) | |
4 | 1, 2, 3 | gsumsnf 19901 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ 𝑉 ∧ 𝐶 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 {csn 4624 ↦ cmpt 5225 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 Σg cgsu 17415 Mndcmnd 18687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-n0 12497 df-z 12583 df-uz 12847 df-fz 13511 df-fzo 13654 df-seq 13993 df-hash 14316 df-0g 17416 df-gsum 17417 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mulg 19017 df-cntz 19261 |
This theorem is referenced by: gsumpr 19903 gsumpt 19910 dpjidcl 20008 srgbinomlem3 20161 srgbinomlem4 20162 srgbinom 20164 islindf4 21765 psrlidm 21898 psrridm 21899 mplmonmul 21967 ply1coe 22210 mat1dimmul 22371 mdet0pr 22487 m1detdiag 22492 mdetdiaglem 22493 mdetrlin 22497 mdetrsca 22498 m2detleib 22526 chfacfscmulgsum 22755 chfacfpmmulgsum 22759 tmdgsum 23992 tsmsxplem1 24050 tsmsxplem2 24051 imasdsf1olem 24272 tdeglem4 25988 tdeglem4OLD 25989 tdeglem2 25990 amgm 26916 wilthlem2 26994 gsumle 32798 signstf0 34194 aks6d1c1 41581 evlsbagval 41793 selvvvval 41812 sge0tsms 45762 lincvalsng 47478 snlindsntor 47533 |
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