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Mirrors > Home > MPE Home > Th. List > gsummgp0 | Structured version Visualization version GIF version |
Description: If one factor in a finite group sum of the multiplicative group of a commutative ring is 0, the whole "sum" (i.e. product) is 0. (Contributed by AV, 3-Jan-2019.) |
Ref | Expression |
---|---|
gsummgp0.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
gsummgp0.0 | ⊢ 0 = (0g‘𝑅) |
gsummgp0.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
gsummgp0.n | ⊢ (𝜑 → 𝑁 ∈ Fin) |
gsummgp0.a | ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅)) |
gsummgp0.e | ⊢ ((𝜑 ∧ 𝑛 = 𝑖) → 𝐴 = 𝐵) |
gsummgp0.b | ⊢ (𝜑 → ∃𝑖 ∈ 𝑁 𝐵 = 0 ) |
Ref | Expression |
---|---|
gsummgp0 | ⊢ (𝜑 → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ 𝐴)) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummgp0.b | . 2 ⊢ (𝜑 → ∃𝑖 ∈ 𝑁 𝐵 = 0 ) | |
2 | difsnid 4814 | . . . . . . 7 ⊢ (𝑖 ∈ 𝑁 → ((𝑁 ∖ {𝑖}) ∪ {𝑖}) = 𝑁) | |
3 | 2 | eqcomd 2739 | . . . . . 6 ⊢ (𝑖 ∈ 𝑁 → 𝑁 = ((𝑁 ∖ {𝑖}) ∪ {𝑖})) |
4 | 3 | ad2antrl 727 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 𝑁 = ((𝑁 ∖ {𝑖}) ∪ {𝑖})) |
5 | 4 | mpteq1d 5244 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝑛 ∈ 𝑁 ↦ 𝐴) = (𝑛 ∈ ((𝑁 ∖ {𝑖}) ∪ {𝑖}) ↦ 𝐴)) |
6 | 5 | oveq2d 7425 | . . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ 𝐴)) = (𝐺 Σg (𝑛 ∈ ((𝑁 ∖ {𝑖}) ∪ {𝑖}) ↦ 𝐴))) |
7 | gsummgp0.g | . . . . 5 ⊢ 𝐺 = (mulGrp‘𝑅) | |
8 | eqid 2733 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 7, 8 | mgpbas 19993 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝐺) |
10 | eqid 2733 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
11 | 7, 10 | mgpplusg 19991 | . . . 4 ⊢ (.r‘𝑅) = (+g‘𝐺) |
12 | gsummgp0.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
13 | 7 | crngmgp 20064 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd) |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
15 | 14 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 𝐺 ∈ CMnd) |
16 | gsummgp0.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
17 | diffi 9179 | . . . . . 6 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝑖}) ∈ Fin) | |
18 | 16, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑁 ∖ {𝑖}) ∈ Fin) |
19 | 18 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝑁 ∖ {𝑖}) ∈ Fin) |
20 | simpl 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 𝜑) | |
21 | eldifi 4127 | . . . . 5 ⊢ (𝑛 ∈ (𝑁 ∖ {𝑖}) → 𝑛 ∈ 𝑁) | |
22 | gsummgp0.a | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅)) | |
23 | 20, 21, 22 | syl2an 597 | . . . 4 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) ∧ 𝑛 ∈ (𝑁 ∖ {𝑖})) → 𝐴 ∈ (Base‘𝑅)) |
24 | simprl 770 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 𝑖 ∈ 𝑁) | |
25 | neldifsnd 4797 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → ¬ 𝑖 ∈ (𝑁 ∖ {𝑖})) | |
26 | crngring 20068 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
27 | 12, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) |
28 | ringmnd 20066 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) | |
29 | gsummgp0.0 | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
30 | 8, 29 | mndidcl 18640 | . . . . . . 7 ⊢ (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅)) |
31 | 27, 28, 30 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 0 ∈ (Base‘𝑅)) |
32 | 31 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 0 ∈ (Base‘𝑅)) |
33 | eleq1 2822 | . . . . . 6 ⊢ (𝐵 = 0 → (𝐵 ∈ (Base‘𝑅) ↔ 0 ∈ (Base‘𝑅))) | |
34 | 33 | ad2antll 728 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝐵 ∈ (Base‘𝑅) ↔ 0 ∈ (Base‘𝑅))) |
35 | 32, 34 | mpbird 257 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 𝐵 ∈ (Base‘𝑅)) |
36 | gsummgp0.e | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 = 𝑖) → 𝐴 = 𝐵) | |
37 | 36 | adantlr 714 | . . . 4 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) ∧ 𝑛 = 𝑖) → 𝐴 = 𝐵) |
38 | 9, 11, 15, 19, 23, 24, 25, 35, 37 | gsumunsnd 19826 | . . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝐺 Σg (𝑛 ∈ ((𝑁 ∖ {𝑖}) ∪ {𝑖}) ↦ 𝐴)) = ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅)𝐵)) |
39 | oveq2 7417 | . . . . 5 ⊢ (𝐵 = 0 → ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅)𝐵) = ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅) 0 )) | |
40 | 39 | ad2antll 728 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅)𝐵) = ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅) 0 )) |
41 | 21, 22 | sylan2 594 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑁 ∖ {𝑖})) → 𝐴 ∈ (Base‘𝑅)) |
42 | 41 | ralrimiva 3147 | . . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ (𝑁 ∖ {𝑖})𝐴 ∈ (Base‘𝑅)) |
43 | 9, 14, 18, 42 | gsummptcl 19835 | . . . . . 6 ⊢ (𝜑 → (𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴)) ∈ (Base‘𝑅)) |
44 | 43 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴)) ∈ (Base‘𝑅)) |
45 | 8, 10, 29 | ringrz 20108 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴)) ∈ (Base‘𝑅)) → ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅) 0 ) = 0 ) |
46 | 27, 44, 45 | syl2an2r 684 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅) 0 ) = 0 ) |
47 | 40, 46 | eqtrd 2773 | . . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅)𝐵) = 0 ) |
48 | 6, 38, 47 | 3eqtrd 2777 | . 2 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ 𝐴)) = 0 ) |
49 | 1, 48 | rexlimddv 3162 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ 𝐴)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 ∖ cdif 3946 ∪ cun 3947 {csn 4629 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7409 Fincfn 8939 Basecbs 17144 .rcmulr 17198 0gc0g 17385 Σg cgsu 17386 Mndcmnd 18625 CMndccmn 19648 mulGrpcmgp 19987 Ringcrg 20056 CRingccrg 20057 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-seq 13967 df-hash 14291 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-0g 17387 df-gsum 17388 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-grp 18822 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-mgp 19988 df-ring 20058 df-cring 20059 |
This theorem is referenced by: smadiadetlem0 22163 |
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