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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 4813 | . . . . . . 7 ⊢ (𝑖 ∈ 𝑁 → ((𝑁 ∖ {𝑖}) ∪ {𝑖}) = 𝑁) | |
3 | 2 | eqcomd 2737 | . . . . . 6 ⊢ (𝑖 ∈ 𝑁 → 𝑁 = ((𝑁 ∖ {𝑖}) ∪ {𝑖})) |
4 | 3 | ad2antrl 725 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 𝑁 = ((𝑁 ∖ {𝑖}) ∪ {𝑖})) |
5 | 4 | mpteq1d 5243 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝑛 ∈ 𝑁 ↦ 𝐴) = (𝑛 ∈ ((𝑁 ∖ {𝑖}) ∪ {𝑖}) ↦ 𝐴)) |
6 | 5 | oveq2d 7428 | . . 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 2731 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 7, 8 | mgpbas 20041 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝐺) |
10 | eqid 2731 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
11 | 7, 10 | mgpplusg 20039 | . . . 4 ⊢ (.r‘𝑅) = (+g‘𝐺) |
12 | gsummgp0.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
13 | 7 | crngmgp 20142 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd) |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
15 | 14 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 𝐺 ∈ CMnd) |
16 | gsummgp0.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
17 | diffi 9185 | . . . . . 6 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝑖}) ∈ Fin) | |
18 | 16, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑁 ∖ {𝑖}) ∈ Fin) |
19 | 18 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝑁 ∖ {𝑖}) ∈ Fin) |
20 | simpl 482 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 𝜑) | |
21 | eldifi 4126 | . . . . 5 ⊢ (𝑛 ∈ (𝑁 ∖ {𝑖}) → 𝑛 ∈ 𝑁) | |
22 | gsummgp0.a | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅)) | |
23 | 20, 21, 22 | syl2an 595 | . . . 4 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) ∧ 𝑛 ∈ (𝑁 ∖ {𝑖})) → 𝐴 ∈ (Base‘𝑅)) |
24 | simprl 768 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 𝑖 ∈ 𝑁) | |
25 | neldifsnd 4796 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → ¬ 𝑖 ∈ (𝑁 ∖ {𝑖})) | |
26 | crngring 20146 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
27 | 12, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) |
28 | ringmnd 20144 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) | |
29 | gsummgp0.0 | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
30 | 8, 29 | mndidcl 18680 | . . . . . . 7 ⊢ (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅)) |
31 | 27, 28, 30 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 0 ∈ (Base‘𝑅)) |
32 | 31 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 0 ∈ (Base‘𝑅)) |
33 | eleq1 2820 | . . . . . 6 ⊢ (𝐵 = 0 → (𝐵 ∈ (Base‘𝑅) ↔ 0 ∈ (Base‘𝑅))) | |
34 | 33 | ad2antll 726 | . . . . 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 712 | . . . 4 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) ∧ 𝑛 = 𝑖) → 𝐴 = 𝐵) |
38 | 9, 11, 15, 19, 23, 24, 25, 35, 37 | gsumunsnd 19874 | . . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝐺 Σg (𝑛 ∈ ((𝑁 ∖ {𝑖}) ∪ {𝑖}) ↦ 𝐴)) = ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅)𝐵)) |
39 | oveq2 7420 | . . . . 5 ⊢ (𝐵 = 0 → ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅)𝐵) = ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅) 0 )) | |
40 | 39 | ad2antll 726 | . . . 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 592 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑁 ∖ {𝑖})) → 𝐴 ∈ (Base‘𝑅)) |
42 | 41 | ralrimiva 3145 | . . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ (𝑁 ∖ {𝑖})𝐴 ∈ (Base‘𝑅)) |
43 | 9, 14, 18, 42 | gsummptcl 19883 | . . . . . 6 ⊢ (𝜑 → (𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴)) ∈ (Base‘𝑅)) |
44 | 43 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴)) ∈ (Base‘𝑅)) |
45 | 8, 10, 29 | ringrz 20189 | . . . . 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 682 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅) 0 ) = 0 ) |
47 | 40, 46 | eqtrd 2771 | . . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅)𝐵) = 0 ) |
48 | 6, 38, 47 | 3eqtrd 2775 | . 2 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ 𝐴)) = 0 ) |
49 | 1, 48 | rexlimddv 3160 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ 𝐴)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 ∖ cdif 3945 ∪ cun 3946 {csn 4628 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7412 Fincfn 8945 Basecbs 17151 .rcmulr 17205 0gc0g 17392 Σg cgsu 17393 Mndcmnd 18665 CMndccmn 19696 mulGrpcmgp 20035 Ringcrg 20134 CRingccrg 20135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-fzo 13635 df-seq 13974 df-hash 14298 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-0g 17394 df-gsum 17395 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-mulg 18994 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 |
This theorem is referenced by: smadiadetlem0 22483 |
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