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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 4810 | . . . . . . 7 ⊢ (𝑖 ∈ 𝑁 → ((𝑁 ∖ {𝑖}) ∪ {𝑖}) = 𝑁) | |
| 3 | 2 | eqcomd 2743 | . . . . . 6 ⊢ (𝑖 ∈ 𝑁 → 𝑁 = ((𝑁 ∖ {𝑖}) ∪ {𝑖})) | 
| 4 | 3 | ad2antrl 728 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 𝑁 = ((𝑁 ∖ {𝑖}) ∪ {𝑖})) | 
| 5 | 4 | mpteq1d 5237 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝑛 ∈ 𝑁 ↦ 𝐴) = (𝑛 ∈ ((𝑁 ∖ {𝑖}) ∪ {𝑖}) ↦ 𝐴)) | 
| 6 | 5 | oveq2d 7447 | . . 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 2737 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 9 | 7, 8 | mgpbas 20142 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝐺) | 
| 10 | eqid 2737 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 11 | 7, 10 | mgpplusg 20141 | . . . 4 ⊢ (.r‘𝑅) = (+g‘𝐺) | 
| 12 | gsummgp0.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
| 13 | 7 | crngmgp 20238 | . . . . . 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 9215 | . . . . . 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 4131 | . . . . 5 ⊢ (𝑛 ∈ (𝑁 ∖ {𝑖}) → 𝑛 ∈ 𝑁) | |
| 22 | gsummgp0.a | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅)) | |
| 23 | 20, 21, 22 | syl2an 596 | . . . 4 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) ∧ 𝑛 ∈ (𝑁 ∖ {𝑖})) → 𝐴 ∈ (Base‘𝑅)) | 
| 24 | simprl 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 𝑖 ∈ 𝑁) | |
| 25 | neldifsnd 4793 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → ¬ 𝑖 ∈ (𝑁 ∖ {𝑖})) | |
| 26 | crngring 20242 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
| 27 | 12, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) | 
| 28 | ringmnd 20240 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) | |
| 29 | gsummgp0.0 | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
| 30 | 8, 29 | mndidcl 18762 | . . . . . . 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 2829 | . . . . . 6 ⊢ (𝐵 = 0 → (𝐵 ∈ (Base‘𝑅) ↔ 0 ∈ (Base‘𝑅))) | |
| 34 | 33 | ad2antll 729 | . . . . 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 715 | . . . 4 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) ∧ 𝑛 = 𝑖) → 𝐴 = 𝐵) | 
| 38 | 9, 11, 15, 19, 23, 24, 25, 35, 37 | gsumunsnd 19976 | . . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝐺 Σg (𝑛 ∈ ((𝑁 ∖ {𝑖}) ∪ {𝑖}) ↦ 𝐴)) = ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅)𝐵)) | 
| 39 | oveq2 7439 | . . . . 5 ⊢ (𝐵 = 0 → ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅)𝐵) = ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅) 0 )) | |
| 40 | 39 | ad2antll 729 | . . . 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 593 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑁 ∖ {𝑖})) → 𝐴 ∈ (Base‘𝑅)) | 
| 42 | 41 | ralrimiva 3146 | . . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ (𝑁 ∖ {𝑖})𝐴 ∈ (Base‘𝑅)) | 
| 43 | 9, 14, 18, 42 | gsummptcl 19985 | . . . . . 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 20291 | . . . . 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 685 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅) 0 ) = 0 ) | 
| 47 | 40, 46 | eqtrd 2777 | . . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅)𝐵) = 0 ) | 
| 48 | 6, 38, 47 | 3eqtrd 2781 | . 2 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ 𝐴)) = 0 ) | 
| 49 | 1, 48 | rexlimddv 3161 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ 𝐴)) = 0 ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ∖ cdif 3948 ∪ cun 3949 {csn 4626 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 Basecbs 17247 .rcmulr 17298 0gc0g 17484 Σg cgsu 17485 Mndcmnd 18747 CMndccmn 19798 mulGrpcmgp 20137 Ringcrg 20230 CRingccrg 20231 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-0g 17486 df-gsum 17487 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 | 
| This theorem is referenced by: smadiadetlem0 22667 | 
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