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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mgpsumz | Structured version Visualization version GIF version |
Description: If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the zero of the ring, the group sum itself is zero. (Contributed by AV, 29-Dec-2018.) |
Ref | Expression |
---|---|
mgpsumunsn.m | ⊢ 𝑀 = (mulGrp‘𝑅) |
mgpsumunsn.t | ⊢ · = (.r‘𝑅) |
mgpsumunsn.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
mgpsumunsn.n | ⊢ (𝜑 → 𝑁 ∈ Fin) |
mgpsumunsn.i | ⊢ (𝜑 → 𝐼 ∈ 𝑁) |
mgpsumunsn.a | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅)) |
mgpsumz.z | ⊢ 0 = (0g‘𝑅) |
mgpsumz.0 | ⊢ (𝑘 = 𝐼 → 𝐴 = 0 ) |
Ref | Expression |
---|---|
mgpsumz | ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgpsumunsn.m | . . 3 ⊢ 𝑀 = (mulGrp‘𝑅) | |
2 | mgpsumunsn.t | . . 3 ⊢ · = (.r‘𝑅) | |
3 | mgpsumunsn.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
4 | mgpsumunsn.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
5 | mgpsumunsn.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑁) | |
6 | mgpsumunsn.a | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅)) | |
7 | crngring 20228 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
8 | ringmnd 20226 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) | |
9 | eqid 2726 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
10 | mgpsumz.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
11 | 9, 10 | mndidcl 18742 | . . . 4 ⊢ (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅)) |
12 | 3, 7, 8, 11 | 4syl 19 | . . 3 ⊢ (𝜑 → 0 ∈ (Base‘𝑅)) |
13 | mgpsumz.0 | . . 3 ⊢ (𝑘 = 𝐼 → 𝐴 = 0 ) | |
14 | 1, 2, 3, 4, 5, 6, 12, 13 | mgpsumunsn 47740 | . 2 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 0 )) |
15 | 3, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) |
16 | 1, 9 | mgpbas 20123 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑀) |
17 | 1 | crngmgp 20224 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ CMnd) |
18 | 3, 17 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ CMnd) |
19 | diffi 9213 | . . . . 5 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐼}) ∈ Fin) | |
20 | 4, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑁 ∖ {𝐼}) ∈ Fin) |
21 | eldifi 4126 | . . . . . 6 ⊢ (𝑘 ∈ (𝑁 ∖ {𝐼}) → 𝑘 ∈ 𝑁) | |
22 | 21, 6 | sylan2 591 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑁 ∖ {𝐼})) → 𝐴 ∈ (Base‘𝑅)) |
23 | 22 | ralrimiva 3136 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (𝑁 ∖ {𝐼})𝐴 ∈ (Base‘𝑅)) |
24 | 16, 18, 20, 23 | gsummptcl 19965 | . . 3 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) ∈ (Base‘𝑅)) |
25 | 9, 2, 10 | ringrz 20273 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) ∈ (Base‘𝑅)) → ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 0 ) = 0 ) |
26 | 15, 24, 25 | syl2anc 582 | . 2 ⊢ (𝜑 → ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 0 ) = 0 ) |
27 | 14, 26 | eqtrd 2766 | 1 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∖ cdif 3944 {csn 4633 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 Fincfn 8974 Basecbs 17213 .rcmulr 17267 0gc0g 17454 Σg cgsu 17455 Mndcmnd 18727 CMndccmn 19778 mulGrpcmgp 20117 Ringcrg 20216 CRingccrg 20217 |
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 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-fzo 13682 df-seq 14022 df-hash 14348 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-0g 17456 df-gsum 17457 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-grp 18931 df-minusg 18932 df-mulg 19062 df-cntz 19311 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-cring 20219 |
This theorem is referenced by: (None) |
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