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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 19302 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
8 | ringmnd 19300 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Mnd) |
10 | 3, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd) |
11 | eqid 2798 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
12 | mgpsumz.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
13 | 11, 12 | mndidcl 17918 | . . . 4 ⊢ (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅)) |
14 | 10, 13 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ (Base‘𝑅)) |
15 | mgpsumz.0 | . . 3 ⊢ (𝑘 = 𝐼 → 𝐴 = 0 ) | |
16 | 1, 2, 3, 4, 5, 6, 14, 15 | mgpsumunsn 44763 | . 2 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 0 )) |
17 | 3, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) |
18 | 1, 11 | mgpbas 19238 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑀) |
19 | 1 | crngmgp 19298 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ CMnd) |
20 | 3, 19 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ CMnd) |
21 | diffi 8734 | . . . . 5 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐼}) ∈ Fin) | |
22 | 4, 21 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑁 ∖ {𝐼}) ∈ Fin) |
23 | eldifi 4054 | . . . . . 6 ⊢ (𝑘 ∈ (𝑁 ∖ {𝐼}) → 𝑘 ∈ 𝑁) | |
24 | 23, 6 | sylan2 595 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑁 ∖ {𝐼})) → 𝐴 ∈ (Base‘𝑅)) |
25 | 24 | ralrimiva 3149 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (𝑁 ∖ {𝐼})𝐴 ∈ (Base‘𝑅)) |
26 | 18, 20, 22, 25 | gsummptcl 19080 | . . 3 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) ∈ (Base‘𝑅)) |
27 | 11, 2, 12 | ringrz 19334 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) ∈ (Base‘𝑅)) → ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 0 ) = 0 ) |
28 | 17, 26, 27 | syl2anc 587 | . 2 ⊢ (𝜑 → ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 0 ) = 0 ) |
29 | 16, 28 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 {csn 4525 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 Basecbs 16475 .rcmulr 16558 0gc0g 16705 Σg cgsu 16706 Mndcmnd 17903 CMndccmn 18898 mulGrpcmgp 19232 Ringcrg 19290 CRingccrg 19291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-mgp 19233 df-ring 19292 df-cring 19293 |
This theorem is referenced by: (None) |
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