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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mgpsumn | 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 one of the ring, this summand/ factor can be removed from the group sum. (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)) | 
| mgpsumn.n | ⊢ 1 = (1r‘𝑅) | 
| mgpsumn.1 | ⊢ (𝑘 = 𝐼 → 𝐴 = 1 ) | 
| Ref | Expression | 
|---|---|
| mgpsumn | ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴))) | 
| 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 20243 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
| 8 | 3, 7 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | 
| 9 | eqid 2736 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 10 | mgpsumn.n | . . . . 5 ⊢ 1 = (1r‘𝑅) | |
| 11 | 9, 10 | ringidcl 20263 | . . . 4 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) | 
| 12 | 8, 11 | syl 17 | . . 3 ⊢ (𝜑 → 1 ∈ (Base‘𝑅)) | 
| 13 | mgpsumn.1 | . . 3 ⊢ (𝑘 = 𝐼 → 𝐴 = 1 ) | |
| 14 | 1, 2, 3, 4, 5, 6, 12, 13 | mgpsumunsn 48282 | . 2 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 1 )) | 
| 15 | 1, 9 | mgpbas 20143 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑀) | 
| 16 | 1 | crngmgp 20239 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ CMnd) | 
| 17 | 3, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ CMnd) | 
| 18 | diffi 9216 | . . . . 5 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐼}) ∈ Fin) | |
| 19 | 4, 18 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑁 ∖ {𝐼}) ∈ Fin) | 
| 20 | eldifi 4130 | . . . . . 6 ⊢ (𝑘 ∈ (𝑁 ∖ {𝐼}) → 𝑘 ∈ 𝑁) | |
| 21 | 20, 6 | sylan2 593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑁 ∖ {𝐼})) → 𝐴 ∈ (Base‘𝑅)) | 
| 22 | 21 | ralrimiva 3145 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (𝑁 ∖ {𝐼})𝐴 ∈ (Base‘𝑅)) | 
| 23 | 15, 17, 19, 22 | gsummptcl 19986 | . . 3 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) ∈ (Base‘𝑅)) | 
| 24 | 9, 2, 10 | ringridm 20268 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) ∈ (Base‘𝑅)) → ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 1 ) = (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴))) | 
| 25 | 8, 23, 24 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 1 ) = (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴))) | 
| 26 | 14, 25 | eqtrd 2776 | 1 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∖ cdif 3947 {csn 4625 ↦ cmpt 5224 ‘cfv 6560 (class class class)co 7432 Fincfn 8986 Basecbs 17248 .rcmulr 17299 Σg cgsu 17486 CMndccmn 19799 mulGrpcmgp 20138 1rcur 20179 Ringcrg 20231 CRingccrg 20232 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-fzo 13696 df-seq 14044 df-hash 14371 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-0g 17487 df-gsum 17488 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-mgp 20139 df-ur 20180 df-ring 20233 df-cring 20234 | 
| This theorem is referenced by: (None) | 
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