Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mgpsumunsn | Structured version Visualization version GIF version |
Description: Extract a summand/factor from the group sum for the multiplicative group of a unital ring. (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)) |
mgpsumunsn.x | ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅)) |
mgpsumunsn.e | ⊢ (𝑘 = 𝐼 → 𝐴 = 𝑋) |
Ref | Expression |
---|---|
mgpsumunsn | ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgpsumunsn.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑁) | |
2 | difsnid 4736 | . . . . . 6 ⊢ (𝐼 ∈ 𝑁 → ((𝑁 ∖ {𝐼}) ∪ {𝐼}) = 𝑁) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝑁 ∖ {𝐼}) ∪ {𝐼}) = 𝑁) |
4 | 3 | eqcomd 2827 | . . . 4 ⊢ (𝜑 → 𝑁 = ((𝑁 ∖ {𝐼}) ∪ {𝐼})) |
5 | 4 | mpteq1d 5147 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑁 ↦ 𝐴) = (𝑘 ∈ ((𝑁 ∖ {𝐼}) ∪ {𝐼}) ↦ 𝐴)) |
6 | 5 | oveq2d 7166 | . 2 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = (𝑀 Σg (𝑘 ∈ ((𝑁 ∖ {𝐼}) ∪ {𝐼}) ↦ 𝐴))) |
7 | mgpsumunsn.m | . . . 4 ⊢ 𝑀 = (mulGrp‘𝑅) | |
8 | eqid 2821 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 7, 8 | mgpbas 19239 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑀) |
10 | mgpsumunsn.t | . . . 4 ⊢ · = (.r‘𝑅) | |
11 | 7, 10 | mgpplusg 19237 | . . 3 ⊢ · = (+g‘𝑀) |
12 | mgpsumunsn.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
13 | 7 | crngmgp 19299 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ CMnd) |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 ∈ CMnd) |
15 | mgpsumunsn.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
16 | diffi 8744 | . . . 4 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐼}) ∈ Fin) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ (𝜑 → (𝑁 ∖ {𝐼}) ∈ Fin) |
18 | eldifi 4102 | . . . 4 ⊢ (𝑘 ∈ (𝑁 ∖ {𝐼}) → 𝑘 ∈ 𝑁) | |
19 | mgpsumunsn.a | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅)) | |
20 | 18, 19 | sylan2 594 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑁 ∖ {𝐼})) → 𝐴 ∈ (Base‘𝑅)) |
21 | neldifsnd 4719 | . . 3 ⊢ (𝜑 → ¬ 𝐼 ∈ (𝑁 ∖ {𝐼})) | |
22 | mgpsumunsn.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅)) | |
23 | mgpsumunsn.e | . . 3 ⊢ (𝑘 = 𝐼 → 𝐴 = 𝑋) | |
24 | 9, 11, 14, 17, 20, 1, 21, 22, 23 | gsumunsn 19074 | . 2 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ ((𝑁 ∖ {𝐼}) ∪ {𝐼}) ↦ 𝐴)) = ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 𝑋)) |
25 | 6, 24 | eqtrd 2856 | 1 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∖ cdif 3932 ∪ cun 3933 {csn 4560 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7150 Fincfn 8503 Basecbs 16477 .rcmulr 16560 Σg cgsu 16708 CMndccmn 18900 mulGrpcmgp 19233 CRingccrg 19292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-0g 16709 df-gsum 16710 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-mgp 19234 df-cring 19294 |
This theorem is referenced by: mgpsumz 44404 mgpsumn 44405 |
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