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 4341 | . . . . . 6 ⊢ (𝐼 ∈ 𝑁 → ((𝑁 ∖ {𝐼}) ∪ {𝐼}) = 𝑁) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝑁 ∖ {𝐼}) ∪ {𝐼}) = 𝑁) |
4 | 3 | eqcomd 2628 | . . . 4 ⊢ (𝜑 → 𝑁 = ((𝑁 ∖ {𝐼}) ∪ {𝐼})) |
5 | 4 | mpteq1d 4738 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑁 ↦ 𝐴) = (𝑘 ∈ ((𝑁 ∖ {𝐼}) ∪ {𝐼}) ↦ 𝐴)) |
6 | 5 | oveq2d 6666 | . 2 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = (𝑀 Σg (𝑘 ∈ ((𝑁 ∖ {𝐼}) ∪ {𝐼}) ↦ 𝐴))) |
7 | mgpsumunsn.m | . . . 4 ⊢ 𝑀 = (mulGrp‘𝑅) | |
8 | eqid 2622 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 7, 8 | mgpbas 18495 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑀) |
10 | mgpsumunsn.t | . . . 4 ⊢ · = (.r‘𝑅) | |
11 | 7, 10 | mgpplusg 18493 | . . 3 ⊢ · = (+g‘𝑀) |
12 | mgpsumunsn.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
13 | 7 | crngmgp 18555 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ CMnd) |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 ∈ CMnd) |
15 | mgpsumunsn.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
16 | diffi 8192 | . . . 4 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐼}) ∈ Fin) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ (𝜑 → (𝑁 ∖ {𝐼}) ∈ Fin) |
18 | eldifi 3732 | . . . 4 ⊢ (𝑘 ∈ (𝑁 ∖ {𝐼}) → 𝑘 ∈ 𝑁) | |
19 | mgpsumunsn.a | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅)) | |
20 | 18, 19 | sylan2 491 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑁 ∖ {𝐼})) → 𝐴 ∈ (Base‘𝑅)) |
21 | neldifsnd 4322 | . . 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 18359 | . 2 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ ((𝑁 ∖ {𝐼}) ∪ {𝐼}) ↦ 𝐴)) = ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 𝑋)) |
25 | 6, 24 | eqtrd 2656 | 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 384 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ∪ cun 3572 {csn 4177 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 Basecbs 15857 .rcmulr 15942 Σg cgsu 16101 CMndccmn 18193 mulGrpcmgp 18489 CRingccrg 18548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-gsum 16103 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-mgp 18490 df-cring 18550 |
This theorem is referenced by: mgpsumz 42141 mgpsumn 42142 |
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