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 4282 | . . . . . 6 ⊢ (𝐼 ∈ 𝑁 → ((𝑁 ∖ {𝐼}) ∪ {𝐼}) = 𝑁) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝑁 ∖ {𝐼}) ∪ {𝐼}) = 𝑁) |
4 | 3 | eqcomd 2616 | . . . 4 ⊢ (𝜑 → 𝑁 = ((𝑁 ∖ {𝐼}) ∪ {𝐼})) |
5 | 4 | mpteq1d 4666 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑁 ↦ 𝐴) = (𝑘 ∈ ((𝑁 ∖ {𝐼}) ∪ {𝐼}) ↦ 𝐴)) |
6 | 5 | oveq2d 6565 | . 2 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = (𝑀 Σg (𝑘 ∈ ((𝑁 ∖ {𝐼}) ∪ {𝐼}) ↦ 𝐴))) |
7 | mgpsumunsn.m | . . . 4 ⊢ 𝑀 = (mulGrp‘𝑅) | |
8 | eqid 2610 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 7, 8 | mgpbas 18318 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑀) |
10 | mgpsumunsn.t | . . . 4 ⊢ · = (.r‘𝑅) | |
11 | 7, 10 | mgpplusg 18316 | . . 3 ⊢ · = (+g‘𝑀) |
12 | mgpsumunsn.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
13 | 7 | crngmgp 18378 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ CMnd) |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 ∈ CMnd) |
15 | mgpsumunsn.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
16 | diffi 8077 | . . . 4 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐼}) ∈ Fin) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ (𝜑 → (𝑁 ∖ {𝐼}) ∈ Fin) |
18 | eldifi 3694 | . . . 4 ⊢ (𝑘 ∈ (𝑁 ∖ {𝐼}) → 𝑘 ∈ 𝑁) | |
19 | mgpsumunsn.a | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅)) | |
20 | 18, 19 | sylan2 490 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑁 ∖ {𝐼})) → 𝐴 ∈ (Base‘𝑅)) |
21 | neldifsnd 4263 | . . 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 18182 | . 2 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ ((𝑁 ∖ {𝐼}) ∪ {𝐼}) ↦ 𝐴)) = ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 𝑋)) |
25 | 6, 24 | eqtrd 2644 | 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 383 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∪ cun 3538 {csn 4125 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 Basecbs 15695 .rcmulr 15769 Σg cgsu 15924 CMndccmn 18016 mulGrpcmgp 18312 CRingccrg 18371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-mgp 18313 df-cring 18373 |
This theorem is referenced by: mgpsumz 41934 mgpsumn 41935 |
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