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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 19308 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
8 | 3, 7 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
9 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
10 | mgpsumn.n | . . . . 5 ⊢ 1 = (1r‘𝑅) | |
11 | 9, 10 | ringidcl 19318 | . . . 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 44429 | . 2 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 1 )) |
15 | 1, 9 | mgpbas 19245 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑀) |
16 | 1 | crngmgp 19305 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ CMnd) |
17 | 3, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ CMnd) |
18 | diffi 8750 | . . . . 5 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐼}) ∈ Fin) | |
19 | 4, 18 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑁 ∖ {𝐼}) ∈ Fin) |
20 | eldifi 4103 | . . . . . 6 ⊢ (𝑘 ∈ (𝑁 ∖ {𝐼}) → 𝑘 ∈ 𝑁) | |
21 | 20, 6 | sylan2 594 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑁 ∖ {𝐼})) → 𝐴 ∈ (Base‘𝑅)) |
22 | 21 | ralrimiva 3182 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (𝑁 ∖ {𝐼})𝐴 ∈ (Base‘𝑅)) |
23 | 15, 17, 19, 22 | gsummptcl 19087 | . . 3 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) ∈ (Base‘𝑅)) |
24 | 9, 2, 10 | ringridm 19322 | . . 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 586 | . 2 ⊢ (𝜑 → ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 1 ) = (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴))) |
26 | 14, 25 | eqtrd 2856 | 1 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3933 {csn 4567 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 Basecbs 16483 .rcmulr 16566 Σg cgsu 16714 CMndccmn 18906 mulGrpcmgp 19239 1rcur 19251 Ringcrg 19297 CRingccrg 19298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-gsum 16716 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 |
This theorem is referenced by: (None) |
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