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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 20263 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
8 | 3, 7 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
9 | eqid 2735 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
10 | mgpsumn.n | . . . . 5 ⊢ 1 = (1r‘𝑅) | |
11 | 9, 10 | ringidcl 20280 | . . . 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 48206 | . 2 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 1 )) |
15 | 1, 9 | mgpbas 20158 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑀) |
16 | 1 | crngmgp 20259 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ CMnd) |
17 | 3, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ CMnd) |
18 | diffi 9214 | . . . . 5 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐼}) ∈ Fin) | |
19 | 4, 18 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑁 ∖ {𝐼}) ∈ Fin) |
20 | eldifi 4141 | . . . . . 6 ⊢ (𝑘 ∈ (𝑁 ∖ {𝐼}) → 𝑘 ∈ 𝑁) | |
21 | 20, 6 | sylan2 593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑁 ∖ {𝐼})) → 𝐴 ∈ (Base‘𝑅)) |
22 | 21 | ralrimiva 3144 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (𝑁 ∖ {𝐼})𝐴 ∈ (Base‘𝑅)) |
23 | 15, 17, 19, 22 | gsummptcl 20000 | . . 3 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) ∈ (Base‘𝑅)) |
24 | 9, 2, 10 | ringridm 20284 | . . 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 2775 | 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 1537 ∈ wcel 2106 ∖ cdif 3960 {csn 4631 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 Basecbs 17245 .rcmulr 17299 Σg cgsu 17487 CMndccmn 19813 mulGrpcmgp 20152 1rcur 20199 Ringcrg 20251 CRingccrg 20252 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-gsum 17489 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-mgp 20153 df-ur 20200 df-ring 20253 df-cring 20254 |
This theorem is referenced by: (None) |
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