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Mirrors > Home > MPE Home > Th. List > gsummgp0 | Structured version Visualization version GIF version |
Description: If one factor in a finite group sum of the multiplicative group of a commutative ring is 0, the whole "sum" (i.e. product) is 0. (Contributed by AV, 3-Jan-2019.) |
Ref | Expression |
---|---|
gsummgp0.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
gsummgp0.0 | ⊢ 0 = (0g‘𝑅) |
gsummgp0.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
gsummgp0.n | ⊢ (𝜑 → 𝑁 ∈ Fin) |
gsummgp0.a | ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅)) |
gsummgp0.e | ⊢ ((𝜑 ∧ 𝑛 = 𝑖) → 𝐴 = 𝐵) |
gsummgp0.b | ⊢ (𝜑 → ∃𝑖 ∈ 𝑁 𝐵 = 0 ) |
Ref | Expression |
---|---|
gsummgp0 | ⊢ (𝜑 → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ 𝐴)) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummgp0.b | . 2 ⊢ (𝜑 → ∃𝑖 ∈ 𝑁 𝐵 = 0 ) | |
2 | difsnid 4815 | . . . . . . 7 ⊢ (𝑖 ∈ 𝑁 → ((𝑁 ∖ {𝑖}) ∪ {𝑖}) = 𝑁) | |
3 | 2 | eqcomd 2741 | . . . . . 6 ⊢ (𝑖 ∈ 𝑁 → 𝑁 = ((𝑁 ∖ {𝑖}) ∪ {𝑖})) |
4 | 3 | ad2antrl 728 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 𝑁 = ((𝑁 ∖ {𝑖}) ∪ {𝑖})) |
5 | 4 | mpteq1d 5243 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝑛 ∈ 𝑁 ↦ 𝐴) = (𝑛 ∈ ((𝑁 ∖ {𝑖}) ∪ {𝑖}) ↦ 𝐴)) |
6 | 5 | oveq2d 7447 | . . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ 𝐴)) = (𝐺 Σg (𝑛 ∈ ((𝑁 ∖ {𝑖}) ∪ {𝑖}) ↦ 𝐴))) |
7 | gsummgp0.g | . . . . 5 ⊢ 𝐺 = (mulGrp‘𝑅) | |
8 | eqid 2735 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 7, 8 | mgpbas 20158 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝐺) |
10 | eqid 2735 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
11 | 7, 10 | mgpplusg 20156 | . . . 4 ⊢ (.r‘𝑅) = (+g‘𝐺) |
12 | gsummgp0.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
13 | 7 | crngmgp 20259 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd) |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
15 | 14 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 𝐺 ∈ CMnd) |
16 | gsummgp0.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
17 | diffi 9214 | . . . . . 6 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝑖}) ∈ Fin) | |
18 | 16, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑁 ∖ {𝑖}) ∈ Fin) |
19 | 18 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝑁 ∖ {𝑖}) ∈ Fin) |
20 | simpl 482 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 𝜑) | |
21 | eldifi 4141 | . . . . 5 ⊢ (𝑛 ∈ (𝑁 ∖ {𝑖}) → 𝑛 ∈ 𝑁) | |
22 | gsummgp0.a | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅)) | |
23 | 20, 21, 22 | syl2an 596 | . . . 4 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) ∧ 𝑛 ∈ (𝑁 ∖ {𝑖})) → 𝐴 ∈ (Base‘𝑅)) |
24 | simprl 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 𝑖 ∈ 𝑁) | |
25 | neldifsnd 4798 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → ¬ 𝑖 ∈ (𝑁 ∖ {𝑖})) | |
26 | crngring 20263 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
27 | 12, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) |
28 | ringmnd 20261 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) | |
29 | gsummgp0.0 | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
30 | 8, 29 | mndidcl 18775 | . . . . . . 7 ⊢ (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅)) |
31 | 27, 28, 30 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 0 ∈ (Base‘𝑅)) |
32 | 31 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 0 ∈ (Base‘𝑅)) |
33 | eleq1 2827 | . . . . . 6 ⊢ (𝐵 = 0 → (𝐵 ∈ (Base‘𝑅) ↔ 0 ∈ (Base‘𝑅))) | |
34 | 33 | ad2antll 729 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝐵 ∈ (Base‘𝑅) ↔ 0 ∈ (Base‘𝑅))) |
35 | 32, 34 | mpbird 257 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → 𝐵 ∈ (Base‘𝑅)) |
36 | gsummgp0.e | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 = 𝑖) → 𝐴 = 𝐵) | |
37 | 36 | adantlr 715 | . . . 4 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) ∧ 𝑛 = 𝑖) → 𝐴 = 𝐵) |
38 | 9, 11, 15, 19, 23, 24, 25, 35, 37 | gsumunsnd 19991 | . . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝐺 Σg (𝑛 ∈ ((𝑁 ∖ {𝑖}) ∪ {𝑖}) ↦ 𝐴)) = ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅)𝐵)) |
39 | oveq2 7439 | . . . . 5 ⊢ (𝐵 = 0 → ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅)𝐵) = ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅) 0 )) | |
40 | 39 | ad2antll 729 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅)𝐵) = ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅) 0 )) |
41 | 21, 22 | sylan2 593 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑁 ∖ {𝑖})) → 𝐴 ∈ (Base‘𝑅)) |
42 | 41 | ralrimiva 3144 | . . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ (𝑁 ∖ {𝑖})𝐴 ∈ (Base‘𝑅)) |
43 | 9, 14, 18, 42 | gsummptcl 20000 | . . . . . 6 ⊢ (𝜑 → (𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴)) ∈ (Base‘𝑅)) |
44 | 43 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴)) ∈ (Base‘𝑅)) |
45 | 8, 10, 29 | ringrz 20308 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴)) ∈ (Base‘𝑅)) → ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅) 0 ) = 0 ) |
46 | 27, 44, 45 | syl2an2r 685 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅) 0 ) = 0 ) |
47 | 40, 46 | eqtrd 2775 | . . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝑖}) ↦ 𝐴))(.r‘𝑅)𝐵) = 0 ) |
48 | 6, 38, 47 | 3eqtrd 2779 | . 2 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑁 ∧ 𝐵 = 0 )) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ 𝐴)) = 0 ) |
49 | 1, 48 | rexlimddv 3159 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ 𝐴)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ∖ cdif 3960 ∪ cun 3961 {csn 4631 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 Basecbs 17245 .rcmulr 17299 0gc0g 17486 Σg cgsu 17487 Mndcmnd 18760 CMndccmn 19813 mulGrpcmgp 20152 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-grp 18967 df-minusg 18968 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 |
This theorem is referenced by: smadiadetlem0 22683 |
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