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Mirrors > Home > MPE Home > Th. List > gsummpt1n0 | Structured version Visualization version GIF version |
Description: If only one summand in a finite group sum is not zero, the whole sum equals this summand. More general version of gsummptif1n0 18842. (Contributed by AV, 11-Oct-2019.) |
Ref | Expression |
---|---|
gsummpt1n0.0 | ⊢ 0 = (0g‘𝐺) |
gsummpt1n0.g | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
gsummpt1n0.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
gsummpt1n0.x | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
gsummpt1n0.f | ⊢ 𝐹 = (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) |
gsummpt1n0.a | ⊢ (𝜑 → ∀𝑛 ∈ 𝐼 𝐴 ∈ (Base‘𝐺)) |
Ref | Expression |
---|---|
gsummpt1n0 | ⊢ (𝜑 → (𝐺 Σg 𝐹) = ⦋𝑋 / 𝑛⦌𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | gsummpt1n0.0 | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | gsummpt1n0.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
4 | gsummpt1n0.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | gsummpt1n0.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
6 | gsummpt1n0.a | . . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ 𝐼 𝐴 ∈ (Base‘𝐺)) | |
7 | 6 | r19.21bi 3158 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → 𝐴 ∈ (Base‘𝐺)) |
8 | 1, 2 | mndidcl 17779 | . . . . . . 7 ⊢ (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺)) |
9 | 3, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
10 | 9 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → 0 ∈ (Base‘𝐺)) |
11 | 7, 10 | ifcld 4396 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → if(𝑛 = 𝑋, 𝐴, 0 ) ∈ (Base‘𝐺)) |
12 | gsummpt1n0.f | . . . 4 ⊢ 𝐹 = (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) | |
13 | 11, 12 | fmptd 6703 | . . 3 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺)) |
14 | 12 | oveq1i 6988 | . . . 4 ⊢ (𝐹 supp 0 ) = ((𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) supp 0 ) |
15 | eldifsni 4597 | . . . . . . 7 ⊢ (𝑛 ∈ (𝐼 ∖ {𝑋}) → 𝑛 ≠ 𝑋) | |
16 | 15 | adantl 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐼 ∖ {𝑋})) → 𝑛 ≠ 𝑋) |
17 | ifnefalse 4363 | . . . . . 6 ⊢ (𝑛 ≠ 𝑋 → if(𝑛 = 𝑋, 𝐴, 0 ) = 0 ) | |
18 | 16, 17 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐼 ∖ {𝑋})) → if(𝑛 = 𝑋, 𝐴, 0 ) = 0 ) |
19 | 18, 4 | suppss2 7669 | . . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) supp 0 ) ⊆ {𝑋}) |
20 | 14, 19 | syl5eqss 3907 | . . 3 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ {𝑋}) |
21 | 1, 2, 3, 4, 5, 13, 20 | gsumpt 18838 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐹‘𝑋)) |
22 | nfcv 2932 | . . . . 5 ⊢ Ⅎ𝑦if(𝑛 = 𝑋, 𝐴, 0 ) | |
23 | nfv 1873 | . . . . . 6 ⊢ Ⅎ𝑛 𝑦 = 𝑋 | |
24 | nfcsb1v 3806 | . . . . . 6 ⊢ Ⅎ𝑛⦋𝑦 / 𝑛⦌𝐴 | |
25 | nfcv 2932 | . . . . . 6 ⊢ Ⅎ𝑛 0 | |
26 | 23, 24, 25 | nfif 4380 | . . . . 5 ⊢ Ⅎ𝑛if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 ) |
27 | eqeq1 2782 | . . . . . 6 ⊢ (𝑛 = 𝑦 → (𝑛 = 𝑋 ↔ 𝑦 = 𝑋)) | |
28 | csbeq1a 3797 | . . . . . 6 ⊢ (𝑛 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑛⦌𝐴) | |
29 | 27, 28 | ifbieq1d 4374 | . . . . 5 ⊢ (𝑛 = 𝑦 → if(𝑛 = 𝑋, 𝐴, 0 ) = if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 )) |
30 | 22, 26, 29 | cbvmpt 5028 | . . . 4 ⊢ (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 )) |
31 | 12, 30 | eqtri 2802 | . . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 )) |
32 | iftrue 4357 | . . . 4 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 ) = ⦋𝑦 / 𝑛⦌𝐴) | |
33 | csbeq1 3791 | . . . 4 ⊢ (𝑦 = 𝑋 → ⦋𝑦 / 𝑛⦌𝐴 = ⦋𝑋 / 𝑛⦌𝐴) | |
34 | 32, 33 | eqtrd 2814 | . . 3 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 ) = ⦋𝑋 / 𝑛⦌𝐴) |
35 | rspcsbela 4272 | . . . 4 ⊢ ((𝑋 ∈ 𝐼 ∧ ∀𝑛 ∈ 𝐼 𝐴 ∈ (Base‘𝐺)) → ⦋𝑋 / 𝑛⦌𝐴 ∈ (Base‘𝐺)) | |
36 | 5, 6, 35 | syl2anc 576 | . . 3 ⊢ (𝜑 → ⦋𝑋 / 𝑛⦌𝐴 ∈ (Base‘𝐺)) |
37 | 31, 34, 5, 36 | fvmptd3 6619 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) = ⦋𝑋 / 𝑛⦌𝐴) |
38 | 21, 37 | eqtrd 2814 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ⦋𝑋 / 𝑛⦌𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ≠ wne 2967 ∀wral 3088 ⦋csb 3788 ∖ cdif 3828 ifcif 4351 {csn 4442 ↦ cmpt 5009 ‘cfv 6190 (class class class)co 6978 supp csupp 7635 Basecbs 16342 0gc0g 16572 Σg cgsu 16573 Mndcmnd 17765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-iin 4796 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-supp 7636 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-fsupp 8631 df-oi 8771 df-card 9164 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-2 11506 df-n0 11711 df-z 11797 df-uz 12062 df-fz 12712 df-fzo 12853 df-seq 13188 df-hash 13509 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-0g 16574 df-gsum 16575 df-mre 16718 df-mrc 16719 df-acs 16721 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-submnd 17807 df-mulg 18015 df-cntz 18221 df-cmn 18671 |
This theorem is referenced by: gsummptif1n0 18842 gsummoncoe1 20178 scmatscm 20829 idpm2idmp 21116 mp2pm2mplem4 21124 monmat2matmon 21139 |
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