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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 19751. (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 2733 | . . 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 3233 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → 𝐴 ∈ (Base‘𝐺)) |
8 | 1, 2 | mndidcl 18579 | . . . . . . 7 ⊢ (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺)) |
9 | 3, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
10 | 9 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → 0 ∈ (Base‘𝐺)) |
11 | 7, 10 | ifcld 4536 | . . . 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 7066 | . . 3 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺)) |
14 | 12 | oveq1i 7371 | . . . 4 ⊢ (𝐹 supp 0 ) = ((𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) supp 0 ) |
15 | eldifsni 4754 | . . . . . . 7 ⊢ (𝑛 ∈ (𝐼 ∖ {𝑋}) → 𝑛 ≠ 𝑋) | |
16 | 15 | adantl 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐼 ∖ {𝑋})) → 𝑛 ≠ 𝑋) |
17 | ifnefalse 4502 | . . . . . 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 8135 | . . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) supp 0 ) ⊆ {𝑋}) |
20 | 14, 19 | eqsstrid 3996 | . . 3 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ {𝑋}) |
21 | 1, 2, 3, 4, 5, 13, 20 | gsumpt 19747 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐹‘𝑋)) |
22 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑦if(𝑛 = 𝑋, 𝐴, 0 ) | |
23 | nfv 1918 | . . . . . 6 ⊢ Ⅎ𝑛 𝑦 = 𝑋 | |
24 | nfcsb1v 3884 | . . . . . 6 ⊢ Ⅎ𝑛⦋𝑦 / 𝑛⦌𝐴 | |
25 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑛 0 | |
26 | 23, 24, 25 | nfif 4520 | . . . . 5 ⊢ Ⅎ𝑛if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 ) |
27 | eqeq1 2737 | . . . . . 6 ⊢ (𝑛 = 𝑦 → (𝑛 = 𝑋 ↔ 𝑦 = 𝑋)) | |
28 | csbeq1a 3873 | . . . . . 6 ⊢ (𝑛 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑛⦌𝐴) | |
29 | 27, 28 | ifbieq1d 4514 | . . . . 5 ⊢ (𝑛 = 𝑦 → if(𝑛 = 𝑋, 𝐴, 0 ) = if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 )) |
30 | 22, 26, 29 | cbvmpt 5220 | . . . 4 ⊢ (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 )) |
31 | 12, 30 | eqtri 2761 | . . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 )) |
32 | iftrue 4496 | . . . 4 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 ) = ⦋𝑦 / 𝑛⦌𝐴) | |
33 | csbeq1 3862 | . . . 4 ⊢ (𝑦 = 𝑋 → ⦋𝑦 / 𝑛⦌𝐴 = ⦋𝑋 / 𝑛⦌𝐴) | |
34 | 32, 33 | eqtrd 2773 | . . 3 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 ) = ⦋𝑋 / 𝑛⦌𝐴) |
35 | rspcsbela 4399 | . . . 4 ⊢ ((𝑋 ∈ 𝐼 ∧ ∀𝑛 ∈ 𝐼 𝐴 ∈ (Base‘𝐺)) → ⦋𝑋 / 𝑛⦌𝐴 ∈ (Base‘𝐺)) | |
36 | 5, 6, 35 | syl2anc 585 | . . 3 ⊢ (𝜑 → ⦋𝑋 / 𝑛⦌𝐴 ∈ (Base‘𝐺)) |
37 | 31, 34, 5, 36 | fvmptd3 6975 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) = ⦋𝑋 / 𝑛⦌𝐴) |
38 | 21, 37 | eqtrd 2773 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ⦋𝑋 / 𝑛⦌𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∀wral 3061 ⦋csb 3859 ∖ cdif 3911 ifcif 4490 {csn 4590 ↦ cmpt 5192 ‘cfv 6500 (class class class)co 7361 supp csupp 8096 Basecbs 17091 0gc0g 17329 Σg cgsu 17330 Mndcmnd 18564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-seq 13916 df-hash 14240 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-0g 17331 df-gsum 17332 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-mulg 18881 df-cntz 19105 df-cmn 19572 |
This theorem is referenced by: gsummptif1n0 19751 gsummoncoe1 21698 scmatscm 21885 idpm2idmp 22173 mp2pm2mplem4 22181 monmat2matmon 22196 |
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