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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 19323. (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 2734 | . . 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 3123 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → 𝐴 ∈ (Base‘𝐺)) |
8 | 1, 2 | mndidcl 18160 | . . . . . . 7 ⊢ (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺)) |
9 | 3, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
10 | 9 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → 0 ∈ (Base‘𝐺)) |
11 | 7, 10 | ifcld 4475 | . . . 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 6920 | . . 3 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺)) |
14 | 12 | oveq1i 7212 | . . . 4 ⊢ (𝐹 supp 0 ) = ((𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) supp 0 ) |
15 | eldifsni 4693 | . . . . . . 7 ⊢ (𝑛 ∈ (𝐼 ∖ {𝑋}) → 𝑛 ≠ 𝑋) | |
16 | 15 | adantl 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐼 ∖ {𝑋})) → 𝑛 ≠ 𝑋) |
17 | ifnefalse 4441 | . . . . . 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 7931 | . . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) supp 0 ) ⊆ {𝑋}) |
20 | 14, 19 | eqsstrid 3939 | . . 3 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ {𝑋}) |
21 | 1, 2, 3, 4, 5, 13, 20 | gsumpt 19319 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐹‘𝑋)) |
22 | nfcv 2900 | . . . . 5 ⊢ Ⅎ𝑦if(𝑛 = 𝑋, 𝐴, 0 ) | |
23 | nfv 1922 | . . . . . 6 ⊢ Ⅎ𝑛 𝑦 = 𝑋 | |
24 | nfcsb1v 3827 | . . . . . 6 ⊢ Ⅎ𝑛⦋𝑦 / 𝑛⦌𝐴 | |
25 | nfcv 2900 | . . . . . 6 ⊢ Ⅎ𝑛 0 | |
26 | 23, 24, 25 | nfif 4459 | . . . . 5 ⊢ Ⅎ𝑛if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 ) |
27 | eqeq1 2738 | . . . . . 6 ⊢ (𝑛 = 𝑦 → (𝑛 = 𝑋 ↔ 𝑦 = 𝑋)) | |
28 | csbeq1a 3816 | . . . . . 6 ⊢ (𝑛 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑛⦌𝐴) | |
29 | 27, 28 | ifbieq1d 4453 | . . . . 5 ⊢ (𝑛 = 𝑦 → if(𝑛 = 𝑋, 𝐴, 0 ) = if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 )) |
30 | 22, 26, 29 | cbvmpt 5145 | . . . 4 ⊢ (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 )) |
31 | 12, 30 | eqtri 2762 | . . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 )) |
32 | iftrue 4435 | . . . 4 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 ) = ⦋𝑦 / 𝑛⦌𝐴) | |
33 | csbeq1 3805 | . . . 4 ⊢ (𝑦 = 𝑋 → ⦋𝑦 / 𝑛⦌𝐴 = ⦋𝑋 / 𝑛⦌𝐴) | |
34 | 32, 33 | eqtrd 2774 | . . 3 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 ) = ⦋𝑋 / 𝑛⦌𝐴) |
35 | rspcsbela 4340 | . . . 4 ⊢ ((𝑋 ∈ 𝐼 ∧ ∀𝑛 ∈ 𝐼 𝐴 ∈ (Base‘𝐺)) → ⦋𝑋 / 𝑛⦌𝐴 ∈ (Base‘𝐺)) | |
36 | 5, 6, 35 | syl2anc 587 | . . 3 ⊢ (𝜑 → ⦋𝑋 / 𝑛⦌𝐴 ∈ (Base‘𝐺)) |
37 | 31, 34, 5, 36 | fvmptd3 6830 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) = ⦋𝑋 / 𝑛⦌𝐴) |
38 | 21, 37 | eqtrd 2774 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ⦋𝑋 / 𝑛⦌𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∀wral 3054 ⦋csb 3802 ∖ cdif 3854 ifcif 4429 {csn 4531 ↦ cmpt 5124 ‘cfv 6369 (class class class)co 7202 supp csupp 7892 Basecbs 16684 0gc0g 16916 Σg cgsu 16917 Mndcmnd 18145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-fzo 13222 df-seq 13558 df-hash 13880 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-0g 16918 df-gsum 16919 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-mulg 18461 df-cntz 18683 df-cmn 19144 |
This theorem is referenced by: gsummptif1n0 19323 gsummoncoe1 21197 scmatscm 21382 idpm2idmp 21670 mp2pm2mplem4 21678 monmat2matmon 21693 |
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