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Mirrors > Home > MPE Home > Th. List > gsummptif1n0 | 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. (Contributed by AV, 17-Feb-2019.) (Proof shortened 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 )) |
gsummptif1n0.a | ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐺)) |
Ref | Expression |
---|---|
gsummptif1n0 | ⊢ (𝜑 → (𝐺 Σg 𝐹) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummpt1n0.0 | . . 3 ⊢ 0 = (0g‘𝐺) | |
2 | gsummpt1n0.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
3 | gsummpt1n0.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
4 | gsummpt1n0.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
5 | gsummpt1n0.f | . . 3 ⊢ 𝐹 = (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) | |
6 | gsummptif1n0.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐺)) | |
7 | 6 | ralrimivw 3150 | . . 3 ⊢ (𝜑 → ∀𝑛 ∈ 𝐼 𝐴 ∈ (Base‘𝐺)) |
8 | 1, 2, 3, 4, 5, 7 | gsummpt1n0 19832 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ⦋𝑋 / 𝑛⦌𝐴) |
9 | csbconstg 3912 | . . 3 ⊢ (𝑋 ∈ 𝐼 → ⦋𝑋 / 𝑛⦌𝐴 = 𝐴) | |
10 | 4, 9 | syl 17 | . 2 ⊢ (𝜑 → ⦋𝑋 / 𝑛⦌𝐴 = 𝐴) |
11 | 8, 10 | eqtrd 2772 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⦋csb 3893 ifcif 4528 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7408 Basecbs 17143 0gc0g 17384 Σg cgsu 17385 Mndcmnd 18624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 df-fzo 13627 df-seq 13966 df-hash 14290 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-0g 17386 df-gsum 17387 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-submnd 18671 df-mulg 18950 df-cntz 19180 df-cmn 19649 |
This theorem is referenced by: mhpvarcl 21690 1mavmul 22049 mulmarep1gsum1 22074 mdetdiag 22100 elrspunsn 32542 |
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