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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 19886. (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 2726 | . . 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 3242 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → 𝐴 ∈ (Base‘𝐺)) |
8 | 1, 2 | mndidcl 18682 | . . . . . . 7 ⊢ (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺)) |
9 | 3, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
10 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → 0 ∈ (Base‘𝐺)) |
11 | 7, 10 | ifcld 4569 | . . . 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 7109 | . . 3 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺)) |
14 | 12 | oveq1i 7415 | . . . 4 ⊢ (𝐹 supp 0 ) = ((𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) supp 0 ) |
15 | eldifsni 4788 | . . . . . . 7 ⊢ (𝑛 ∈ (𝐼 ∖ {𝑋}) → 𝑛 ≠ 𝑋) | |
16 | 15 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐼 ∖ {𝑋})) → 𝑛 ≠ 𝑋) |
17 | ifnefalse 4535 | . . . . . 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 8186 | . . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) supp 0 ) ⊆ {𝑋}) |
20 | 14, 19 | eqsstrid 4025 | . . 3 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ {𝑋}) |
21 | 1, 2, 3, 4, 5, 13, 20 | gsumpt 19882 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐹‘𝑋)) |
22 | nfcv 2897 | . . . . 5 ⊢ Ⅎ𝑦if(𝑛 = 𝑋, 𝐴, 0 ) | |
23 | nfv 1909 | . . . . . 6 ⊢ Ⅎ𝑛 𝑦 = 𝑋 | |
24 | nfcsb1v 3913 | . . . . . 6 ⊢ Ⅎ𝑛⦋𝑦 / 𝑛⦌𝐴 | |
25 | nfcv 2897 | . . . . . 6 ⊢ Ⅎ𝑛 0 | |
26 | 23, 24, 25 | nfif 4553 | . . . . 5 ⊢ Ⅎ𝑛if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 ) |
27 | eqeq1 2730 | . . . . . 6 ⊢ (𝑛 = 𝑦 → (𝑛 = 𝑋 ↔ 𝑦 = 𝑋)) | |
28 | csbeq1a 3902 | . . . . . 6 ⊢ (𝑛 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑛⦌𝐴) | |
29 | 27, 28 | ifbieq1d 4547 | . . . . 5 ⊢ (𝑛 = 𝑦 → if(𝑛 = 𝑋, 𝐴, 0 ) = if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 )) |
30 | 22, 26, 29 | cbvmpt 5252 | . . . 4 ⊢ (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 )) |
31 | 12, 30 | eqtri 2754 | . . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 )) |
32 | iftrue 4529 | . . . 4 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 ) = ⦋𝑦 / 𝑛⦌𝐴) | |
33 | csbeq1 3891 | . . . 4 ⊢ (𝑦 = 𝑋 → ⦋𝑦 / 𝑛⦌𝐴 = ⦋𝑋 / 𝑛⦌𝐴) | |
34 | 32, 33 | eqtrd 2766 | . . 3 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 ) = ⦋𝑋 / 𝑛⦌𝐴) |
35 | rspcsbela 4430 | . . . 4 ⊢ ((𝑋 ∈ 𝐼 ∧ ∀𝑛 ∈ 𝐼 𝐴 ∈ (Base‘𝐺)) → ⦋𝑋 / 𝑛⦌𝐴 ∈ (Base‘𝐺)) | |
36 | 5, 6, 35 | syl2anc 583 | . . 3 ⊢ (𝜑 → ⦋𝑋 / 𝑛⦌𝐴 ∈ (Base‘𝐺)) |
37 | 31, 34, 5, 36 | fvmptd3 7015 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) = ⦋𝑋 / 𝑛⦌𝐴) |
38 | 21, 37 | eqtrd 2766 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ⦋𝑋 / 𝑛⦌𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 ⦋csb 3888 ∖ cdif 3940 ifcif 4523 {csn 4623 ↦ cmpt 5224 ‘cfv 6537 (class class class)co 7405 supp csupp 8146 Basecbs 17153 0gc0g 17394 Σg cgsu 17395 Mndcmnd 18667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-0g 17396 df-gsum 17397 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 |
This theorem is referenced by: gsummptif1n0 19886 gsummoncoe1 22182 scmatscm 22370 idpm2idmp 22658 mp2pm2mplem4 22666 monmat2matmon 22681 |
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