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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 18188. (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 2610 | . . 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 2916 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → 𝐴 ∈ (Base‘𝐺)) |
8 | 1, 2 | mndidcl 17131 | . . . . . . 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 4081 | . . . 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 6292 | . . 3 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺)) |
14 | 12 | oveq1i 6559 | . . . 4 ⊢ (𝐹 supp 0 ) = ((𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) supp 0 ) |
15 | eldifsni 4261 | . . . . . . 7 ⊢ (𝑛 ∈ (𝐼 ∖ {𝑋}) → 𝑛 ≠ 𝑋) | |
16 | 15 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐼 ∖ {𝑋})) → 𝑛 ≠ 𝑋) |
17 | ifnefalse 4048 | . . . . . 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 7216 | . . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) supp 0 ) ⊆ {𝑋}) |
20 | 14, 19 | syl5eqss 3612 | . . 3 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ {𝑋}) |
21 | 1, 2, 3, 4, 5, 13, 20 | gsumpt 18184 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐹‘𝑋)) |
22 | rspcsbela 3958 | . . . 4 ⊢ ((𝑋 ∈ 𝐼 ∧ ∀𝑛 ∈ 𝐼 𝐴 ∈ (Base‘𝐺)) → ⦋𝑋 / 𝑛⦌𝐴 ∈ (Base‘𝐺)) | |
23 | 5, 6, 22 | syl2anc 691 | . . 3 ⊢ (𝜑 → ⦋𝑋 / 𝑛⦌𝐴 ∈ (Base‘𝐺)) |
24 | iftrue 4042 | . . . . 5 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 ) = ⦋𝑦 / 𝑛⦌𝐴) | |
25 | csbeq1 3502 | . . . . 5 ⊢ (𝑦 = 𝑋 → ⦋𝑦 / 𝑛⦌𝐴 = ⦋𝑋 / 𝑛⦌𝐴) | |
26 | 24, 25 | eqtrd 2644 | . . . 4 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 ) = ⦋𝑋 / 𝑛⦌𝐴) |
27 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑦if(𝑛 = 𝑋, 𝐴, 0 ) | |
28 | nfv 1830 | . . . . . . 7 ⊢ Ⅎ𝑛 𝑦 = 𝑋 | |
29 | nfcsb1v 3515 | . . . . . . 7 ⊢ Ⅎ𝑛⦋𝑦 / 𝑛⦌𝐴 | |
30 | nfcv 2751 | . . . . . . 7 ⊢ Ⅎ𝑛 0 | |
31 | 28, 29, 30 | nfif 4065 | . . . . . 6 ⊢ Ⅎ𝑛if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 ) |
32 | eqeq1 2614 | . . . . . . 7 ⊢ (𝑛 = 𝑦 → (𝑛 = 𝑋 ↔ 𝑦 = 𝑋)) | |
33 | csbeq1a 3508 | . . . . . . 7 ⊢ (𝑛 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑛⦌𝐴) | |
34 | 32, 33 | ifbieq1d 4059 | . . . . . 6 ⊢ (𝑛 = 𝑦 → if(𝑛 = 𝑋, 𝐴, 0 ) = if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 )) |
35 | 27, 31, 34 | cbvmpt 4677 | . . . . 5 ⊢ (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 )) |
36 | 12, 35 | eqtri 2632 | . . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, ⦋𝑦 / 𝑛⦌𝐴, 0 )) |
37 | 26, 36 | fvmptg 6189 | . . 3 ⊢ ((𝑋 ∈ 𝐼 ∧ ⦋𝑋 / 𝑛⦌𝐴 ∈ (Base‘𝐺)) → (𝐹‘𝑋) = ⦋𝑋 / 𝑛⦌𝐴) |
38 | 5, 23, 37 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) = ⦋𝑋 / 𝑛⦌𝐴) |
39 | 21, 38 | eqtrd 2644 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ⦋𝑋 / 𝑛⦌𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ⦋csb 3499 ∖ cdif 3537 ifcif 4036 {csn 4125 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 supp csupp 7182 Basecbs 15695 0gc0g 15923 Σg cgsu 15924 Mndcmnd 17117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 |
This theorem is referenced by: gsummptif1n0 18188 gsummoncoe1 19495 scmatscm 20138 idpm2idmp 20425 mp2pm2mplem4 20433 monmat2matmon 20448 |
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