Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gsummptres | Structured version Visualization version GIF version |
Description: Extend a finite group sum by padding outside with zeroes. Proof generated using OpenAI's proof assistant. (Contributed by Thierry Arnoux, 11-Jul-2020.) |
Ref | Expression |
---|---|
gsummptres.0 | ⊢ 𝐵 = (Base‘𝐺) |
gsummptres.1 | ⊢ 0 = (0g‘𝐺) |
gsummptres.2 | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptres.3 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
gsummptres.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
gsummptres.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐷)) → 𝐶 = 0 ) |
Ref | Expression |
---|---|
gsummptres | ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptres.0 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsummptres.1 | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2737 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | gsummptres.2 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
5 | gsummptres.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
6 | gsummptres.4 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) | |
7 | eqid 2737 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
8 | 2 | fvexi 6731 | . . . . 5 ⊢ 0 ∈ V |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
10 | 7, 5, 6, 9 | fsuppmptdm 8996 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) finSupp 0 ) |
11 | inindif 30582 | . . . 4 ⊢ ((𝐴 ∩ 𝐷) ∩ (𝐴 ∖ 𝐷)) = ∅ | |
12 | 11 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐷) ∩ (𝐴 ∖ 𝐷)) = ∅) |
13 | inundif 4393 | . . . . 5 ⊢ ((𝐴 ∩ 𝐷) ∪ (𝐴 ∖ 𝐷)) = 𝐴 | |
14 | 13 | eqcomi 2746 | . . . 4 ⊢ 𝐴 = ((𝐴 ∩ 𝐷) ∪ (𝐴 ∖ 𝐷)) |
15 | 14 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐴 = ((𝐴 ∩ 𝐷) ∪ (𝐴 ∖ 𝐷))) |
16 | 1, 2, 3, 4, 5, 6, 10, 12, 15 | gsumsplit2 19314 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = ((𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))(+g‘𝐺)(𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶)))) |
17 | gsummptres.5 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐷)) → 𝐶 = 0 ) | |
18 | 17 | mpteq2dva 5150 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶) = (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 )) |
19 | 18 | oveq2d 7229 | . . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 ))) |
20 | cmnmnd 19186 | . . . . . . 7 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
21 | 4, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
22 | diffi 8906 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ 𝐷) ∈ Fin) | |
23 | 5, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ∈ Fin) |
24 | 2 | gsumz 18262 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∖ 𝐷) ∈ Fin) → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 )) = 0 ) |
25 | 21, 23, 24 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 )) = 0 ) |
26 | 19, 25 | eqtrd 2777 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶)) = 0 ) |
27 | 26 | oveq2d 7229 | . . 3 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))(+g‘𝐺)(𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶))) = ((𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))(+g‘𝐺) 0 )) |
28 | infi 8899 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ∩ 𝐷) ∈ Fin) | |
29 | 5, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐷) ∈ Fin) |
30 | inss1 4143 | . . . . . . . 8 ⊢ (𝐴 ∩ 𝐷) ⊆ 𝐴 | |
31 | 30 | sseli 3896 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐷) → 𝑥 ∈ 𝐴) |
32 | 31, 6 | sylan2 596 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐷)) → 𝐶 ∈ 𝐵) |
33 | 32 | ralrimiva 3105 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐷)𝐶 ∈ 𝐵) |
34 | 1, 4, 29, 33 | gsummptcl 19352 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶)) ∈ 𝐵) |
35 | 1, 3, 2 | mndrid 18194 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶)) ∈ 𝐵) → ((𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))(+g‘𝐺) 0 ) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))) |
36 | 21, 34, 35 | syl2anc 587 | . . 3 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))(+g‘𝐺) 0 ) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))) |
37 | 27, 36 | eqtrd 2777 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))(+g‘𝐺)(𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶))) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))) |
38 | 16, 37 | eqtrd 2777 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ∖ cdif 3863 ∪ cun 3864 ∩ cin 3865 ∅c0 4237 ↦ cmpt 5135 ‘cfv 6380 (class class class)co 7213 Fincfn 8626 Basecbs 16760 +gcplusg 16802 0gc0g 16944 Σg cgsu 16945 Mndcmnd 18173 CMndccmn 19170 |
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 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-fzo 13239 df-seq 13575 df-hash 13897 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-0g 16946 df-gsum 16947 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-cntz 18711 df-cmn 19172 |
This theorem is referenced by: (None) |
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