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 2736 | . . 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 2736 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
8 | 2 | fvexi 6814 | . . . . 5 ⊢ 0 ∈ V |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
10 | 7, 5, 6, 9 | fsuppmptdm 9179 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) finSupp 0 ) |
11 | inindif 30904 | . . . 4 ⊢ ((𝐴 ∩ 𝐷) ∩ (𝐴 ∖ 𝐷)) = ∅ | |
12 | 11 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐷) ∩ (𝐴 ∖ 𝐷)) = ∅) |
13 | inundif 4418 | . . . . 5 ⊢ ((𝐴 ∩ 𝐷) ∪ (𝐴 ∖ 𝐷)) = 𝐴 | |
14 | 13 | eqcomi 2745 | . . . 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 19571 | . 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 5181 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶) = (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 )) |
19 | 18 | oveq2d 7319 | . . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 ))) |
20 | cmnmnd 19443 | . . . . . . 7 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
21 | 4, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
22 | diffi 8996 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ 𝐷) ∈ Fin) | |
23 | 5, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ∈ Fin) |
24 | 2 | gsumz 18515 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∖ 𝐷) ∈ Fin) → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 )) = 0 ) |
25 | 21, 23, 24 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 )) = 0 ) |
26 | 19, 25 | eqtrd 2776 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶)) = 0 ) |
27 | 26 | oveq2d 7319 | . . 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 9083 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ∩ 𝐷) ∈ Fin) | |
29 | 5, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐷) ∈ Fin) |
30 | inss1 4168 | . . . . . . . 8 ⊢ (𝐴 ∩ 𝐷) ⊆ 𝐴 | |
31 | 30 | sseli 3922 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐷) → 𝑥 ∈ 𝐴) |
32 | 31, 6 | sylan2 594 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐷)) → 𝐶 ∈ 𝐵) |
33 | 32 | ralrimiva 3140 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐷)𝐶 ∈ 𝐵) |
34 | 1, 4, 29, 33 | gsummptcl 19609 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶)) ∈ 𝐵) |
35 | 1, 3, 2 | mndrid 18447 | . . . 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 585 | . . 3 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))(+g‘𝐺) 0 ) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))) |
37 | 27, 36 | eqtrd 2776 | . 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 2776 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1539 ∈ wcel 2104 Vcvv 3437 ∖ cdif 3889 ∪ cun 3890 ∩ cin 3891 ∅c0 4262 ↦ cmpt 5164 ‘cfv 6454 (class class class)co 7303 Fincfn 8760 Basecbs 16953 +gcplusg 17003 0gc0g 17191 Σg cgsu 17192 Mndcmnd 18426 CMndccmn 19427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-se 5552 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-isom 6463 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-of 7561 df-om 7741 df-1st 7859 df-2nd 7860 df-supp 8005 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-1o 8324 df-er 8525 df-en 8761 df-dom 8762 df-sdom 8763 df-fin 8764 df-fsupp 9169 df-oi 9309 df-card 9737 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-nn 12016 df-2 12078 df-n0 12276 df-z 12362 df-uz 12625 df-fz 13282 df-fzo 13425 df-seq 13764 df-hash 14087 df-sets 16906 df-slot 16924 df-ndx 16936 df-base 16954 df-ress 16983 df-plusg 17016 df-0g 17193 df-gsum 17194 df-mre 17336 df-mrc 17337 df-acs 17339 df-mgm 18367 df-sgrp 18416 df-mnd 18427 df-submnd 18472 df-cntz 18964 df-cmn 19429 |
This theorem is referenced by: (None) |
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