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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gsummptres2 | Structured version Visualization version GIF version |
Description: Extend a finite group sum by padding outside with zeroes. (Contributed by Thierry Arnoux, 22-Jun-2024.) |
Ref | Expression |
---|---|
gsummptres2.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptres2.z | ⊢ 0 = (0g‘𝐺) |
gsummptres2.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptres2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsummptres2.0 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝑆)) → 𝑌 = 0 ) |
gsummptres2.1 | ⊢ (𝜑 → 𝑆 ∈ Fin) |
gsummptres2.y | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐵) |
gsummptres2.2 | ⊢ (𝜑 → 𝑆 ⊆ 𝐴) |
Ref | Expression |
---|---|
gsummptres2 | ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝑌)) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptres2.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsummptres2.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2759 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | gsummptres2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
5 | gsummptres2.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | gsummptres2.y | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐵) | |
7 | 5 | mptexd 6971 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑌) ∈ V) |
8 | funmpt 6366 | . . . . 5 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝑌) | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → Fun (𝑥 ∈ 𝐴 ↦ 𝑌)) |
10 | 2 | fvexi 6665 | . . . . 5 ⊢ 0 ∈ V |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
12 | gsummptres2.1 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Fin) | |
13 | gsummptres2.0 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝑆)) → 𝑌 = 0 ) | |
14 | 13, 5 | suppss2 7867 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝑌) supp 0 ) ⊆ 𝑆) |
15 | suppssfifsupp 8866 | . . . 4 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝑌) ∈ V ∧ Fun (𝑥 ∈ 𝐴 ↦ 𝑌) ∧ 0 ∈ V) ∧ (𝑆 ∈ Fin ∧ ((𝑥 ∈ 𝐴 ↦ 𝑌) supp 0 ) ⊆ 𝑆)) → (𝑥 ∈ 𝐴 ↦ 𝑌) finSupp 0 ) | |
16 | 7, 9, 11, 12, 14, 15 | syl32anc 1376 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑌) finSupp 0 ) |
17 | disjdif 4361 | . . . 4 ⊢ (𝑆 ∩ (𝐴 ∖ 𝑆)) = ∅ | |
18 | 17 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑆 ∩ (𝐴 ∖ 𝑆)) = ∅) |
19 | gsummptres2.2 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐴) | |
20 | undif 4371 | . . . . 5 ⊢ (𝑆 ⊆ 𝐴 ↔ (𝑆 ∪ (𝐴 ∖ 𝑆)) = 𝐴) | |
21 | 19, 20 | sylib 221 | . . . 4 ⊢ (𝜑 → (𝑆 ∪ (𝐴 ∖ 𝑆)) = 𝐴) |
22 | 21 | eqcomd 2765 | . . 3 ⊢ (𝜑 → 𝐴 = (𝑆 ∪ (𝐴 ∖ 𝑆))) |
23 | 1, 2, 3, 4, 5, 6, 16, 18, 22 | gsumsplit2 19102 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝑌)) = ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺)(𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌)))) |
24 | 13 | mpteq2dva 5120 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌) = (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 )) |
25 | 24 | oveq2d 7159 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 ))) |
26 | 4 | cmnmndd 18981 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
27 | difexg 5190 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝑆) ∈ V) | |
28 | 5, 27 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 ∖ 𝑆) ∈ V) |
29 | 2 | gsumz 18051 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∖ 𝑆) ∈ V) → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 )) = 0 ) |
30 | 26, 28, 29 | syl2anc 588 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 )) = 0 ) |
31 | 25, 30 | eqtrd 2794 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌)) = 0 ) |
32 | 31 | oveq2d 7159 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺)(𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌))) = ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺) 0 )) |
33 | 6 | ralrimiva 3111 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑌 ∈ 𝐵) |
34 | ssralv 3954 | . . . . 5 ⊢ (𝑆 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 𝑌 ∈ 𝐵 → ∀𝑥 ∈ 𝑆 𝑌 ∈ 𝐵)) | |
35 | 19, 33, 34 | sylc 65 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝑌 ∈ 𝐵) |
36 | 1, 4, 12, 35 | gsummptcl 19140 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌)) ∈ 𝐵) |
37 | 1, 3, 2 | mndrid 17983 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌)) ∈ 𝐵) → ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺) 0 ) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) |
38 | 26, 36, 37 | syl2anc 588 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺) 0 ) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) |
39 | 23, 32, 38 | 3eqtrd 2798 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝑌)) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 ∀wral 3068 Vcvv 3407 ∖ cdif 3851 ∪ cun 3852 ∩ cin 3853 ⊆ wss 3854 ∅c0 4221 class class class wbr 5025 ↦ cmpt 5105 Fun wfun 6322 ‘cfv 6328 (class class class)co 7143 supp csupp 7828 Fincfn 8520 finSupp cfsupp 8851 Basecbs 16526 +gcplusg 16608 0gc0g 16756 Σg cgsu 16757 Mndcmnd 17962 CMndccmn 18958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-iin 4879 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-se 5477 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-isom 6337 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-of 7398 df-om 7573 df-1st 7686 df-2nd 7687 df-supp 7829 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-fsupp 8852 df-oi 8992 df-card 9386 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-n0 11920 df-z 12006 df-uz 12268 df-fz 12925 df-fzo 13068 df-seq 13404 df-hash 13726 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-ress 16534 df-plusg 16621 df-0g 16758 df-gsum 16759 df-mre 16900 df-mrc 16901 df-acs 16903 df-mgm 17903 df-sgrp 17952 df-mnd 17963 df-submnd 18008 df-cntz 18499 df-cmn 18960 |
This theorem is referenced by: elrspunidl 31112 |
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