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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 2736 | . . 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 7245 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑌) ∈ V) | 
| 8 | funmpt 6603 | . . . . 5 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝑌) | |
| 9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → Fun (𝑥 ∈ 𝐴 ↦ 𝑌)) | 
| 10 | 2 | fvexi 6919 | . . . . 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 8226 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝑌) supp 0 ) ⊆ 𝑆) | 
| 15 | suppssfifsupp 9421 | . . . 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 1379 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑌) finSupp 0 ) | 
| 17 | disjdif 4471 | . . . 4 ⊢ (𝑆 ∩ (𝐴 ∖ 𝑆)) = ∅ | |
| 18 | 17 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑆 ∩ (𝐴 ∖ 𝑆)) = ∅) | 
| 19 | gsummptres2.2 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐴) | |
| 20 | undif 4481 | . . . . 5 ⊢ (𝑆 ⊆ 𝐴 ↔ (𝑆 ∪ (𝐴 ∖ 𝑆)) = 𝐴) | |
| 21 | 19, 20 | sylib 218 | . . . 4 ⊢ (𝜑 → (𝑆 ∪ (𝐴 ∖ 𝑆)) = 𝐴) | 
| 22 | 21 | eqcomd 2742 | . . 3 ⊢ (𝜑 → 𝐴 = (𝑆 ∪ (𝐴 ∖ 𝑆))) | 
| 23 | 1, 2, 3, 4, 5, 6, 16, 18, 22 | gsumsplit2 19948 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝑌)) = ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺)(𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌)))) | 
| 24 | 13 | mpteq2dva 5241 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌) = (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 )) | 
| 25 | 24 | oveq2d 7448 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 ))) | 
| 26 | 4 | cmnmndd 19823 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Mnd) | 
| 27 | 5 | difexd 5330 | . . . . 5 ⊢ (𝜑 → (𝐴 ∖ 𝑆) ∈ V) | 
| 28 | 2 | gsumz 18850 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∖ 𝑆) ∈ V) → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 )) = 0 ) | 
| 29 | 26, 27, 28 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 )) = 0 ) | 
| 30 | 25, 29 | eqtrd 2776 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌)) = 0 ) | 
| 31 | 30 | oveq2d 7448 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺)(𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌))) = ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺) 0 )) | 
| 32 | 6 | ralrimiva 3145 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑌 ∈ 𝐵) | 
| 33 | ssralv 4051 | . . . . 5 ⊢ (𝑆 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 𝑌 ∈ 𝐵 → ∀𝑥 ∈ 𝑆 𝑌 ∈ 𝐵)) | |
| 34 | 19, 32, 33 | sylc 65 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝑌 ∈ 𝐵) | 
| 35 | 1, 4, 12, 34 | gsummptcl 19986 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌)) ∈ 𝐵) | 
| 36 | 1, 3, 2 | mndrid 18769 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌)) ∈ 𝐵) → ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺) 0 ) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) | 
| 37 | 26, 35, 36 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺) 0 ) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) | 
| 38 | 23, 31, 37 | 3eqtrd 2780 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝑌)) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 ∖ cdif 3947 ∪ cun 3948 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 class class class wbr 5142 ↦ cmpt 5224 Fun wfun 6554 ‘cfv 6560 (class class class)co 7432 supp csupp 8186 Fincfn 8986 finSupp cfsupp 9402 Basecbs 17248 +gcplusg 17298 0gc0g 17485 Σg cgsu 17486 Mndcmnd 18748 CMndccmn 19799 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-fzo 13696 df-seq 14044 df-hash 14371 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-0g 17487 df-gsum 17488 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-cntz 19336 df-cmn 19801 | 
| This theorem is referenced by: gsumfs2d 33059 elrgspnlem4 33250 elrgspnsubrunlem1 33252 elrgspnsubrunlem2 33253 elrspunidl 33457 gsummoncoe1fzo 33619 fldextrspunlsp 33725 | 
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