![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 2733 | . . 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 7226 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑌) ∈ V) |
8 | funmpt 6587 | . . . . 5 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝑌) | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → Fun (𝑥 ∈ 𝐴 ↦ 𝑌)) |
10 | 2 | fvexi 6906 | . . . . 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 8185 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝑌) supp 0 ) ⊆ 𝑆) |
15 | suppssfifsupp 9378 | . . . 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 4472 | . . . 4 ⊢ (𝑆 ∩ (𝐴 ∖ 𝑆)) = ∅ | |
18 | 17 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑆 ∩ (𝐴 ∖ 𝑆)) = ∅) |
19 | gsummptres2.2 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐴) | |
20 | undif 4482 | . . . . 5 ⊢ (𝑆 ⊆ 𝐴 ↔ (𝑆 ∪ (𝐴 ∖ 𝑆)) = 𝐴) | |
21 | 19, 20 | sylib 217 | . . . 4 ⊢ (𝜑 → (𝑆 ∪ (𝐴 ∖ 𝑆)) = 𝐴) |
22 | 21 | eqcomd 2739 | . . 3 ⊢ (𝜑 → 𝐴 = (𝑆 ∪ (𝐴 ∖ 𝑆))) |
23 | 1, 2, 3, 4, 5, 6, 16, 18, 22 | gsumsplit2 19797 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝑌)) = ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺)(𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌)))) |
24 | 13 | mpteq2dva 5249 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌) = (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 )) |
25 | 24 | oveq2d 7425 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 ))) |
26 | 4 | cmnmndd 19672 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
27 | 5 | difexd 5330 | . . . . 5 ⊢ (𝜑 → (𝐴 ∖ 𝑆) ∈ V) |
28 | 2 | gsumz 18717 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∖ 𝑆) ∈ V) → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 )) = 0 ) |
29 | 26, 27, 28 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 )) = 0 ) |
30 | 25, 29 | eqtrd 2773 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌)) = 0 ) |
31 | 30 | oveq2d 7425 | . 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 3147 | . . . . 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 19835 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌)) ∈ 𝐵) |
36 | 1, 3, 2 | mndrid 18646 | . . 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 585 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺) 0 ) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) |
38 | 23, 31, 37 | 3eqtrd 2777 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝑌)) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ∖ cdif 3946 ∪ cun 3947 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 class class class wbr 5149 ↦ cmpt 5232 Fun wfun 6538 ‘cfv 6544 (class class class)co 7409 supp csupp 8146 Fincfn 8939 finSupp cfsupp 9361 Basecbs 17144 +gcplusg 17197 0gc0g 17385 Σg cgsu 17386 Mndcmnd 18625 CMndccmn 19648 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-seq 13967 df-hash 14291 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-0g 17387 df-gsum 17388 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-cntz 19181 df-cmn 19650 |
This theorem is referenced by: elrspunidl 32546 gsummoncoe1fzo 32668 |
Copyright terms: Public domain | W3C validator |