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 2734 | . . 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 7029 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑌) ∈ V) |
8 | funmpt 6407 | . . . . 5 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝑌) | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → Fun (𝑥 ∈ 𝐴 ↦ 𝑌)) |
10 | 2 | fvexi 6720 | . . . . 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 7931 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝑌) supp 0 ) ⊆ 𝑆) |
15 | suppssfifsupp 8989 | . . . 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 1380 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑌) finSupp 0 ) |
17 | disjdif 4376 | . . . 4 ⊢ (𝑆 ∩ (𝐴 ∖ 𝑆)) = ∅ | |
18 | 17 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑆 ∩ (𝐴 ∖ 𝑆)) = ∅) |
19 | gsummptres2.2 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐴) | |
20 | undif 4386 | . . . . 5 ⊢ (𝑆 ⊆ 𝐴 ↔ (𝑆 ∪ (𝐴 ∖ 𝑆)) = 𝐴) | |
21 | 19, 20 | sylib 221 | . . . 4 ⊢ (𝜑 → (𝑆 ∪ (𝐴 ∖ 𝑆)) = 𝐴) |
22 | 21 | eqcomd 2740 | . . 3 ⊢ (𝜑 → 𝐴 = (𝑆 ∪ (𝐴 ∖ 𝑆))) |
23 | 1, 2, 3, 4, 5, 6, 16, 18, 22 | gsumsplit2 19286 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝑌)) = ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺)(𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌)))) |
24 | 13 | mpteq2dva 5139 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌) = (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 )) |
25 | 24 | oveq2d 7218 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 ))) |
26 | 4 | cmnmndd 19165 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
27 | 5 | difexd 5211 | . . . . 5 ⊢ (𝜑 → (𝐴 ∖ 𝑆) ∈ V) |
28 | 2 | gsumz 18234 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∖ 𝑆) ∈ V) → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 )) = 0 ) |
29 | 26, 27, 28 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 )) = 0 ) |
30 | 25, 29 | eqtrd 2774 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌)) = 0 ) |
31 | 30 | oveq2d 7218 | . 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 3098 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑌 ∈ 𝐵) |
33 | ssralv 3957 | . . . . 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 19324 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌)) ∈ 𝐵) |
36 | 1, 3, 2 | mndrid 18166 | . . 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 587 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺) 0 ) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) |
38 | 23, 31, 37 | 3eqtrd 2778 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝑌)) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3054 Vcvv 3401 ∖ cdif 3854 ∪ cun 3855 ∩ cin 3856 ⊆ wss 3857 ∅c0 4227 class class class wbr 5043 ↦ cmpt 5124 Fun wfun 6363 ‘cfv 6369 (class class class)co 7202 supp csupp 7892 Fincfn 8615 finSupp cfsupp 8974 Basecbs 16684 +gcplusg 16767 0gc0g 16916 Σg cgsu 16917 Mndcmnd 18145 CMndccmn 19142 |
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 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
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 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-fzo 13222 df-seq 13558 df-hash 13880 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-0g 16918 df-gsum 16919 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-cntz 18683 df-cmn 19144 |
This theorem is referenced by: elrspunidl 31292 |
Copyright terms: Public domain | W3C validator |