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 2824 | . . 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 2824 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
8 | 2 | fvexi 6687 | . . . . 5 ⊢ 0 ∈ V |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
10 | 7, 5, 6, 9 | fsuppmptdm 8847 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) finSupp 0 ) |
11 | inindif 30281 | . . . 4 ⊢ ((𝐴 ∩ 𝐷) ∩ (𝐴 ∖ 𝐷)) = ∅ | |
12 | 11 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐷) ∩ (𝐴 ∖ 𝐷)) = ∅) |
13 | inundif 4430 | . . . . 5 ⊢ ((𝐴 ∩ 𝐷) ∪ (𝐴 ∖ 𝐷)) = 𝐴 | |
14 | 13 | eqcomi 2833 | . . . 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 19052 | . 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 5164 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶) = (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 )) |
19 | 18 | oveq2d 7175 | . . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 ))) |
20 | cmnmnd 18925 | . . . . . . 7 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
21 | 4, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
22 | diffi 8753 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ 𝐷) ∈ Fin) | |
23 | 5, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ∈ Fin) |
24 | 2 | gsumz 18003 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∖ 𝐷) ∈ Fin) → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 )) = 0 ) |
25 | 21, 23, 24 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 )) = 0 ) |
26 | 19, 25 | eqtrd 2859 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶)) = 0 ) |
27 | 26 | oveq2d 7175 | . . 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 8745 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ∩ 𝐷) ∈ Fin) | |
29 | 5, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐷) ∈ Fin) |
30 | inss1 4208 | . . . . . . . 8 ⊢ (𝐴 ∩ 𝐷) ⊆ 𝐴 | |
31 | 30 | sseli 3966 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐷) → 𝑥 ∈ 𝐴) |
32 | 31, 6 | sylan2 594 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐷)) → 𝐶 ∈ 𝐵) |
33 | 32 | ralrimiva 3185 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐷)𝐶 ∈ 𝐵) |
34 | 1, 4, 29, 33 | gsummptcl 19090 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶)) ∈ 𝐵) |
35 | 1, 3, 2 | mndrid 17935 | . . . 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 586 | . . 3 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))(+g‘𝐺) 0 ) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))) |
37 | 27, 36 | eqtrd 2859 | . 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 2859 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∖ cdif 3936 ∪ cun 3937 ∩ cin 3938 ∅c0 4294 ↦ cmpt 5149 ‘cfv 6358 (class class class)co 7159 Fincfn 8512 Basecbs 16486 +gcplusg 16568 0gc0g 16716 Σg cgsu 16717 Mndcmnd 17914 CMndccmn 18909 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-0g 16718 df-gsum 16719 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-cntz 18450 df-cmn 18911 |
This theorem is referenced by: (None) |
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