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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 2726 | . . 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 2726 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
8 | 2 | fvexi 6915 | . . . . 5 ⊢ 0 ∈ V |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
10 | 7, 5, 6, 9 | fsuppmptdm 9419 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) finSupp 0 ) |
11 | inindif 32443 | . . . 4 ⊢ ((𝐴 ∩ 𝐷) ∩ (𝐴 ∖ 𝐷)) = ∅ | |
12 | 11 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐷) ∩ (𝐴 ∖ 𝐷)) = ∅) |
13 | inundif 4483 | . . . . 5 ⊢ ((𝐴 ∩ 𝐷) ∪ (𝐴 ∖ 𝐷)) = 𝐴 | |
14 | 13 | eqcomi 2735 | . . . 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 19927 | . 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 5253 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶) = (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 )) |
19 | 18 | oveq2d 7440 | . . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 ))) |
20 | cmnmnd 19795 | . . . . . . 7 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
21 | 4, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
22 | diffi 9213 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ 𝐷) ∈ Fin) | |
23 | 5, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ∈ Fin) |
24 | 2 | gsumz 18826 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∖ 𝐷) ∈ Fin) → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 )) = 0 ) |
25 | 21, 23, 24 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 )) = 0 ) |
26 | 19, 25 | eqtrd 2766 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶)) = 0 ) |
27 | 26 | oveq2d 7440 | . . 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 9302 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ∩ 𝐷) ∈ Fin) | |
29 | 5, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐷) ∈ Fin) |
30 | inss1 4230 | . . . . . . . 8 ⊢ (𝐴 ∩ 𝐷) ⊆ 𝐴 | |
31 | 30 | sseli 3975 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐷) → 𝑥 ∈ 𝐴) |
32 | 31, 6 | sylan2 591 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐷)) → 𝐶 ∈ 𝐵) |
33 | 32 | ralrimiva 3136 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐷)𝐶 ∈ 𝐵) |
34 | 1, 4, 29, 33 | gsummptcl 19965 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶)) ∈ 𝐵) |
35 | 1, 3, 2 | mndrid 18748 | . . . 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 582 | . . 3 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))(+g‘𝐺) 0 ) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))) |
37 | 27, 36 | eqtrd 2766 | . 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 2766 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∖ cdif 3944 ∪ cun 3945 ∩ cin 3946 ∅c0 4325 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 Fincfn 8974 Basecbs 17213 +gcplusg 17266 0gc0g 17454 Σg cgsu 17455 Mndcmnd 18727 CMndccmn 19778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-fzo 13682 df-seq 14022 df-hash 14348 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-0g 17456 df-gsum 17457 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-cntz 19311 df-cmn 19780 |
This theorem is referenced by: (None) |
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