Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 2610 | . . 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 2610 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
8 | fvex 6113 | . . . . . 6 ⊢ (0g‘𝐺) ∈ V | |
9 | 2, 8 | eqeltri 2684 | . . . . 5 ⊢ 0 ∈ V |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
11 | 7, 5, 6, 10 | fsuppmptdm 8169 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) finSupp 0 ) |
12 | inindif 28738 | . . . 4 ⊢ ((𝐴 ∩ 𝐷) ∩ (𝐴 ∖ 𝐷)) = ∅ | |
13 | 12 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐷) ∩ (𝐴 ∖ 𝐷)) = ∅) |
14 | inundif 3998 | . . . . 5 ⊢ ((𝐴 ∩ 𝐷) ∪ (𝐴 ∖ 𝐷)) = 𝐴 | |
15 | 14 | eqcomi 2619 | . . . 4 ⊢ 𝐴 = ((𝐴 ∩ 𝐷) ∪ (𝐴 ∖ 𝐷)) |
16 | 15 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐴 = ((𝐴 ∩ 𝐷) ∪ (𝐴 ∖ 𝐷))) |
17 | 1, 2, 3, 4, 5, 6, 11, 13, 16 | gsumsplit2 18152 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = ((𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))(+g‘𝐺)(𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶)))) |
18 | gsummptres.5 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐷)) → 𝐶 = 0 ) | |
19 | 18 | mpteq2dva 4672 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶) = (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 )) |
20 | 19 | oveq2d 6565 | . . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 ))) |
21 | cmnmnd 18031 | . . . . . . 7 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
22 | 4, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
23 | diffi 8077 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ 𝐷) ∈ Fin) | |
24 | 5, 23 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ∈ Fin) |
25 | 2 | gsumz 17197 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∖ 𝐷) ∈ Fin) → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 )) = 0 ) |
26 | 22, 24, 25 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 0 )) = 0 ) |
27 | 20, 26 | eqtrd 2644 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶)) = 0 ) |
28 | 27 | oveq2d 6565 | . . 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 )) |
29 | infi 8069 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ∩ 𝐷) ∈ Fin) | |
30 | 5, 29 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐷) ∈ Fin) |
31 | inss1 3795 | . . . . . . . 8 ⊢ (𝐴 ∩ 𝐷) ⊆ 𝐴 | |
32 | 31 | sseli 3564 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐷) → 𝑥 ∈ 𝐴) |
33 | 32, 6 | sylan2 490 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐷)) → 𝐶 ∈ 𝐵) |
34 | 33 | ralrimiva 2949 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐷)𝐶 ∈ 𝐵) |
35 | 1, 4, 30, 34 | gsummptcl 18189 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶)) ∈ 𝐵) |
36 | 1, 3, 2 | mndrid 17135 | . . . 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))) |
37 | 22, 35, 36 | syl2anc 691 | . . 3 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))(+g‘𝐺) 0 ) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))) |
38 | 28, 37 | eqtrd 2644 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))(+g‘𝐺)(𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶))) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))) |
39 | 17, 38 | eqtrd 2644 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Σg cgsu 15924 Mndcmnd 17117 CMndccmn 18016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-cntz 17573 df-cmn 18018 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |