![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > gsummptfzcl | Structured version Visualization version GIF version |
Description: Closure of a finite group sum over a finite set of sequential integers as map. (Contributed by AV, 14-Dec-2018.) |
Ref | Expression |
---|---|
gsummptfzcl.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptfzcl.g | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
gsummptfzcl.n | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
gsummptfzcl.i | ⊢ (𝜑 → 𝐼 = (𝑀...𝑁)) |
gsummptfzcl.e | ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
gsummptfzcl | ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ 𝐼 ↦ 𝑋)) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptfzcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2733 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | gsummptfzcl.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
4 | gsummptfzcl.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
5 | gsummptfzcl.e | . . . 4 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 𝑋 ∈ 𝐵) | |
6 | eqid 2733 | . . . . . 6 ⊢ (𝑖 ∈ 𝐼 ↦ 𝑋) = (𝑖 ∈ 𝐼 ↦ 𝑋) | |
7 | 6 | fmpt 7062 | . . . . 5 ⊢ (∀𝑖 ∈ 𝐼 𝑋 ∈ 𝐵 ↔ (𝑖 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵) |
8 | gsummptfzcl.i | . . . . . 6 ⊢ (𝜑 → 𝐼 = (𝑀...𝑁)) | |
9 | 8 | feq2d 6658 | . . . . 5 ⊢ (𝜑 → ((𝑖 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵 ↔ (𝑖 ∈ 𝐼 ↦ 𝑋):(𝑀...𝑁)⟶𝐵)) |
10 | 7, 9 | bitrid 283 | . . . 4 ⊢ (𝜑 → (∀𝑖 ∈ 𝐼 𝑋 ∈ 𝐵 ↔ (𝑖 ∈ 𝐼 ↦ 𝑋):(𝑀...𝑁)⟶𝐵)) |
11 | 5, 10 | mpbid 231 | . . 3 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ 𝑋):(𝑀...𝑁)⟶𝐵) |
12 | 1, 2, 3, 4, 11 | gsumval2 18549 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ 𝐼 ↦ 𝑋)) = (seq𝑀((+g‘𝐺), (𝑖 ∈ 𝐼 ↦ 𝑋))‘𝑁)) |
13 | 5 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ∀𝑖 ∈ 𝐼 𝑋 ∈ 𝐵) |
14 | 13, 7 | sylib 217 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑖 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵) |
15 | 8 | eqcomd 2739 | . . . . . 6 ⊢ (𝜑 → (𝑀...𝑁) = 𝐼) |
16 | 15 | eleq2d 2820 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑥 ∈ 𝐼)) |
17 | 16 | biimpa 478 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ 𝐼) |
18 | 14, 17 | ffvelcdmd 7040 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑖 ∈ 𝐼 ↦ 𝑋)‘𝑥) ∈ 𝐵) |
19 | 3 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Mnd) |
20 | simprl 770 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵) | |
21 | simprr 772 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵) | |
22 | 1, 2 | mndcl 18572 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
23 | 19, 20, 21, 22 | syl3anc 1372 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
24 | 4, 18, 23 | seqcl 13937 | . 2 ⊢ (𝜑 → (seq𝑀((+g‘𝐺), (𝑖 ∈ 𝐼 ↦ 𝑋))‘𝑁) ∈ 𝐵) |
25 | 12, 24 | eqeltrd 2834 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ 𝐼 ↦ 𝑋)) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ↦ cmpt 5192 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 ℤ≥cuz 12771 ...cfz 13433 seqcseq 13915 Basecbs 17091 +gcplusg 17141 Σg cgsu 17330 Mndcmnd 18564 |
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-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-seq 13916 df-0g 17331 df-gsum 17332 df-mgm 18505 df-sgrp 18554 df-mnd 18565 |
This theorem is referenced by: m2detleiblem2 22000 |
Copyright terms: Public domain | W3C validator |