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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 7107 | . . . . 5 ⊢ (∀𝑖 ∈ 𝐼 𝑋 ∈ 𝐵 ↔ (𝑖 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵) |
8 | gsummptfzcl.i | . . . . . 6 ⊢ (𝜑 → 𝐼 = (𝑀...𝑁)) | |
9 | 8 | feq2d 6701 | . . . . 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 18602 | . 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 7085 | . . 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 18630 | . . . 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 13985 | . 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 3062 ↦ cmpt 5231 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ℤ≥cuz 12819 ...cfz 13481 seqcseq 13963 Basecbs 17141 +gcplusg 17194 Σg cgsu 17383 Mndcmnd 18622 |
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 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-seq 13964 df-0g 17384 df-gsum 17385 df-mgm 18558 df-sgrp 18607 df-mnd 18623 |
This theorem is referenced by: m2detleiblem2 22122 |
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