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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 2798 | . . 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 2798 | . . . . . 6 ⊢ (𝑖 ∈ 𝐼 ↦ 𝑋) = (𝑖 ∈ 𝐼 ↦ 𝑋) | |
7 | 6 | fmpt 6851 | . . . . 5 ⊢ (∀𝑖 ∈ 𝐼 𝑋 ∈ 𝐵 ↔ (𝑖 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵) |
8 | gsummptfzcl.i | . . . . . 6 ⊢ (𝜑 → 𝐼 = (𝑀...𝑁)) | |
9 | 8 | feq2d 6473 | . . . . 5 ⊢ (𝜑 → ((𝑖 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵 ↔ (𝑖 ∈ 𝐼 ↦ 𝑋):(𝑀...𝑁)⟶𝐵)) |
10 | 7, 9 | syl5bb 286 | . . . 4 ⊢ (𝜑 → (∀𝑖 ∈ 𝐼 𝑋 ∈ 𝐵 ↔ (𝑖 ∈ 𝐼 ↦ 𝑋):(𝑀...𝑁)⟶𝐵)) |
11 | 5, 10 | mpbid 235 | . . 3 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ 𝑋):(𝑀...𝑁)⟶𝐵) |
12 | 1, 2, 3, 4, 11 | gsumval2 17888 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ 𝐼 ↦ 𝑋)) = (seq𝑀((+g‘𝐺), (𝑖 ∈ 𝐼 ↦ 𝑋))‘𝑁)) |
13 | 5 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ∀𝑖 ∈ 𝐼 𝑋 ∈ 𝐵) |
14 | 13, 7 | sylib 221 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑖 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵) |
15 | 8 | eqcomd 2804 | . . . . . 6 ⊢ (𝜑 → (𝑀...𝑁) = 𝐼) |
16 | 15 | eleq2d 2875 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑥 ∈ 𝐼)) |
17 | 16 | biimpa 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ 𝐼) |
18 | 14, 17 | ffvelrnd 6829 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑖 ∈ 𝐼 ↦ 𝑋)‘𝑥) ∈ 𝐵) |
19 | 3 | adantr 484 | . . . 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 17911 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
23 | 19, 20, 21, 22 | syl3anc 1368 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
24 | 4, 18, 23 | seqcl 13386 | . 2 ⊢ (𝜑 → (seq𝑀((+g‘𝐺), (𝑖 ∈ 𝐼 ↦ 𝑋))‘𝑁) ∈ 𝐵) |
25 | 12, 24 | eqeltrd 2890 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ 𝐼 ↦ 𝑋)) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ↦ cmpt 5110 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℤ≥cuz 12231 ...cfz 12885 seqcseq 13364 Basecbs 16475 +gcplusg 16557 Σg cgsu 16706 Mndcmnd 17903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-seq 13365 df-0g 16707 df-gsum 16708 df-mgm 17844 df-sgrp 17893 df-mnd 17904 |
This theorem is referenced by: m2detleiblem2 21233 |
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