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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 2821 | . . 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 2821 | . . . . . 6 ⊢ (𝑖 ∈ 𝐼 ↦ 𝑋) = (𝑖 ∈ 𝐼 ↦ 𝑋) | |
7 | 6 | fmpt 6874 | . . . . 5 ⊢ (∀𝑖 ∈ 𝐼 𝑋 ∈ 𝐵 ↔ (𝑖 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵) |
8 | gsummptfzcl.i | . . . . . 6 ⊢ (𝜑 → 𝐼 = (𝑀...𝑁)) | |
9 | 8 | feq2d 6500 | . . . . 5 ⊢ (𝜑 → ((𝑖 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵 ↔ (𝑖 ∈ 𝐼 ↦ 𝑋):(𝑀...𝑁)⟶𝐵)) |
10 | 7, 9 | syl5bb 285 | . . . 4 ⊢ (𝜑 → (∀𝑖 ∈ 𝐼 𝑋 ∈ 𝐵 ↔ (𝑖 ∈ 𝐼 ↦ 𝑋):(𝑀...𝑁)⟶𝐵)) |
11 | 5, 10 | mpbid 234 | . . 3 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ 𝑋):(𝑀...𝑁)⟶𝐵) |
12 | 1, 2, 3, 4, 11 | gsumval2 17896 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ 𝐼 ↦ 𝑋)) = (seq𝑀((+g‘𝐺), (𝑖 ∈ 𝐼 ↦ 𝑋))‘𝑁)) |
13 | 5 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ∀𝑖 ∈ 𝐼 𝑋 ∈ 𝐵) |
14 | 13, 7 | sylib 220 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑖 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵) |
15 | 8 | eqcomd 2827 | . . . . . 6 ⊢ (𝜑 → (𝑀...𝑁) = 𝐼) |
16 | 15 | eleq2d 2898 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑥 ∈ 𝐼)) |
17 | 16 | biimpa 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ 𝐼) |
18 | 14, 17 | ffvelrnd 6852 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑖 ∈ 𝐼 ↦ 𝑋)‘𝑥) ∈ 𝐵) |
19 | 3 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Mnd) |
20 | simprl 769 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵) | |
21 | simprr 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵) | |
22 | 1, 2 | mndcl 17919 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
23 | 19, 20, 21, 22 | syl3anc 1367 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
24 | 4, 18, 23 | seqcl 13391 | . 2 ⊢ (𝜑 → (seq𝑀((+g‘𝐺), (𝑖 ∈ 𝐼 ↦ 𝑋))‘𝑁) ∈ 𝐵) |
25 | 12, 24 | eqeltrd 2913 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ 𝐼 ↦ 𝑋)) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ↦ cmpt 5146 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℤ≥cuz 12244 ...cfz 12893 seqcseq 13370 Basecbs 16483 +gcplusg 16565 Σg cgsu 16714 Mndcmnd 17911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-seq 13371 df-0g 16715 df-gsum 16716 df-mgm 17852 df-sgrp 17901 df-mnd 17912 |
This theorem is referenced by: m2detleiblem2 21237 |
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