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Mirrors > Home > MPE Home > Th. List > odf | Structured version Visualization version GIF version |
Description: Functionality of the group element order. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 5-Oct-2020.) |
Ref | Expression |
---|---|
odcl.1 | ⊢ 𝑋 = (Base‘𝐺) |
odcl.2 | ⊢ 𝑂 = (od‘𝐺) |
Ref | Expression |
---|---|
odf | ⊢ 𝑂:𝑋⟶ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 10623 | . . . . 5 ⊢ 0 ∈ V | |
2 | ltso 10709 | . . . . . 6 ⊢ < Or ℝ | |
3 | 2 | infex 8945 | . . . . 5 ⊢ inf(𝑤, ℝ, < ) ∈ V |
4 | 1, 3 | ifex 4511 | . . . 4 ⊢ if(𝑤 = ∅, 0, inf(𝑤, ℝ, < )) ∈ V |
5 | 4 | csbex 5206 | . . 3 ⊢ ⦋{𝑧 ∈ ℕ ∣ (𝑧(.g‘𝐺)𝑦) = (0g‘𝐺)} / 𝑤⦌if(𝑤 = ∅, 0, inf(𝑤, ℝ, < )) ∈ V |
6 | odcl.1 | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
7 | eqid 2818 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
8 | eqid 2818 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
9 | odcl.2 | . . . 4 ⊢ 𝑂 = (od‘𝐺) | |
10 | 6, 7, 8, 9 | odfval 18589 | . . 3 ⊢ 𝑂 = (𝑦 ∈ 𝑋 ↦ ⦋{𝑧 ∈ ℕ ∣ (𝑧(.g‘𝐺)𝑦) = (0g‘𝐺)} / 𝑤⦌if(𝑤 = ∅, 0, inf(𝑤, ℝ, < ))) |
11 | 5, 10 | fnmpti 6484 | . 2 ⊢ 𝑂 Fn 𝑋 |
12 | 6, 9 | odcl 18593 | . . 3 ⊢ (𝑥 ∈ 𝑋 → (𝑂‘𝑥) ∈ ℕ0) |
13 | 12 | rgen 3145 | . 2 ⊢ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ ℕ0 |
14 | ffnfv 6874 | . 2 ⊢ (𝑂:𝑋⟶ℕ0 ↔ (𝑂 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ ℕ0)) | |
15 | 11, 13, 14 | mpbir2an 707 | 1 ⊢ 𝑂:𝑋⟶ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 ∀wral 3135 {crab 3139 ⦋csb 3880 ∅c0 4288 ifcif 4463 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 infcinf 8893 ℝcr 10524 0cc0 10525 < clt 10663 ℕcn 11626 ℕ0cn0 11885 Basecbs 16471 0gc0g 16701 .gcmg 18162 odcod 18581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-od 18585 |
This theorem is referenced by: gexex 18902 torsubg 18903 proot1mul 39677 proot1hash 39678 proot1ex 39679 |
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