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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 10624 | . . . . 5 ⊢ 0 ∈ V | |
2 | ltso 10710 | . . . . . 6 ⊢ < Or ℝ | |
3 | 2 | infex 8941 | . . . . 5 ⊢ inf(𝑤, ℝ, < ) ∈ V |
4 | 1, 3 | ifex 4473 | . . . 4 ⊢ if(𝑤 = ∅, 0, inf(𝑤, ℝ, < )) ∈ V |
5 | 4 | csbex 5179 | . . 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 2798 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
8 | eqid 2798 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
9 | odcl.2 | . . . 4 ⊢ 𝑂 = (od‘𝐺) | |
10 | 6, 7, 8, 9 | odfval 18652 | . . 3 ⊢ 𝑂 = (𝑦 ∈ 𝑋 ↦ ⦋{𝑧 ∈ ℕ ∣ (𝑧(.g‘𝐺)𝑦) = (0g‘𝐺)} / 𝑤⦌if(𝑤 = ∅, 0, inf(𝑤, ℝ, < ))) |
11 | 5, 10 | fnmpti 6463 | . 2 ⊢ 𝑂 Fn 𝑋 |
12 | 6, 9 | odcl 18656 | . . 3 ⊢ (𝑥 ∈ 𝑋 → (𝑂‘𝑥) ∈ ℕ0) |
13 | 12 | rgen 3116 | . 2 ⊢ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ ℕ0 |
14 | ffnfv 6859 | . 2 ⊢ (𝑂:𝑋⟶ℕ0 ↔ (𝑂 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ ℕ0)) | |
15 | 11, 13, 14 | mpbir2an 710 | 1 ⊢ 𝑂:𝑋⟶ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ∀wral 3106 {crab 3110 ⦋csb 3828 ∅c0 4243 ifcif 4425 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 infcinf 8889 ℝcr 10525 0cc0 10526 < clt 10664 ℕcn 11625 ℕ0cn0 11885 Basecbs 16475 0gc0g 16705 .gcmg 18216 odcod 18644 |
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-fal 1551 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-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 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-od 18648 |
This theorem is referenced by: gexex 18966 torsubg 18967 proot1mul 40143 proot1hash 40144 proot1ex 40145 |
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