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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 10637 | . . . . 5 ⊢ 0 ∈ V | |
2 | ltso 10723 | . . . . . 6 ⊢ < Or ℝ | |
3 | 2 | infex 8959 | . . . . 5 ⊢ inf(𝑤, ℝ, < ) ∈ V |
4 | 1, 3 | ifex 4517 | . . . 4 ⊢ if(𝑤 = ∅, 0, inf(𝑤, ℝ, < )) ∈ V |
5 | 4 | csbex 5217 | . . 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 2823 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
8 | eqid 2823 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
9 | odcl.2 | . . . 4 ⊢ 𝑂 = (od‘𝐺) | |
10 | 6, 7, 8, 9 | odfval 18662 | . . 3 ⊢ 𝑂 = (𝑦 ∈ 𝑋 ↦ ⦋{𝑧 ∈ ℕ ∣ (𝑧(.g‘𝐺)𝑦) = (0g‘𝐺)} / 𝑤⦌if(𝑤 = ∅, 0, inf(𝑤, ℝ, < ))) |
11 | 5, 10 | fnmpti 6493 | . 2 ⊢ 𝑂 Fn 𝑋 |
12 | 6, 9 | odcl 18666 | . . 3 ⊢ (𝑥 ∈ 𝑋 → (𝑂‘𝑥) ∈ ℕ0) |
13 | 12 | rgen 3150 | . 2 ⊢ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ ℕ0 |
14 | ffnfv 6884 | . 2 ⊢ (𝑂:𝑋⟶ℕ0 ↔ (𝑂 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ ℕ0)) | |
15 | 11, 13, 14 | mpbir2an 709 | 1 ⊢ 𝑂:𝑋⟶ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 ⦋csb 3885 ∅c0 4293 ifcif 4469 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 infcinf 8907 ℝcr 10538 0cc0 10539 < clt 10677 ℕcn 11640 ℕ0cn0 11900 Basecbs 16485 0gc0g 16715 .gcmg 18226 odcod 18654 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-od 18658 |
This theorem is referenced by: gexex 18975 torsubg 18976 proot1mul 39806 proot1hash 39807 proot1ex 39808 |
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