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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 10320 | . . . . 5 ⊢ 0 ∈ V | |
2 | ltso 10406 | . . . . . 6 ⊢ < Or ℝ | |
3 | 2 | infex 8639 | . . . . 5 ⊢ inf(𝑤, ℝ, < ) ∈ V |
4 | 1, 3 | ifex 4323 | . . . 4 ⊢ if(𝑤 = ∅, 0, inf(𝑤, ℝ, < )) ∈ V |
5 | 4 | csbex 4986 | . . 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 2797 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
8 | eqid 2797 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
9 | odcl.2 | . . . 4 ⊢ 𝑂 = (od‘𝐺) | |
10 | 6, 7, 8, 9 | odfval 18261 | . . 3 ⊢ 𝑂 = (𝑦 ∈ 𝑋 ↦ ⦋{𝑧 ∈ ℕ ∣ (𝑧(.g‘𝐺)𝑦) = (0g‘𝐺)} / 𝑤⦌if(𝑤 = ∅, 0, inf(𝑤, ℝ, < ))) |
11 | 5, 10 | fnmpti 6231 | . 2 ⊢ 𝑂 Fn 𝑋 |
12 | 6, 9 | odcl 18264 | . . 3 ⊢ (𝑥 ∈ 𝑋 → (𝑂‘𝑥) ∈ ℕ0) |
13 | 12 | rgen 3101 | . 2 ⊢ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ ℕ0 |
14 | ffnfv 6612 | . 2 ⊢ (𝑂:𝑋⟶ℕ0 ↔ (𝑂 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ ℕ0)) | |
15 | 11, 13, 14 | mpbir2an 703 | 1 ⊢ 𝑂:𝑋⟶ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 ∀wral 3087 {crab 3091 ⦋csb 3726 ∅c0 4113 ifcif 4275 Fn wfn 6094 ⟶wf 6095 ‘cfv 6099 (class class class)co 6876 infcinf 8587 ℝcr 10221 0cc0 10222 < clt 10361 ℕcn 11310 ℕ0cn0 11576 Basecbs 16180 0gc0g 16411 .gcmg 17852 odcod 18253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-sup 8588 df-inf 8589 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-n0 11577 df-z 11663 df-uz 11927 df-od 18257 |
This theorem is referenced by: gexex 18567 torsubg 18568 proot1mul 38549 proot1hash 38550 proot1ex 38551 |
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