| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 11255 | . . . . 5 ⊢ 0 ∈ V | |
| 2 | ltso 11341 | . . . . . 6 ⊢ < Or ℝ | |
| 3 | 2 | infex 9533 | . . . . 5 ⊢ inf(𝑤, ℝ, < ) ∈ V |
| 4 | 1, 3 | ifex 4576 | . . . 4 ⊢ if(𝑤 = ∅, 0, inf(𝑤, ℝ, < )) ∈ V |
| 5 | 4 | csbex 5311 | . . 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 2737 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
| 8 | eqid 2737 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 9 | odcl.2 | . . . 4 ⊢ 𝑂 = (od‘𝐺) | |
| 10 | 6, 7, 8, 9 | odfval 19550 | . . 3 ⊢ 𝑂 = (𝑦 ∈ 𝑋 ↦ ⦋{𝑧 ∈ ℕ ∣ (𝑧(.g‘𝐺)𝑦) = (0g‘𝐺)} / 𝑤⦌if(𝑤 = ∅, 0, inf(𝑤, ℝ, < ))) |
| 11 | 5, 10 | fnmpti 6711 | . 2 ⊢ 𝑂 Fn 𝑋 |
| 12 | 6, 9 | odcl 19554 | . . 3 ⊢ (𝑥 ∈ 𝑋 → (𝑂‘𝑥) ∈ ℕ0) |
| 13 | 12 | rgen 3063 | . 2 ⊢ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ ℕ0 |
| 14 | ffnfv 7139 | . 2 ⊢ (𝑂:𝑋⟶ℕ0 ↔ (𝑂 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ ℕ0)) | |
| 15 | 11, 13, 14 | mpbir2an 711 | 1 ⊢ 𝑂:𝑋⟶ℕ0 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ⦋csb 3899 ∅c0 4333 ifcif 4525 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 infcinf 9481 ℝcr 11154 0cc0 11155 < clt 11295 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 0gc0g 17484 .gcmg 19085 odcod 19542 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-od 19546 |
| This theorem is referenced by: gexex 19871 torsubg 19872 proot1mul 43206 proot1hash 43207 proot1ex 43208 |
| Copyright terms: Public domain | W3C validator |