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Mirrors > Home > MPE Home > Th. List > odid | Structured version Visualization version GIF version |
Description: Any element to the power of its order is the identity. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
odcl.1 | ⊢ 𝑋 = (Base‘𝐺) |
odcl.2 | ⊢ 𝑂 = (od‘𝐺) |
odid.3 | ⊢ · = (.g‘𝐺) |
odid.4 | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
odid | ⊢ (𝐴 ∈ 𝑋 → ((𝑂‘𝐴) · 𝐴) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7364 | . . . 4 ⊢ ((𝑂‘𝐴) = 0 → ((𝑂‘𝐴) · 𝐴) = (0 · 𝐴)) | |
2 | odcl.1 | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
3 | odid.4 | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
4 | odid.3 | . . . . 5 ⊢ · = (.g‘𝐺) | |
5 | 2, 3, 4 | mulg0 18879 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → (0 · 𝐴) = 0 ) |
6 | 1, 5 | sylan9eqr 2798 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ((𝑂‘𝐴) · 𝐴) = 0 ) |
7 | 6 | adantrr 715 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ ((𝑂‘𝐴) = 0 ∧ {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 } = ∅)) → ((𝑂‘𝐴) · 𝐴) = 0 ) |
8 | oveq1 7364 | . . . . . 6 ⊢ (𝑦 = (𝑂‘𝐴) → (𝑦 · 𝐴) = ((𝑂‘𝐴) · 𝐴)) | |
9 | 8 | eqeq1d 2738 | . . . . 5 ⊢ (𝑦 = (𝑂‘𝐴) → ((𝑦 · 𝐴) = 0 ↔ ((𝑂‘𝐴) · 𝐴) = 0 )) |
10 | 9 | elrab 3645 | . . . 4 ⊢ ((𝑂‘𝐴) ∈ {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 } ↔ ((𝑂‘𝐴) ∈ ℕ ∧ ((𝑂‘𝐴) · 𝐴) = 0 )) |
11 | 10 | simprbi 497 | . . 3 ⊢ ((𝑂‘𝐴) ∈ {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 } → ((𝑂‘𝐴) · 𝐴) = 0 ) |
12 | 11 | adantl 482 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) ∈ {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 }) → ((𝑂‘𝐴) · 𝐴) = 0 ) |
13 | odcl.2 | . . 3 ⊢ 𝑂 = (od‘𝐺) | |
14 | eqid 2736 | . . 3 ⊢ {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 } = {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 } | |
15 | 2, 4, 3, 13, 14 | odlem1 19317 | . 2 ⊢ (𝐴 ∈ 𝑋 → (((𝑂‘𝐴) = 0 ∧ {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 } = ∅) ∨ (𝑂‘𝐴) ∈ {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 })) |
16 | 7, 12, 15 | mpjaodan 957 | 1 ⊢ (𝐴 ∈ 𝑋 → ((𝑂‘𝐴) · 𝐴) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3407 ∅c0 4282 ‘cfv 6496 (class class class)co 7357 0cc0 11051 ℕcn 12153 Basecbs 17083 0gc0g 17321 .gcmg 18872 odcod 19306 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-z 12500 df-uz 12764 df-seq 13907 df-mulg 18873 df-od 19310 |
This theorem is referenced by: odmodnn0 19322 mndodconglem 19323 odmod 19328 odeq 19332 odm1inv 19335 odeq1 19342 odf1 19344 chrid 20930 |
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