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Theorem odval2 18178

Description: A non-conditional definition of the group order. (Contributed by Stefan O'Rear, 6-Sep-2015.)

**Hypotheses**
Ref	Expression
odcl.1	⊢ 𝑋 = (Base‘𝐺)
odcl.2	⊢ 𝑂 = (od‘𝐺)
odid.3	⊢ · = (._g‘𝐺)
odid.4	⊢ 0 = (0_g‘𝐺)

**Assertion**
Ref	Expression
odval2	⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = (℩𝑥 ∈ ℕ₀ ∀𝑦 ∈ ℕ₀ (𝑥 ∥ 𝑦 ↔ (𝑦 · 𝐴) = 0 )))

Distinct variable groups: 𝑦, 0 𝑥,𝑦,𝐴 𝑥,𝑂,𝑦 𝑦, · 𝑥,𝐺,𝑦 𝑥,𝑋,𝑦 Allowed substitution hints: · (𝑥) 0 (𝑥)

**Proof of Theorem odval2**
Step	Hyp	Ref	Expression
1		odcl.1	. . . . 5 ⊢ 𝑋 = (Base‘𝐺)
2		odcl.2	. . . . 5 ⊢ 𝑂 = (od‘𝐺)
3	1, 2	odcl 18163	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
4	3	adantl 467	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ₀)
5		odid.3	. . . . . 6 ⊢ · = (._g‘𝐺)
6		odid.4	. . . . . 6 ⊢ 0 = (0_g‘𝐺)
7	1, 2, 5, 6	odeq 18177	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑥 ∈ ℕ₀) → (𝑥 = (𝑂‘𝐴) ↔ ∀𝑦 ∈ ℕ₀ (𝑥 ∥ 𝑦 ↔ (𝑦 · 𝐴) = 0 )))
8	7	3expa 1111	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ ℕ₀) → (𝑥 = (𝑂‘𝐴) ↔ ∀𝑦 ∈ ℕ₀ (𝑥 ∥ 𝑦 ↔ (𝑦 · 𝐴) = 0 )))
9	8	bicomd 213	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ ℕ₀) → (∀𝑦 ∈ ℕ₀ (𝑥 ∥ 𝑦 ↔ (𝑦 · 𝐴) = 0 ) ↔ 𝑥 = (𝑂‘𝐴)))
10	4, 9	riota5 6781	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (℩𝑥 ∈ ℕ₀ ∀𝑦 ∈ ℕ₀ (𝑥 ∥ 𝑦 ↔ (𝑦 · 𝐴) = 0 )) = (𝑂‘𝐴))
11	10	eqcomd 2777	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = (℩𝑥 ∈ ℕ₀ ∀𝑦 ∈ ℕ₀ (𝑥 ∥ 𝑦 ↔ (𝑦 · 𝐴) = 0 )))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∀wral 3061 class class class wbr 4787 ‘cfv 6032 ℩crio 6754 (class class class)co 6794 ℕ₀cn0 11495 ∥ cdvds 15190 Basecbs 16065 0_gc0g 16309 Grpcgrp 17631 ._gcmg 17749 odcod 18152

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 ax-inf2 8703 ax-cnex 10195 ax-resscn 10196 ax-1cn 10197 ax-icn 10198 ax-addcl 10199 ax-addrcl 10200 ax-mulcl 10201 ax-mulrcl 10202 ax-mulcom 10203 ax-addass 10204 ax-mulass 10205 ax-distr 10206 ax-i2m1 10207 ax-1ne0 10208 ax-1rid 10209 ax-rnegex 10210 ax-rrecex 10211 ax-cnre 10212 ax-pre-lttri 10213 ax-pre-lttrn 10214 ax-pre-ltadd 10215 ax-pre-mulgt0 10216 ax-pre-sup 10217

This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-riota 6755 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-om 7214 df-1st 7316 df-2nd 7317 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-er 7897 df-en 8111 df-dom 8112 df-sdom 8113 df-sup 8505 df-inf 8506 df-pnf 10279 df-mnf 10280 df-xr 10281 df-ltxr 10282 df-le 10283 df-sub 10471 df-neg 10472 df-div 10888 df-nn 11224 df-2 11282 df-3 11283 df-n0 11496 df-z 11581 df-uz 11890 df-rp 12037 df-fz 12535 df-fl 12802 df-mod 12878 df-seq 13010 df-exp 13069 df-cj 14048 df-re 14049 df-im 14050 df-sqrt 14184 df-abs 14185 df-dvds 15191 df-0g 16311 df-mgm 17451 df-sgrp 17493 df-mnd 17504 df-grp 17634 df-minusg 17635 df-sbg 17636 df-mulg 17750 df-od 18156

This theorem is referenced by: (None)

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