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Theorem odval2 19471

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,𝐺 𝑦, 0 𝑥,𝑦,𝐴 𝑥,𝑂,𝑦 𝑦, · 𝑥,𝐺 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 19456	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
4	3	adantl 481	. . 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 19470	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑥 ∈ ℕ₀) → (𝑥 = (𝑂‘𝐴) ↔ ∀𝑦 ∈ ℕ₀ (𝑥 ∥ 𝑦 ↔ (𝑦 · 𝐴) = 0 )))
8	7	3expa 1118	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ ℕ₀) → (𝑥 = (𝑂‘𝐴) ↔ ∀𝑦 ∈ ℕ₀ (𝑥 ∥ 𝑦 ↔ (𝑦 · 𝐴) = 0 )))
9	8	bicomd 223	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ ℕ₀) → (∀𝑦 ∈ ℕ₀ (𝑥 ∥ 𝑦 ↔ (𝑦 · 𝐴) = 0 ) ↔ 𝑥 = (𝑂‘𝐴)))
10	4, 9	riota5 7341	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (℩𝑥 ∈ ℕ₀ ∀𝑦 ∈ ℕ₀ (𝑥 ∥ 𝑦 ↔ (𝑦 · 𝐴) = 0 )) = (𝑂‘𝐴))
11	10	eqcomd 2739	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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3048 class class class wbr 5095 ‘cfv 6489 ℩crio 7311 (class class class)co 7355 ℕ₀cn0 12392 ∥ cdvds 16170 Basecbs 17127 0_gc0g 17350 Grpcgrp 18854 ._gcmg 18988 odcod 19444

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-er 8631 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9337 df-inf 9338 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-n0 12393 df-z 12480 df-uz 12743 df-rp 12897 df-fz 13415 df-fl 13703 df-mod 13781 df-seq 13916 df-exp 13976 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-dvds 16171 df-0g 17352 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-grp 18857 df-minusg 18858 df-sbg 18859 df-mulg 18989 df-od 19448

This theorem is referenced by: (None)

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