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Theorem odval2 19411

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 19396	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
4	3	adantl 483	. . 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 19410	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑥 ∈ ℕ₀) → (𝑥 = (𝑂‘𝐴) ↔ ∀𝑦 ∈ ℕ₀ (𝑥 ∥ 𝑦 ↔ (𝑦 · 𝐴) = 0 )))
8	7	3expa 1119	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ ℕ₀) → (𝑥 = (𝑂‘𝐴) ↔ ∀𝑦 ∈ ℕ₀ (𝑥 ∥ 𝑦 ↔ (𝑦 · 𝐴) = 0 )))
9	8	bicomd 222	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ ℕ₀) → (∀𝑦 ∈ ℕ₀ (𝑥 ∥ 𝑦 ↔ (𝑦 · 𝐴) = 0 ) ↔ 𝑥 = (𝑂‘𝐴)))
10	4, 9	riota5 7389	. 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 class class class wbr 5146 ‘cfv 6539 ℩crio 7358 (class class class)co 7403 ℕ₀cn0 12467 ∥ cdvds 16192 Basecbs 17139 0_gc0g 17380 Grpcgrp 18814 ._gcmg 18943 odcod 19384

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-sup 9432 df-inf 9433 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-div 11867 df-nn 12208 df-2 12270 df-3 12271 df-n0 12468 df-z 12554 df-uz 12818 df-rp 12970 df-fz 13480 df-fl 13752 df-mod 13830 df-seq 13962 df-exp 14023 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-dvds 16193 df-0g 17382 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-grp 18817 df-minusg 18818 df-sbg 18819 df-mulg 18944 df-od 19388

This theorem is referenced by: (None)

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