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Theorem gexval 18967
Description: Value of the exponent of a group. (Contributed by Mario Carneiro, 23-Apr-2016.) (Revised by AV, 26-Sep-2020.)
Hypotheses
Ref Expression
gexval.1 𝑋 = (Base‘𝐺)
gexval.2 · = (.g𝐺)
gexval.3 0 = (0g𝐺)
gexval.4 𝐸 = (gEx‘𝐺)
gexval.i 𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥𝑋 (𝑦 · 𝑥) = 0 }
Assertion
Ref Expression
gexval (𝐺𝑉𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
Distinct variable groups:   𝑥,𝑦, 0   𝑥,𝐺,𝑦   𝑥,𝑉,𝑦   𝑥, · ,𝑦   𝑥,𝑋
Allowed substitution hints:   𝐸(𝑥,𝑦)   𝐼(𝑥,𝑦)   𝑋(𝑦)

Proof of Theorem gexval
Dummy variables 𝑔 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gexval.4 . 2 𝐸 = (gEx‘𝐺)
2 df-gex 18921 . . 3 gEx = (𝑔 ∈ V ↦ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
3 nnex 11836 . . . . . 6 ℕ ∈ V
43rabex 5225 . . . . 5 {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} ∈ V
54a1i 11 . . . 4 ((𝐺𝑉𝑔 = 𝐺) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} ∈ V)
6 simpr 488 . . . . . . . . . . . . 13 ((𝐺𝑉𝑔 = 𝐺) → 𝑔 = 𝐺)
76fveq2d 6721 . . . . . . . . . . . 12 ((𝐺𝑉𝑔 = 𝐺) → (Base‘𝑔) = (Base‘𝐺))
8 gexval.1 . . . . . . . . . . . 12 𝑋 = (Base‘𝐺)
97, 8eqtr4di 2796 . . . . . . . . . . 11 ((𝐺𝑉𝑔 = 𝐺) → (Base‘𝑔) = 𝑋)
106fveq2d 6721 . . . . . . . . . . . . . 14 ((𝐺𝑉𝑔 = 𝐺) → (.g𝑔) = (.g𝐺))
11 gexval.2 . . . . . . . . . . . . . 14 · = (.g𝐺)
1210, 11eqtr4di 2796 . . . . . . . . . . . . 13 ((𝐺𝑉𝑔 = 𝐺) → (.g𝑔) = · )
1312oveqd 7230 . . . . . . . . . . . 12 ((𝐺𝑉𝑔 = 𝐺) → (𝑦(.g𝑔)𝑥) = (𝑦 · 𝑥))
146fveq2d 6721 . . . . . . . . . . . . 13 ((𝐺𝑉𝑔 = 𝐺) → (0g𝑔) = (0g𝐺))
15 gexval.3 . . . . . . . . . . . . 13 0 = (0g𝐺)
1614, 15eqtr4di 2796 . . . . . . . . . . . 12 ((𝐺𝑉𝑔 = 𝐺) → (0g𝑔) = 0 )
1713, 16eqeq12d 2753 . . . . . . . . . . 11 ((𝐺𝑉𝑔 = 𝐺) → ((𝑦(.g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 · 𝑥) = 0 ))
189, 17raleqbidv 3313 . . . . . . . . . 10 ((𝐺𝑉𝑔 = 𝐺) → (∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔) ↔ ∀𝑥𝑋 (𝑦 · 𝑥) = 0 ))
1918rabbidv 3390 . . . . . . . . 9 ((𝐺𝑉𝑔 = 𝐺) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} = {𝑦 ∈ ℕ ∣ ∀𝑥𝑋 (𝑦 · 𝑥) = 0 })
20 gexval.i . . . . . . . . 9 𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥𝑋 (𝑦 · 𝑥) = 0 }
2119, 20eqtr4di 2796 . . . . . . . 8 ((𝐺𝑉𝑔 = 𝐺) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} = 𝐼)
2221eqeq2d 2748 . . . . . . 7 ((𝐺𝑉𝑔 = 𝐺) → (𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} ↔ 𝑖 = 𝐼))
2322biimpa 480 . . . . . 6 (((𝐺𝑉𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)}) → 𝑖 = 𝐼)
2423eqeq1d 2739 . . . . 5 (((𝐺𝑉𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)}) → (𝑖 = ∅ ↔ 𝐼 = ∅))
2523infeq1d 9093 . . . . 5 (((𝐺𝑉𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)}) → inf(𝑖, ℝ, < ) = inf(𝐼, ℝ, < ))
2624, 25ifbieq2d 4465 . . . 4 (((𝐺𝑉𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)}) → if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
275, 26csbied 3849 . . 3 ((𝐺𝑉𝑔 = 𝐺) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
28 elex 3426 . . 3 (𝐺𝑉𝐺 ∈ V)
29 c0ex 10827 . . . . 5 0 ∈ V
30 ltso 10913 . . . . . 6 < Or ℝ
3130infex 9109 . . . . 5 inf(𝐼, ℝ, < ) ∈ V
3229, 31ifex 4489 . . . 4 if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V
3332a1i 11 . . 3 (𝐺𝑉 → if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V)
342, 27, 28, 33fvmptd2 6826 . 2 (𝐺𝑉 → (gEx‘𝐺) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
351, 34syl5eq 2790 1 (𝐺𝑉𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  wral 3061  {crab 3065  Vcvv 3408  csb 3811  c0 4237  ifcif 4439  cfv 6380  (class class class)co 7213  infcinf 9057  cr 10728  0cc0 10729   < clt 10867  cn 11830  Basecbs 16760  0gc0g 16944  .gcmg 18488  gExcgex 18917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-mulcl 10791  ax-i2m1 10797  ax-pre-lttri 10803  ax-pre-lttrn 10804
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-om 7645  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-sup 9058  df-inf 9059  df-pnf 10869  df-mnf 10870  df-ltxr 10872  df-nn 11831  df-gex 18921
This theorem is referenced by:  gexlem1  18968  gexlem2  18971
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