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Theorem gexval 19596
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 19547 . . 3 gEx = (𝑔 ∈ V ↦ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
3 nnex 12272 . . . . . 6 ℕ ∈ V
43rabex 5339 . . . . 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 484 . . . . . . . . . . . . 13 ((𝐺𝑉𝑔 = 𝐺) → 𝑔 = 𝐺)
76fveq2d 6910 . . . . . . . . . . . 12 ((𝐺𝑉𝑔 = 𝐺) → (Base‘𝑔) = (Base‘𝐺))
8 gexval.1 . . . . . . . . . . . 12 𝑋 = (Base‘𝐺)
97, 8eqtr4di 2795 . . . . . . . . . . 11 ((𝐺𝑉𝑔 = 𝐺) → (Base‘𝑔) = 𝑋)
106fveq2d 6910 . . . . . . . . . . . . . 14 ((𝐺𝑉𝑔 = 𝐺) → (.g𝑔) = (.g𝐺))
11 gexval.2 . . . . . . . . . . . . . 14 · = (.g𝐺)
1210, 11eqtr4di 2795 . . . . . . . . . . . . 13 ((𝐺𝑉𝑔 = 𝐺) → (.g𝑔) = · )
1312oveqd 7448 . . . . . . . . . . . 12 ((𝐺𝑉𝑔 = 𝐺) → (𝑦(.g𝑔)𝑥) = (𝑦 · 𝑥))
146fveq2d 6910 . . . . . . . . . . . . 13 ((𝐺𝑉𝑔 = 𝐺) → (0g𝑔) = (0g𝐺))
15 gexval.3 . . . . . . . . . . . . 13 0 = (0g𝐺)
1614, 15eqtr4di 2795 . . . . . . . . . . . 12 ((𝐺𝑉𝑔 = 𝐺) → (0g𝑔) = 0 )
1713, 16eqeq12d 2753 . . . . . . . . . . 11 ((𝐺𝑉𝑔 = 𝐺) → ((𝑦(.g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 · 𝑥) = 0 ))
189, 17raleqbidv 3346 . . . . . . . . . 10 ((𝐺𝑉𝑔 = 𝐺) → (∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔) ↔ ∀𝑥𝑋 (𝑦 · 𝑥) = 0 ))
1918rabbidv 3444 . . . . . . . . 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 2795 . . . . . . . 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 476 . . . . . 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 9517 . . . . 5 (((𝐺𝑉𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)}) → inf(𝑖, ℝ, < ) = inf(𝐼, ℝ, < ))
2624, 25ifbieq2d 4552 . . . 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 3935 . . 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 3501 . . 3 (𝐺𝑉𝐺 ∈ V)
29 c0ex 11255 . . . . 5 0 ∈ V
30 ltso 11341 . . . . . 6 < Or ℝ
3130infex 9533 . . . . 5 inf(𝐼, ℝ, < ) ∈ V
3229, 31ifex 4576 . . . 4 if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V
3332a1i 11 . . 3 (𝐺𝑉 → if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V)
342, 27, 28, 33fvmptd2 7024 . 2 (𝐺𝑉 → (gEx‘𝐺) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
351, 34eqtrid 2789 1 (𝐺𝑉𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  csb 3899  c0 4333  ifcif 4525  cfv 6561  (class class class)co 7431  infcinf 9481  cr 11154  0cc0 11155   < clt 11295  cn 12266  Basecbs 17247  0gc0g 17484  .gcmg 19085  gExcgex 19543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-mulcl 11217  ax-i2m1 11223  ax-pre-lttri 11229  ax-pre-lttrn 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-ltxr 11300  df-nn 12267  df-gex 19547
This theorem is referenced by:  gexlem1  19597  gexlem2  19600
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