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Theorem gexval 18200
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 18156 . . . 4 gEx = (𝑔 ∈ V ↦ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
32a1i 11 . . 3 (𝐺𝑉 → gEx = (𝑔 ∈ V ↦ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
4 nnex 11232 . . . . . 6 ℕ ∈ V
54rabex 4947 . . . . 5 {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} ∈ V
65a1i 11 . . . 4 ((𝐺𝑉𝑔 = 𝐺) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} ∈ V)
7 simpr 471 . . . . . . . . . . . . 13 ((𝐺𝑉𝑔 = 𝐺) → 𝑔 = 𝐺)
87fveq2d 6337 . . . . . . . . . . . 12 ((𝐺𝑉𝑔 = 𝐺) → (Base‘𝑔) = (Base‘𝐺))
9 gexval.1 . . . . . . . . . . . 12 𝑋 = (Base‘𝐺)
108, 9syl6eqr 2823 . . . . . . . . . . 11 ((𝐺𝑉𝑔 = 𝐺) → (Base‘𝑔) = 𝑋)
117fveq2d 6337 . . . . . . . . . . . . . 14 ((𝐺𝑉𝑔 = 𝐺) → (.g𝑔) = (.g𝐺))
12 gexval.2 . . . . . . . . . . . . . 14 · = (.g𝐺)
1311, 12syl6eqr 2823 . . . . . . . . . . . . 13 ((𝐺𝑉𝑔 = 𝐺) → (.g𝑔) = · )
1413oveqd 6813 . . . . . . . . . . . 12 ((𝐺𝑉𝑔 = 𝐺) → (𝑦(.g𝑔)𝑥) = (𝑦 · 𝑥))
157fveq2d 6337 . . . . . . . . . . . . 13 ((𝐺𝑉𝑔 = 𝐺) → (0g𝑔) = (0g𝐺))
16 gexval.3 . . . . . . . . . . . . 13 0 = (0g𝐺)
1715, 16syl6eqr 2823 . . . . . . . . . . . 12 ((𝐺𝑉𝑔 = 𝐺) → (0g𝑔) = 0 )
1814, 17eqeq12d 2786 . . . . . . . . . . 11 ((𝐺𝑉𝑔 = 𝐺) → ((𝑦(.g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 · 𝑥) = 0 ))
1910, 18raleqbidv 3301 . . . . . . . . . 10 ((𝐺𝑉𝑔 = 𝐺) → (∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔) ↔ ∀𝑥𝑋 (𝑦 · 𝑥) = 0 ))
2019rabbidv 3339 . . . . . . . . 9 ((𝐺𝑉𝑔 = 𝐺) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} = {𝑦 ∈ ℕ ∣ ∀𝑥𝑋 (𝑦 · 𝑥) = 0 })
21 gexval.i . . . . . . . . 9 𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥𝑋 (𝑦 · 𝑥) = 0 }
2220, 21syl6eqr 2823 . . . . . . . 8 ((𝐺𝑉𝑔 = 𝐺) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} = 𝐼)
2322eqeq2d 2781 . . . . . . 7 ((𝐺𝑉𝑔 = 𝐺) → (𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} ↔ 𝑖 = 𝐼))
2423biimpa 462 . . . . . 6 (((𝐺𝑉𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)}) → 𝑖 = 𝐼)
2524eqeq1d 2773 . . . . 5 (((𝐺𝑉𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)}) → (𝑖 = ∅ ↔ 𝐼 = ∅))
2624infeq1d 8543 . . . . 5 (((𝐺𝑉𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)}) → inf(𝑖, ℝ, < ) = inf(𝐼, ℝ, < ))
2725, 26ifbieq2d 4251 . . . 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, < )))
286, 27csbied 3709 . . 3 ((𝐺𝑉𝑔 = 𝐺) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
29 elex 3364 . . 3 (𝐺𝑉𝐺 ∈ V)
30 c0ex 10240 . . . . 5 0 ∈ V
31 ltso 10324 . . . . . 6 < Or ℝ
3231infex 8559 . . . . 5 inf(𝐼, ℝ, < ) ∈ V
3330, 32ifex 4296 . . . 4 if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V
3433a1i 11 . . 3 (𝐺𝑉 → if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V)
353, 28, 29, 34fvmptd 6432 . 2 (𝐺𝑉 → (gEx‘𝐺) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
361, 35syl5eq 2817 1 (𝐺𝑉𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  csb 3682  c0 4063  ifcif 4226  cmpt 4864  cfv 6030  (class class class)co 6796  infcinf 8507  cr 10141  0cc0 10142   < clt 10280  cn 11226  Basecbs 16064  0gc0g 16308  .gcmg 17748  gExcgex 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-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-i2m1 10210  ax-1ne0 10211  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  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 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-om 7217  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-sup 8508  df-inf 8509  df-pnf 10282  df-mnf 10283  df-ltxr 10285  df-nn 11227  df-gex 18156
This theorem is referenced by:  gexlem1  18201  gexlem2  18204
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