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Theorem gexval 17987
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 17943 . . . 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 11023 . . . . . 6 ℕ ∈ V
54rabex 4811 . . . . 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 477 . . . . . . . . . . . . 13 ((𝐺𝑉𝑔 = 𝐺) → 𝑔 = 𝐺)
87fveq2d 6193 . . . . . . . . . . . 12 ((𝐺𝑉𝑔 = 𝐺) → (Base‘𝑔) = (Base‘𝐺))
9 gexval.1 . . . . . . . . . . . 12 𝑋 = (Base‘𝐺)
108, 9syl6eqr 2673 . . . . . . . . . . 11 ((𝐺𝑉𝑔 = 𝐺) → (Base‘𝑔) = 𝑋)
117fveq2d 6193 . . . . . . . . . . . . . 14 ((𝐺𝑉𝑔 = 𝐺) → (.g𝑔) = (.g𝐺))
12 gexval.2 . . . . . . . . . . . . . 14 · = (.g𝐺)
1311, 12syl6eqr 2673 . . . . . . . . . . . . 13 ((𝐺𝑉𝑔 = 𝐺) → (.g𝑔) = · )
1413oveqd 6664 . . . . . . . . . . . 12 ((𝐺𝑉𝑔 = 𝐺) → (𝑦(.g𝑔)𝑥) = (𝑦 · 𝑥))
157fveq2d 6193 . . . . . . . . . . . . 13 ((𝐺𝑉𝑔 = 𝐺) → (0g𝑔) = (0g𝐺))
16 gexval.3 . . . . . . . . . . . . 13 0 = (0g𝐺)
1715, 16syl6eqr 2673 . . . . . . . . . . . 12 ((𝐺𝑉𝑔 = 𝐺) → (0g𝑔) = 0 )
1814, 17eqeq12d 2636 . . . . . . . . . . 11 ((𝐺𝑉𝑔 = 𝐺) → ((𝑦(.g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 · 𝑥) = 0 ))
1910, 18raleqbidv 3150 . . . . . . . . . 10 ((𝐺𝑉𝑔 = 𝐺) → (∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔) ↔ ∀𝑥𝑋 (𝑦 · 𝑥) = 0 ))
2019rabbidv 3187 . . . . . . . . 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 2673 . . . . . . . 8 ((𝐺𝑉𝑔 = 𝐺) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} = 𝐼)
2322eqeq2d 2631 . . . . . . 7 ((𝐺𝑉𝑔 = 𝐺) → (𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} ↔ 𝑖 = 𝐼))
2423biimpa 501 . . . . . 6 (((𝐺𝑉𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)}) → 𝑖 = 𝐼)
2524eqeq1d 2623 . . . . 5 (((𝐺𝑉𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)}) → (𝑖 = ∅ ↔ 𝐼 = ∅))
2624infeq1d 8380 . . . . 5 (((𝐺𝑉𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)}) → inf(𝑖, ℝ, < ) = inf(𝐼, ℝ, < ))
2725, 26ifbieq2d 4109 . . . 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 3558 . . 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 3210 . . 3 (𝐺𝑉𝐺 ∈ V)
30 c0ex 10031 . . . . 5 0 ∈ V
31 ltso 10115 . . . . . 6 < Or ℝ
3231infex 8396 . . . . 5 inf(𝐼, ℝ, < ) ∈ V
3330, 32ifex 4154 . . . 4 if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V
3433a1i 11 . . 3 (𝐺𝑉 → if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V)
353, 28, 29, 34fvmptd 6286 . 2 (𝐺𝑉 → (gEx‘𝐺) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
361, 35syl5eq 2667 1 (𝐺𝑉𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  wral 2911  {crab 2915  Vcvv 3198  csb 3531  c0 3913  ifcif 4084  cmpt 4727  cfv 5886  (class class class)co 6647  infcinf 8344  cr 9932  0cc0 9933   < clt 10071  cn 11017  Basecbs 15851  0gc0g 16094  .gcmg 17534  gExcgex 17939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-i2m1 10001  ax-1ne0 10002  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-om 7063  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-sup 8345  df-inf 8346  df-pnf 10073  df-mnf 10074  df-ltxr 10076  df-nn 11018  df-gex 17943
This theorem is referenced by:  gexlem1  17988  gexlem2  17991
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