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Theorem gexval 19610
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 19561 . . 3 gEx = (𝑔 ∈ V ↦ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
3 nnex 12269 . . . . . 6 ℕ ∈ V
43rabex 5344 . . . . 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 2792 . . . . . . . . . . 11 ((𝐺𝑉𝑔 = 𝐺) → (Base‘𝑔) = 𝑋)
106fveq2d 6910 . . . . . . . . . . . . . 14 ((𝐺𝑉𝑔 = 𝐺) → (.g𝑔) = (.g𝐺))
11 gexval.2 . . . . . . . . . . . . . 14 · = (.g𝐺)
1210, 11eqtr4di 2792 . . . . . . . . . . . . 13 ((𝐺𝑉𝑔 = 𝐺) → (.g𝑔) = · )
1312oveqd 7447 . . . . . . . . . . . 12 ((𝐺𝑉𝑔 = 𝐺) → (𝑦(.g𝑔)𝑥) = (𝑦 · 𝑥))
146fveq2d 6910 . . . . . . . . . . . . 13 ((𝐺𝑉𝑔 = 𝐺) → (0g𝑔) = (0g𝐺))
15 gexval.3 . . . . . . . . . . . . 13 0 = (0g𝐺)
1614, 15eqtr4di 2792 . . . . . . . . . . . 12 ((𝐺𝑉𝑔 = 𝐺) → (0g𝑔) = 0 )
1713, 16eqeq12d 2750 . . . . . . . . . . 11 ((𝐺𝑉𝑔 = 𝐺) → ((𝑦(.g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 · 𝑥) = 0 ))
189, 17raleqbidv 3343 . . . . . . . . . 10 ((𝐺𝑉𝑔 = 𝐺) → (∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔) ↔ ∀𝑥𝑋 (𝑦 · 𝑥) = 0 ))
1918rabbidv 3440 . . . . . . . . 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 2792 . . . . . . . 8 ((𝐺𝑉𝑔 = 𝐺) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)} = 𝐼)
2221eqeq2d 2745 . . . . . . 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 2736 . . . . 5 (((𝐺𝑉𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)}) → (𝑖 = ∅ ↔ 𝐼 = ∅))
2523infeq1d 9514 . . . . 5 (((𝐺𝑉𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g𝑔)𝑥) = (0g𝑔)}) → inf(𝑖, ℝ, < ) = inf(𝐼, ℝ, < ))
2624, 25ifbieq2d 4556 . . . 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 3945 . . 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 3498 . . 3 (𝐺𝑉𝐺 ∈ V)
29 c0ex 11252 . . . . 5 0 ∈ V
30 ltso 11338 . . . . . 6 < Or ℝ
3130infex 9530 . . . . 5 inf(𝐼, ℝ, < ) ∈ V
3229, 31ifex 4580 . . . 4 if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V
3332a1i 11 . . 3 (𝐺𝑉 → if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V)
342, 27, 28, 33fvmptd2 7023 . 2 (𝐺𝑉 → (gEx‘𝐺) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
351, 34eqtrid 2786 1 (𝐺𝑉𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wral 3058  {crab 3432  Vcvv 3477  csb 3907  c0 4338  ifcif 4530  cfv 6562  (class class class)co 7430  infcinf 9478  cr 11151  0cc0 11152   < clt 11292  cn 12263  Basecbs 17244  0gc0g 17485  .gcmg 19097  gExcgex 19557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-mulcl 11214  ax-i2m1 11220  ax-pre-lttri 11226  ax-pre-lttrn 11227
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-ltxr 11297  df-nn 12264  df-gex 19561
This theorem is referenced by:  gexlem1  19611  gexlem2  19614
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