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Theorem odval 18160
Description: Second substitution for the group order definition. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 5-Sep-2015.) (Revised by AV, 5-Oct-2020.)
Hypotheses
Ref Expression
odval.1 𝑋 = (Base‘𝐺)
odval.2 · = (.g𝐺)
odval.3 0 = (0g𝐺)
odval.4 𝑂 = (od‘𝐺)
odval.i 𝐼 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 }
Assertion
Ref Expression
odval (𝐴𝑋 → (𝑂𝐴) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐺   𝑦, ·   𝑦, 0
Allowed substitution hints:   𝐼(𝑦)   𝑂(𝑦)   𝑋(𝑦)

Proof of Theorem odval
Dummy variables 𝑖 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6801 . . . . . . 7 (𝑥 = 𝐴 → (𝑦 · 𝑥) = (𝑦 · 𝐴))
21eqeq1d 2773 . . . . . 6 (𝑥 = 𝐴 → ((𝑦 · 𝑥) = 0 ↔ (𝑦 · 𝐴) = 0 ))
32rabbidv 3339 . . . . 5 (𝑥 = 𝐴 → {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 })
4 odval.i . . . . 5 𝐼 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 }
53, 4syl6eqr 2823 . . . 4 (𝑥 = 𝐴 → {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = 𝐼)
65csbeq1d 3689 . . 3 (𝑥 = 𝐴{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = 𝐼 / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
7 nnex 11228 . . . . 5 ℕ ∈ V
84, 7rabex2 4948 . . . 4 𝐼 ∈ V
9 eqeq1 2775 . . . . 5 (𝑖 = 𝐼 → (𝑖 = ∅ ↔ 𝐼 = ∅))
10 infeq1 8538 . . . . 5 (𝑖 = 𝐼 → inf(𝑖, ℝ, < ) = inf(𝐼, ℝ, < ))
119, 10ifbieq2d 4250 . . . 4 (𝑖 = 𝐼 → if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
128, 11csbie 3708 . . 3 𝐼 / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))
136, 12syl6eq 2821 . 2 (𝑥 = 𝐴{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
14 odval.1 . . 3 𝑋 = (Base‘𝐺)
15 odval.2 . . 3 · = (.g𝐺)
16 odval.3 . . 3 0 = (0g𝐺)
17 odval.4 . . 3 𝑂 = (od‘𝐺)
1814, 15, 16, 17odfval 18159 . 2 𝑂 = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
19 c0ex 10236 . . 3 0 ∈ V
20 ltso 10320 . . . 4 < Or ℝ
2120infex 8555 . . 3 inf(𝐼, ℝ, < ) ∈ V
2219, 21ifex 4295 . 2 if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V
2313, 18, 22fvmpt 6424 1 (𝐴𝑋 → (𝑂𝐴) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  {crab 3065  csb 3682  c0 4063  ifcif 4225  cfv 6031  (class class class)co 6793  infcinf 8503  cr 10137  0cc0 10138   < clt 10276  cn 11222  Basecbs 16064  0gc0g 16308  .gcmg 17748  odcod 18151
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-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-i2m1 10206  ax-1ne0 10207  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  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 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-sup 8504  df-inf 8505  df-pnf 10278  df-mnf 10279  df-ltxr 10281  df-nn 11223  df-od 18155
This theorem is referenced by:  odlem1  18161  odlem2  18165  submod  18191  ofldchr  30154
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