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Theorem odval 17999
 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 6698 . . . . . . 7 (𝑥 = 𝐴 → (𝑦 · 𝑥) = (𝑦 · 𝐴))
21eqeq1d 2653 . . . . . 6 (𝑥 = 𝐴 → ((𝑦 · 𝑥) = 0 ↔ (𝑦 · 𝐴) = 0 ))
32rabbidv 3220 . . . . 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 2703 . . . 4 (𝑥 = 𝐴 → {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = 𝐼)
65csbeq1d 3573 . . 3 (𝑥 = 𝐴{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = 𝐼 / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
7 nnex 11064 . . . . 5 ℕ ∈ V
84, 7rabex2 4847 . . . 4 𝐼 ∈ V
9 eqeq1 2655 . . . . 5 (𝑖 = 𝐼 → (𝑖 = ∅ ↔ 𝐼 = ∅))
10 infeq1 8423 . . . . 5 (𝑖 = 𝐼 → inf(𝑖, ℝ, < ) = inf(𝐼, ℝ, < ))
119, 10ifbieq2d 4144 . . . 4 (𝑖 = 𝐼 → if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
128, 11csbie 3592 . . 3 𝐼 / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))
136, 12syl6eq 2701 . 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 17998 . 2 𝑂 = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
19 c0ex 10072 . . 3 0 ∈ V
20 ltso 10156 . . . 4 < Or ℝ
2120infex 8440 . . 3 inf(𝐼, ℝ, < ) ∈ V
2219, 21ifex 4189 . 2 if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V
2313, 18, 22fvmpt 6321 1 (𝐴𝑋 → (𝑂𝐴) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  {crab 2945  ⦋csb 3566  ∅c0 3948  ifcif 4119  ‘cfv 5926  (class class class)co 6690  infcinf 8388  ℝcr 9973  0cc0 9974   < clt 10112  ℕcn 11058  Basecbs 15904  0gc0g 16147  .gcmg 17587  odcod 17990 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-i2m1 10042  ax-1ne0 10043  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-ltxr 10117  df-nn 11059  df-od 17994 This theorem is referenced by:  odlem1  18000  odlem2  18004  submod  18030  ofldchr  29942
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