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Theorem odval 18266
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 6886 . . . . . . 7 (𝑥 = 𝐴 → (𝑦 · 𝑥) = (𝑦 · 𝐴))
21eqeq1d 2801 . . . . . 6 (𝑥 = 𝐴 → ((𝑦 · 𝑥) = 0 ↔ (𝑦 · 𝐴) = 0 ))
32rabbidv 3373 . . . . 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 2851 . . . 4 (𝑥 = 𝐴 → {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = 𝐼)
65csbeq1d 3735 . . 3 (𝑥 = 𝐴{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = 𝐼 / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
7 nnex 11319 . . . . 5 ℕ ∈ V
84, 7rabex2 5009 . . . 4 𝐼 ∈ V
9 eqeq1 2803 . . . . 5 (𝑖 = 𝐼 → (𝑖 = ∅ ↔ 𝐼 = ∅))
10 infeq1 8624 . . . . 5 (𝑖 = 𝐼 → inf(𝑖, ℝ, < ) = inf(𝐼, ℝ, < ))
119, 10ifbieq2d 4302 . . . 4 (𝑖 = 𝐼 → if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
128, 11csbie 3754 . . 3 𝐼 / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))
136, 12syl6eq 2849 . 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 18265 . 2 𝑂 = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
19 c0ex 10322 . . 3 0 ∈ V
20 ltso 10408 . . . 4 < Or ℝ
2120infex 8641 . . 3 inf(𝐼, ℝ, < ) ∈ V
2219, 21ifex 4325 . 2 if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V
2313, 18, 22fvmpt 6507 1 (𝐴𝑋 → (𝑂𝐴) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  {crab 3093  csb 3728  c0 4115  ifcif 4277  cfv 6101  (class class class)co 6878  infcinf 8589  cr 10223  0cc0 10224   < clt 10363  cn 11312  Basecbs 16184  0gc0g 16415  .gcmg 17856  odcod 18257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-mulcl 10286  ax-i2m1 10292  ax-pre-lttri 10298  ax-pre-lttrn 10299
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-sup 8590  df-inf 8591  df-pnf 10365  df-mnf 10366  df-ltxr 10368  df-nn 11313  df-od 18261
This theorem is referenced by:  odlem1  18267  odlem2  18271  submod  18297  ofldchr  30330
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