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Theorem odfval 17775
 Description: Value of the order function. (Contributed by Mario Carneiro, 13-Jul-2014.) (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)
Assertion
Ref Expression
odfval 𝑂 = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
Distinct variable groups:   𝑦,𝑖,𝑥   𝑥,𝐺,𝑦   𝑥, · ,𝑦   𝑥, 0 ,𝑦   𝑥,𝑖   𝑥,𝑋
Allowed substitution hints:   · (𝑖)   𝐺(𝑖)   𝑂(𝑥,𝑦,𝑖)   𝑋(𝑦,𝑖)   0 (𝑖)

Proof of Theorem odfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 odval.4 . 2 𝑂 = (od‘𝐺)
2 fveq2 6103 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 odval.1 . . . . . 6 𝑋 = (Base‘𝐺)
42, 3syl6eqr 2662 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
5 fveq2 6103 . . . . . . . . . 10 (𝑔 = 𝐺 → (.g𝑔) = (.g𝐺))
6 odval.2 . . . . . . . . . 10 · = (.g𝐺)
75, 6syl6eqr 2662 . . . . . . . . 9 (𝑔 = 𝐺 → (.g𝑔) = · )
87oveqd 6566 . . . . . . . 8 (𝑔 = 𝐺 → (𝑦(.g𝑔)𝑥) = (𝑦 · 𝑥))
9 fveq2 6103 . . . . . . . . 9 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
10 odval.3 . . . . . . . . 9 0 = (0g𝐺)
119, 10syl6eqr 2662 . . . . . . . 8 (𝑔 = 𝐺 → (0g𝑔) = 0 )
128, 11eqeq12d 2625 . . . . . . 7 (𝑔 = 𝐺 → ((𝑦(.g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 · 𝑥) = 0 ))
1312rabbidv 3164 . . . . . 6 (𝑔 = 𝐺 → {𝑦 ∈ ℕ ∣ (𝑦(.g𝑔)𝑥) = (0g𝑔)} = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 })
1413csbeq1d 3506 . . . . 5 (𝑔 = 𝐺{𝑦 ∈ ℕ ∣ (𝑦(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
154, 14mpteq12dv 4663 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ {𝑦 ∈ ℕ ∣ (𝑦(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
16 df-od 17771 . . . 4 od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ {𝑦 ∈ ℕ ∣ (𝑦(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
17 fvex 6113 . . . . . 6 (Base‘𝐺) ∈ V
183, 17eqeltri 2684 . . . . 5 𝑋 ∈ V
1918mptex 6390 . . . 4 (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) ∈ V
2015, 16, 19fvmpt 6191 . . 3 (𝐺 ∈ V → (od‘𝐺) = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
21 fvprc 6097 . . . 4 𝐺 ∈ V → (od‘𝐺) = ∅)
22 fvprc 6097 . . . . . . 7 𝐺 ∈ V → (Base‘𝐺) = ∅)
233, 22syl5eq 2656 . . . . . 6 𝐺 ∈ V → 𝑋 = ∅)
2423mpteq1d 4666 . . . . 5 𝐺 ∈ V → (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = (𝑥 ∈ ∅ ↦ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
25 mpt0 5934 . . . . 5 (𝑥 ∈ ∅ ↦ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅
2624, 25syl6eq 2660 . . . 4 𝐺 ∈ V → (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅)
2721, 26eqtr4d 2647 . . 3 𝐺 ∈ V → (od‘𝐺) = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
2820, 27pm2.61i 175 . 2 (od‘𝐺) = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
291, 28eqtri 2632 1 𝑂 = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ⦋csb 3499  ∅c0 3874  ifcif 4036   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  infcinf 8230  ℝcr 9814  0cc0 9815   < clt 9953  ℕcn 10897  Basecbs 15695  0gc0g 15923  .gcmg 17363  odcod 17767 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-od 17771 This theorem is referenced by:  odval  17776  odf  17779
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