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Theorem odfval 18158
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 6332 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 odval.1 . . . . . 6 𝑋 = (Base‘𝐺)
42, 3syl6eqr 2822 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
5 fveq2 6332 . . . . . . . . . 10 (𝑔 = 𝐺 → (.g𝑔) = (.g𝐺))
6 odval.2 . . . . . . . . . 10 · = (.g𝐺)
75, 6syl6eqr 2822 . . . . . . . . 9 (𝑔 = 𝐺 → (.g𝑔) = · )
87oveqd 6809 . . . . . . . 8 (𝑔 = 𝐺 → (𝑦(.g𝑔)𝑥) = (𝑦 · 𝑥))
9 fveq2 6332 . . . . . . . . 9 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
10 odval.3 . . . . . . . . 9 0 = (0g𝐺)
119, 10syl6eqr 2822 . . . . . . . 8 (𝑔 = 𝐺 → (0g𝑔) = 0 )
128, 11eqeq12d 2785 . . . . . . 7 (𝑔 = 𝐺 → ((𝑦(.g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 · 𝑥) = 0 ))
1312rabbidv 3338 . . . . . 6 (𝑔 = 𝐺 → {𝑦 ∈ ℕ ∣ (𝑦(.g𝑔)𝑥) = (0g𝑔)} = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 })
1413csbeq1d 3687 . . . . 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 4865 . . . 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 18154 . . . 4 od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ {𝑦 ∈ ℕ ∣ (𝑦(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
17 fvex 6342 . . . . . 6 (Base‘𝐺) ∈ V
183, 17eqeltri 2845 . . . . 5 𝑋 ∈ V
1918mptex 6629 . . . 4 (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) ∈ V
2015, 16, 19fvmpt 6424 . . 3 (𝐺 ∈ V → (od‘𝐺) = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
21 fvprc 6326 . . . 4 𝐺 ∈ V → (od‘𝐺) = ∅)
22 fvprc 6326 . . . . . . 7 𝐺 ∈ V → (Base‘𝐺) = ∅)
233, 22syl5eq 2816 . . . . . 6 𝐺 ∈ V → 𝑋 = ∅)
2423mpteq1d 4870 . . . . 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 6161 . . . . 5 (𝑥 ∈ ∅ ↦ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅
2624, 25syl6eq 2820 . . . 4 𝐺 ∈ V → (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅)
2721, 26eqtr4d 2807 . . 3 𝐺 ∈ V → (od‘𝐺) = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
2820, 27pm2.61i 176 . 2 (od‘𝐺) = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
291, 28eqtri 2792 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 1630  wcel 2144  {crab 3064  Vcvv 3349  csb 3680  c0 4061  ifcif 4223  cmpt 4861  cfv 6031  (class class class)co 6792  infcinf 8502  cr 10136  0cc0 10137   < clt 10275  cn 11221  Basecbs 16063  0gc0g 16307  .gcmg 17747  odcod 18150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  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-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 6795  df-od 18154
This theorem is referenced by:  odval  18159  odf  18162
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