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Theorem odfvalALT 19396
Description: Shorter proof of odfval 19395 using ax-rep 5285. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
odval.1 𝑋 = (Base‘𝐺)
odval.2 · = (.g𝐺)
odval.3 0 = (0g𝐺)
odval.4 𝑂 = (od‘𝐺)
Assertion
Ref Expression
odfvalALT 𝑂 = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
Distinct variable groups:   𝑦,𝑖,𝑥   𝑥,𝐺,𝑦   𝑥, · ,𝑖,𝑦   𝑥, 0 ,𝑦,𝑖   𝑥,𝑋
Allowed substitution hints:   𝐺(𝑖)   𝑂(𝑥,𝑦,𝑖)   𝑋(𝑦,𝑖)

Proof of Theorem odfvalALT
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 odval.4 . 2 𝑂 = (od‘𝐺)
2 fveq2 6889 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 odval.1 . . . . . 6 𝑋 = (Base‘𝐺)
42, 3eqtr4di 2791 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
5 fveq2 6889 . . . . . . . . . 10 (𝑔 = 𝐺 → (.g𝑔) = (.g𝐺))
6 odval.2 . . . . . . . . . 10 · = (.g𝐺)
75, 6eqtr4di 2791 . . . . . . . . 9 (𝑔 = 𝐺 → (.g𝑔) = · )
87oveqd 7423 . . . . . . . 8 (𝑔 = 𝐺 → (𝑦(.g𝑔)𝑥) = (𝑦 · 𝑥))
9 fveq2 6889 . . . . . . . . 9 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
10 odval.3 . . . . . . . . 9 0 = (0g𝐺)
119, 10eqtr4di 2791 . . . . . . . 8 (𝑔 = 𝐺 → (0g𝑔) = 0 )
128, 11eqeq12d 2749 . . . . . . 7 (𝑔 = 𝐺 → ((𝑦(.g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 · 𝑥) = 0 ))
1312rabbidv 3441 . . . . . 6 (𝑔 = 𝐺 → {𝑦 ∈ ℕ ∣ (𝑦(.g𝑔)𝑥) = (0g𝑔)} = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 })
1413csbeq1d 3897 . . . . 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 5239 . . . 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 19391 . . . 4 od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ {𝑦 ∈ ℕ ∣ (𝑦(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
1715, 16, 3mptfvmpt 7227 . . 3 (𝐺 ∈ V → (od‘𝐺) = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
18 fvprc 6881 . . . 4 𝐺 ∈ V → (od‘𝐺) = ∅)
19 fvprc 6881 . . . . . . 7 𝐺 ∈ V → (Base‘𝐺) = ∅)
203, 19eqtrid 2785 . . . . . 6 𝐺 ∈ V → 𝑋 = ∅)
2120mpteq1d 5243 . . . . 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, < ))))
22 mpt0 6690 . . . . 5 (𝑥 ∈ ∅ ↦ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅
2321, 22eqtrdi 2789 . . . 4 𝐺 ∈ V → (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅)
2418, 23eqtr4d 2776 . . 3 𝐺 ∈ V → (od‘𝐺) = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
2517, 24pm2.61i 182 . 2 (od‘𝐺) = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
261, 25eqtri 2761 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 1542  wcel 2107  {crab 3433  Vcvv 3475  csb 3893  c0 4322  ifcif 4528  cmpt 5231  cfv 6541  (class class class)co 7406  infcinf 9433  cr 11106  0cc0 11107   < clt 11245  cn 12209  Basecbs 17141  0gc0g 17382  .gcmg 18945  odcod 19387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-od 19391
This theorem is referenced by: (None)
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