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Theorem odfval 19237
Description: Value of the order function. For a shorter proof using ax-rep 5234, see odfvalALT 19238. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.) Remove dependency on ax-rep 5234. (Revised by Rohan Ridenour, 17-Aug-2023.)
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, · ,𝑖,𝑦   𝑥, 0 ,𝑦,𝑖   𝑥,𝑋
Allowed substitution hints:   𝐺(𝑖)   𝑂(𝑥,𝑦,𝑖)   𝑋(𝑦,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 6830 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 odval.1 . . . . . 6 𝑋 = (Base‘𝐺)
42, 3eqtr4di 2795 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
5 fveq2 6830 . . . . . . . . . 10 (𝑔 = 𝐺 → (.g𝑔) = (.g𝐺))
6 odval.2 . . . . . . . . . 10 · = (.g𝐺)
75, 6eqtr4di 2795 . . . . . . . . 9 (𝑔 = 𝐺 → (.g𝑔) = · )
87oveqd 7359 . . . . . . . 8 (𝑔 = 𝐺 → (𝑦(.g𝑔)𝑥) = (𝑦 · 𝑥))
9 fveq2 6830 . . . . . . . . 9 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
10 odval.3 . . . . . . . . 9 0 = (0g𝐺)
119, 10eqtr4di 2795 . . . . . . . 8 (𝑔 = 𝐺 → (0g𝑔) = 0 )
128, 11eqeq12d 2753 . . . . . . 7 (𝑔 = 𝐺 → ((𝑦(.g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 · 𝑥) = 0 ))
1312rabbidv 3412 . . . . . 6 (𝑔 = 𝐺 → {𝑦 ∈ ℕ ∣ (𝑦(.g𝑔)𝑥) = (0g𝑔)} = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 })
1413csbeq1d 3851 . . . . 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 5188 . . . 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 19233 . . . 4 od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ {𝑦 ∈ ℕ ∣ (𝑦(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
173fvexi 6844 . . . . 5 𝑋 ∈ V
18 nn0ex 12345 . . . . 5 0 ∈ V
19 nnex 12085 . . . . . . . . 9 ℕ ∈ V
2019rabex 5281 . . . . . . . 8 {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ∈ V
21 eqeq1 2741 . . . . . . . . 9 (𝑖 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } → (𝑖 = ∅ ↔ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅))
22 infeq1 9338 . . . . . . . . 9 (𝑖 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } → inf(𝑖, ℝ, < ) = inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ))
2321, 22ifbieq2d 4504 . . . . . . . 8 (𝑖 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } → if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅, 0, inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < )))
2420, 23csbie 3883 . . . . . . 7 {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅, 0, inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ))
25 0nn0 12354 . . . . . . . . . 10 0 ∈ ℕ0
2625a1i 11 . . . . . . . . 9 ((⊤ ∧ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅) → 0 ∈ ℕ0)
27 df-ne 2942 . . . . . . . . . . . 12 ({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ≠ ∅ ↔ ¬ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅)
28 ssrab2 4029 . . . . . . . . . . . . 13 {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ⊆ ℕ
29 nnuz 12727 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
3028, 29sseqtri 3972 . . . . . . . . . . . . . 14 {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ⊆ (ℤ‘1)
31 infssuzcl 12778 . . . . . . . . . . . . . 14 (({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ⊆ (ℤ‘1) ∧ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ≠ ∅) → inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ) ∈ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 })
3230, 31mpan 688 . . . . . . . . . . . . 13 ({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ≠ ∅ → inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ) ∈ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 })
3328, 32sselid 3934 . . . . . . . . . . . 12 ({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ≠ ∅ → inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ) ∈ ℕ)
3427, 33sylbir 234 . . . . . . . . . . 11 (¬ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅ → inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ) ∈ ℕ)
3534nnnn0d 12399 . . . . . . . . . 10 (¬ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅ → inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ) ∈ ℕ0)
3635adantl 483 . . . . . . . . 9 ((⊤ ∧ ¬ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅) → inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ) ∈ ℕ0)
3726, 36ifclda 4513 . . . . . . . 8 (⊤ → if({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅, 0, inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < )) ∈ ℕ0)
3837mptru 1548 . . . . . . 7 if({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅, 0, inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < )) ∈ ℕ0
3924, 38eqeltri 2834 . . . . . 6 {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) ∈ ℕ0
4039rgenw 3066 . . . . 5 𝑥𝑋 {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) ∈ ℕ0
4117, 18, 40mptexw 7868 . . . 4 (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) ∈ V
4215, 16, 41fvmpt 6936 . . 3 (𝐺 ∈ V → (od‘𝐺) = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
43 fvprc 6822 . . . 4 𝐺 ∈ V → (od‘𝐺) = ∅)
44 fvprc 6822 . . . . . . 7 𝐺 ∈ V → (Base‘𝐺) = ∅)
453, 44eqtrid 2789 . . . . . 6 𝐺 ∈ V → 𝑋 = ∅)
4645mpteq1d 5192 . . . . 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, < ))))
47 mpt0 6631 . . . . 5 (𝑥 ∈ ∅ ↦ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅
4846, 47eqtrdi 2793 . . . 4 𝐺 ∈ V → (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅)
4943, 48eqtr4d 2780 . . 3 𝐺 ∈ V → (od‘𝐺) = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
5042, 49pm2.61i 182 . 2 (od‘𝐺) = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
511, 50eqtri 2765 1 𝑂 = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397   = wceq 1541  wtru 1542  wcel 2106  wne 2941  {crab 3404  Vcvv 3442  csb 3847  wss 3902  c0 4274  ifcif 4478  cmpt 5180  cfv 6484  (class class class)co 7342  infcinf 9303  cr 10976  0cc0 10977  1c1 10978   < clt 11115  cn 12079  0cn0 12339  cuz 12688  Basecbs 17010  0gc0g 17248  .gcmg 18797  odcod 19229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655  ax-cnex 11033  ax-resscn 11034  ax-1cn 11035  ax-icn 11036  ax-addcl 11037  ax-addrcl 11038  ax-mulcl 11039  ax-mulrcl 11040  ax-mulcom 11041  ax-addass 11042  ax-mulass 11043  ax-distr 11044  ax-i2m1 11045  ax-1ne0 11046  ax-1rid 11047  ax-rnegex 11048  ax-rrecex 11049  ax-cnre 11050  ax-pre-lttri 11051  ax-pre-lttrn 11052  ax-pre-ltadd 11053  ax-pre-mulgt0 11054
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5181  df-tr 5215  df-id 5523  df-eprel 5529  df-po 5537  df-so 5538  df-fr 5580  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6243  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-riota 7298  df-ov 7345  df-oprab 7346  df-mpo 7347  df-om 7786  df-2nd 7905  df-frecs 8172  df-wrecs 8203  df-recs 8277  df-rdg 8316  df-er 8574  df-en 8810  df-dom 8811  df-sdom 8812  df-sup 9304  df-inf 9305  df-pnf 11117  df-mnf 11118  df-xr 11119  df-ltxr 11120  df-le 11121  df-sub 11313  df-neg 11314  df-nn 12080  df-n0 12340  df-z 12426  df-uz 12689  df-od 19233
This theorem is referenced by:  odval  19239  odf  19242
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