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Theorem odfval 19314
Description: Value of the order function. For a shorter proof using ax-rep 5242, see odfvalALT 19315. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.) Remove dependency on ax-rep 5242. (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 6842 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 odval.1 . . . . . 6 𝑋 = (Base‘𝐺)
42, 3eqtr4di 2794 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
5 fveq2 6842 . . . . . . . . . 10 (𝑔 = 𝐺 → (.g𝑔) = (.g𝐺))
6 odval.2 . . . . . . . . . 10 · = (.g𝐺)
75, 6eqtr4di 2794 . . . . . . . . 9 (𝑔 = 𝐺 → (.g𝑔) = · )
87oveqd 7374 . . . . . . . 8 (𝑔 = 𝐺 → (𝑦(.g𝑔)𝑥) = (𝑦 · 𝑥))
9 fveq2 6842 . . . . . . . . 9 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
10 odval.3 . . . . . . . . 9 0 = (0g𝐺)
119, 10eqtr4di 2794 . . . . . . . 8 (𝑔 = 𝐺 → (0g𝑔) = 0 )
128, 11eqeq12d 2752 . . . . . . 7 (𝑔 = 𝐺 → ((𝑦(.g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 · 𝑥) = 0 ))
1312rabbidv 3415 . . . . . 6 (𝑔 = 𝐺 → {𝑦 ∈ ℕ ∣ (𝑦(.g𝑔)𝑥) = (0g𝑔)} = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 })
1413csbeq1d 3859 . . . . 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 5196 . . . 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 19310 . . . 4 od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ {𝑦 ∈ ℕ ∣ (𝑦(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
173fvexi 6856 . . . . 5 𝑋 ∈ V
18 nn0ex 12419 . . . . 5 0 ∈ V
19 nnex 12159 . . . . . . . . 9 ℕ ∈ V
2019rabex 5289 . . . . . . . 8 {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ∈ V
21 eqeq1 2740 . . . . . . . . 9 (𝑖 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } → (𝑖 = ∅ ↔ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅))
22 infeq1 9412 . . . . . . . . 9 (𝑖 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } → inf(𝑖, ℝ, < ) = inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ))
2321, 22ifbieq2d 4512 . . . . . . . 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 3891 . . . . . . 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 12428 . . . . . . . . . 10 0 ∈ ℕ0
2625a1i 11 . . . . . . . . 9 ((⊤ ∧ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅) → 0 ∈ ℕ0)
27 df-ne 2944 . . . . . . . . . . . 12 ({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ≠ ∅ ↔ ¬ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅)
28 ssrab2 4037 . . . . . . . . . . . . 13 {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ⊆ ℕ
29 nnuz 12806 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
3028, 29sseqtri 3980 . . . . . . . . . . . . . 14 {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ⊆ (ℤ‘1)
31 infssuzcl 12857 . . . . . . . . . . . . . 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 3942 . . . . . . . . . . . 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 12473 . . . . . . . . . 10 (¬ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅ → inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ) ∈ ℕ0)
3635adantl 482 . . . . . . . . 9 ((⊤ ∧ ¬ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅) → inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ) ∈ ℕ0)
3726, 36ifclda 4521 . . . . . . . 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 3068 . . . . 5 𝑥𝑋 {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) ∈ ℕ0
4117, 18, 40mptexw 7885 . . . 4 (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) ∈ V
4215, 16, 41fvmpt 6948 . . 3 (𝐺 ∈ V → (od‘𝐺) = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
43 fvprc 6834 . . . 4 𝐺 ∈ V → (od‘𝐺) = ∅)
44 fvprc 6834 . . . . . . 7 𝐺 ∈ V → (Base‘𝐺) = ∅)
453, 44eqtrid 2788 . . . . . 6 𝐺 ∈ V → 𝑋 = ∅)
4645mpteq1d 5200 . . . . 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 6643 . . . . 5 (𝑥 ∈ ∅ ↦ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅
4846, 47eqtrdi 2792 . . . 4 𝐺 ∈ V → (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅)
4943, 48eqtr4d 2779 . . 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 2764 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 396   = wceq 1541  wtru 1542  wcel 2106  wne 2943  {crab 3407  Vcvv 3445  csb 3855  wss 3910  c0 4282  ifcif 4486  cmpt 5188  cfv 6496  (class class class)co 7357  infcinf 9377  cr 11050  0cc0 11051  1c1 11052   < clt 11189  cn 12153  0cn0 12413  cuz 12763  Basecbs 17083  0gc0g 17321  .gcmg 18872  odcod 19306
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-od 19310
This theorem is referenced by:  odval  19316  odf  19319
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