Theorem odfval 19429

Description: Value of the order function. For a shorter proof using ax-rep 5221, see odfvalALT 19430. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.) Remove dependency on ax-rep 5221. (Revised by Rohan Ridenour, 17-Aug-2023.)

Hypotheses

Ref	Expression
odval.1	⊢ 𝑋 = (Base‘𝐺)
odval.2	⊢ · = (.g‘𝐺)
odval.3	⊢ 0 = (0g‘𝐺)
odval.4	⊢ 𝑂 = (od‘𝐺)

Assertion

Ref	Expression
odfval	⊢ 𝑂 = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(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.

Step	Hyp	Ref	Expression
1		odval.4	. 2 ⊢ 𝑂 = (od‘𝐺)
2		fveq2 6826	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		odval.1	. . . . . 6 ⊢ 𝑋 = (Base‘𝐺)
4	2, 3	eqtr4di 2782	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
5		fveq2 6826	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺))
6		odval.2	. . . . . . . . . 10 ⊢ · = (.g‘𝐺)
7	5, 6	eqtr4di 2782	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · )
8	7	oveqd 7370	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑦(.g‘𝑔)𝑥) = (𝑦 · 𝑥))
9		fveq2 6826	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
10		odval.3	. . . . . . . . 9 ⊢ 0 = (0g‘𝐺)
11	9, 10	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
12	8, 11	eqeq12d 2745	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑦(.g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 · 𝑥) = 0 ))
13	12	rabbidv 3404	. . . . . 6 ⊢ (𝑔 = 𝐺 → {𝑦 ∈ ℕ ∣ (𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 })
14	13	csbeq1d 3857	. . . . 5 ⊢ (𝑔 = 𝐺 → ⦋{𝑦 ∈ ℕ ∣ (𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
15	4, 14	mpteq12dv 5182	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
16		df-od 19425	. . . 4 ⊢ od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
17	3	fvexi 6840	. . . . 5 ⊢ 𝑋 ∈ V
18		nn0ex 12408	. . . . 5 ⊢ ℕ0 ∈ V
19		nnex 12152	. . . . . . . . 9 ⊢ ℕ ∈ V
20	19	rabex 5281	. . . . . . . 8 ⊢ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ∈ V
21		eqeq1 2733	. . . . . . . . 9 ⊢ (𝑖 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } → (𝑖 = ∅ ↔ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅))
22		infeq1 9386	. . . . . . . . 9 ⊢ (𝑖 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } → inf(𝑖, ℝ, < ) = inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ))
23	21, 22	ifbieq2d 4505	. . . . . . . 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, < )))
24	20, 23	csbie 3888	. . . . . . 7 ⊢ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅, 0, inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ))
25		0nn0 12417	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
26	25	a1i 11	. . . . . . . . 9 ⊢ ((⊤ ∧ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅) → 0 ∈ ℕ0)
27		df-ne 2926	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ≠ ∅ ↔ ¬ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅)
28		ssrab2 4033	. . . . . . . . . . . . 13 ⊢ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ⊆ ℕ
29		nnuz 12796	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ≥‘1)
30	28, 29	sseqtri 3986	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ⊆ (ℤ≥‘1)
31		infssuzcl 12851	. . . . . . . . . . . . . 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 })
32	30, 31	mpan 690	. . . . . . . . . . . . 13 ⊢ ({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ≠ ∅ → inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ) ∈ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 })
33	28, 32	sselid 3935	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ≠ ∅ → inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ) ∈ ℕ)
34	27, 33	sylbir 235	. . . . . . . . . . 11 ⊢ (¬ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅ → inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ) ∈ ℕ)
35	34	nnnn0d 12463	. . . . . . . . . 10 ⊢ (¬ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅ → inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ) ∈ ℕ0)
36	35	adantl 481	. . . . . . . . 9 ⊢ ((⊤ ∧ ¬ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅) → inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ) ∈ ℕ0)
37	26, 36	ifclda 4514	. . . . . . . 8 ⊢ (⊤ → if({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅, 0, inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < )) ∈ ℕ0)
38	37	mptru 1547	. . . . . . 7 ⊢ if({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅, 0, inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < )) ∈ ℕ0
39	24, 38	eqeltri 2824	. . . . . 6 ⊢ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) ∈ ℕ0
40	39	rgenw 3048	. . . . 5 ⊢ ∀𝑥 ∈ 𝑋 ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) ∈ ℕ0
41	17, 18, 40	mptexw 7895	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) ∈ V
42	15, 16, 41	fvmpt 6934	. . 3 ⊢ (𝐺 ∈ V → (od‘𝐺) = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
43		fvprc 6818	. . . 4 ⊢ (¬ 𝐺 ∈ V → (od‘𝐺) = ∅)
44		fvprc 6818	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
45	3, 44	eqtrid 2776	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → 𝑋 = ∅)
46	45	mpteq1d 5185	. . . . 5 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = (𝑥 ∈ ∅ ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
47		mpt0 6628	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅
48	46, 47	eqtrdi 2780	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅)
49	43, 48	eqtr4d 2767	. . 3 ⊢ (¬ 𝐺 ∈ V → (od‘𝐺) = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
50	42, 49	pm2.61i 182	. 2 ⊢ (od‘𝐺) = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
51	1, 50	eqtri 2752	1 ⊢ 𝑂 = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109   ≠ wne 2925  {crab 3396  Vcvv 3438  ⦋csb 3853   ⊆ wss 3905  ∅c0 4286  ifcif 4478   ↦ cmpt 5176  ‘cfv 6486  (class class class)co 7353  infcinf 9350  ℝcr 11027  0cc0 11028  1c1 11029   < clt 11168  ℕcn 12146  ℕ₀cn0 12402  ℤ_≥cuz 12753  Basecbs 17138  0_gc0g 17361  ._gcmg 18964  odcod 19421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-sup 9351  df-inf 9352  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-n0 12403  df-z 12490  df-uz 12754  df-od 19425

This theorem is referenced by:  odval  19431  odf  19434