Theorem odfval 19498

Description: Value of the order function. For a shorter proof using ax-rep 5212, see odfvalALT 19499. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.) Remove dependency on ax-rep 5212. (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 6834	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		odval.1	. . . . . 6 ⊢ 𝑋 = (Base‘𝐺)
4	2, 3	eqtr4di 2790	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
5		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺))
6		odval.2	. . . . . . . . . 10 ⊢ · = (.g‘𝐺)
7	5, 6	eqtr4di 2790	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · )
8	7	oveqd 7377	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑦(.g‘𝑔)𝑥) = (𝑦 · 𝑥))
9		fveq2 6834	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
10		odval.3	. . . . . . . . 9 ⊢ 0 = (0g‘𝐺)
11	9, 10	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
12	8, 11	eqeq12d 2753	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑦(.g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 · 𝑥) = 0 ))
13	12	rabbidv 3397	. . . . . 6 ⊢ (𝑔 = 𝐺 → {𝑦 ∈ ℕ ∣ (𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 })
14	13	csbeq1d 3842	. . . . 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 5173	. . . 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 19494	. . . 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 6848	. . . . 5 ⊢ 𝑋 ∈ V
18		nn0ex 12434	. . . . 5 ⊢ ℕ0 ∈ V
19		nnex 12171	. . . . . . . . 9 ⊢ ℕ ∈ V
20	19	rabex 5276	. . . . . . . 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 9383	. . . . . . . . 9 ⊢ (𝑖 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } → inf(𝑖, ℝ, < ) = inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ))
23	21, 22	ifbieq2d 4494	. . . . . . . 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 3873	. . . . . . 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 12443	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
26	25	a1i 11	. . . . . . . . 9 ⊢ ((⊤ ∧ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅) → 0 ∈ ℕ0)
27		df-ne 2934	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ≠ ∅ ↔ ¬ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅)
28		ssrab2 4021	. . . . . . . . . . . . 13 ⊢ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ⊆ ℕ
29		nnuz 12818	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ≥‘1)
30	28, 29	sseqtri 3971	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ⊆ (ℤ≥‘1)
31		infssuzcl 12873	. . . . . . . . . . . . . 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 691	. . . . . . . . . . . . 13 ⊢ ({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ≠ ∅ → inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ) ∈ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 })
33	28, 32	sselid 3920	. . . . . . . . . . . 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 12489	. . . . . . . . . 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 4503	. . . . . . . 8 ⊢ (⊤ → if({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅, 0, inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < )) ∈ ℕ0)
38	37	mptru 1549	. . . . . . 7 ⊢ if({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅, 0, inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < )) ∈ ℕ0
39	24, 38	eqeltri 2833	. . . . . 6 ⊢ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) ∈ ℕ0
40	39	rgenw 3056	. . . . 5 ⊢ ∀𝑥 ∈ 𝑋 ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) ∈ ℕ0
41	17, 18, 40	mptexw 7899	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) ∈ V
42	15, 16, 41	fvmpt 6941	. . 3 ⊢ (𝐺 ∈ V → (od‘𝐺) = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
43		fvprc 6826	. . . 4 ⊢ (¬ 𝐺 ∈ V → (od‘𝐺) = ∅)
44		fvprc 6826	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
45	3, 44	eqtrid 2784	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → 𝑋 = ∅)
46	45	mpteq1d 5176	. . . . 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 6634	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅
48	46, 47	eqtrdi 2788	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅)
49	43, 48	eqtr4d 2775	. . 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 2760	1 ⊢ 𝑂 = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114   ≠ wne 2933  {crab 3390  Vcvv 3430  ⦋csb 3838   ⊆ wss 3890  ∅c0 4274  ifcif 4467   ↦ cmpt 5167  ‘cfv 6492  (class class class)co 7360  infcinf 9347  ℝcr 11028  0cc0 11029  1c1 11030   < clt 11170  ℕcn 12165  ℕ₀cn0 12428  ℤ_≥cuz 12779  Basecbs 17170  0_gc0g 17393  ._gcmg 19034  odcod 19490

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9348  df-inf 9349  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-od 19494

This theorem is referenced by:  odval  19500  odf  19503