Theorem odfval 19469

Description: Value of the order function. For a shorter proof using ax-rep 5237, see odfvalALT 19470. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.) Remove dependency on ax-rep 5237. (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 6861	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		odval.1	. . . . . 6 ⊢ 𝑋 = (Base‘𝐺)
4	2, 3	eqtr4di 2783	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
5		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺))
6		odval.2	. . . . . . . . . 10 ⊢ · = (.g‘𝐺)
7	5, 6	eqtr4di 2783	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · )
8	7	oveqd 7407	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑦(.g‘𝑔)𝑥) = (𝑦 · 𝑥))
9		fveq2 6861	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
10		odval.3	. . . . . . . . 9 ⊢ 0 = (0g‘𝐺)
11	9, 10	eqtr4di 2783	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
12	8, 11	eqeq12d 2746	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑦(.g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 · 𝑥) = 0 ))
13	12	rabbidv 3416	. . . . . 6 ⊢ (𝑔 = 𝐺 → {𝑦 ∈ ℕ ∣ (𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 })
14	13	csbeq1d 3869	. . . . 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 5197	. . . 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 19465	. . . 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 6875	. . . . 5 ⊢ 𝑋 ∈ V
18		nn0ex 12455	. . . . 5 ⊢ ℕ0 ∈ V
19		nnex 12199	. . . . . . . . 9 ⊢ ℕ ∈ V
20	19	rabex 5297	. . . . . . . 8 ⊢ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ∈ V
21		eqeq1 2734	. . . . . . . . 9 ⊢ (𝑖 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } → (𝑖 = ∅ ↔ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅))
22		infeq1 9435	. . . . . . . . 9 ⊢ (𝑖 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } → inf(𝑖, ℝ, < ) = inf({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 }, ℝ, < ))
23	21, 22	ifbieq2d 4518	. . . . . . . 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 3900	. . . . . . 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 12464	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
26	25	a1i 11	. . . . . . . . 9 ⊢ ((⊤ ∧ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅) → 0 ∈ ℕ0)
27		df-ne 2927	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ≠ ∅ ↔ ¬ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = ∅)
28		ssrab2 4046	. . . . . . . . . . . . 13 ⊢ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ⊆ ℕ
29		nnuz 12843	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ≥‘1)
30	28, 29	sseqtri 3998	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } ⊆ (ℤ≥‘1)
31		infssuzcl 12898	. . . . . . . . . . . . . 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 3947	. . . . . . . . . . . 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 12510	. . . . . . . . . 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 4527	. . . . . . . 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 2825	. . . . . 6 ⊢ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) ∈ ℕ0
40	39	rgenw 3049	. . . . 5 ⊢ ∀𝑥 ∈ 𝑋 ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) ∈ ℕ0
41	17, 18, 40	mptexw 7934	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) ∈ V
42	15, 16, 41	fvmpt 6971	. . 3 ⊢ (𝐺 ∈ V → (od‘𝐺) = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
43		fvprc 6853	. . . 4 ⊢ (¬ 𝐺 ∈ V → (od‘𝐺) = ∅)
44		fvprc 6853	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
45	3, 44	eqtrid 2777	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → 𝑋 = ∅)
46	45	mpteq1d 5200	. . . . 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 6663	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅
48	46, 47	eqtrdi 2781	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅)
49	43, 48	eqtr4d 2768	. . 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 2753	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 2926  {crab 3408  Vcvv 3450  ⦋csb 3865   ⊆ wss 3917  ∅c0 4299  ifcif 4491   ↦ cmpt 5191  ‘cfv 6514  (class class class)co 7390  infcinf 9399  ℝcr 11074  0cc0 11075  1c1 11076   < clt 11215  ℕcn 12193  ℕ₀cn0 12449  ℤ_≥cuz 12800  Basecbs 17186  0_gc0g 17409  ._gcmg 19006  odcod 19461

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-sup 9400  df-inf 9401  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-n0 12450  df-z 12537  df-uz 12801  df-od 19465

This theorem is referenced by:  odval  19471  odf  19474