Theorem odfvalALT 19627

Description: Shorter proof of odfval 19626 using ax-rep 5240. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

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
odfvalALT	⊢ 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 odfvalALT

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		odval.4	. 2 ⊢ 𝑂 = (od‘𝐺)
2		fveq2 6885	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		odval.1	. . . . . 6 ⊢ 𝑋 = (Base‘𝐺)
4	2, 3	eqtr4di 2818	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
5		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺))
6		odval.2	. . . . . . . . . 10 ⊢ · = (.g‘𝐺)
7	5, 6	eqtr4di 2818	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · )
8	7	oveqd 7433	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑦(.g‘𝑔)𝑥) = (𝑦 · 𝑥))
9		fveq2 6885	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
10		odval.3	. . . . . . . . 9 ⊢ 0 = (0g‘𝐺)
11	9, 10	eqtr4di 2818	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
12	8, 11	eqeq12d 2781	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑦(.g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 · 𝑥) = 0 ))
13	12	rabbidv 3425	. . . . . 6 ⊢ (𝑔 = 𝐺 → {𝑦 ∈ ℕ ∣ (𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 })
14	13	csbeq1d 3858	. . . . 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 5200	. . . 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 19622	. . . 4 ⊢ od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
17	15, 16, 3	mptfvmpt 7230	. . 3 ⊢ (𝐺 ∈ V → (od‘𝐺) = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
18		fvprc 6877	. . . 4 ⊢ (¬ 𝐺 ∈ V → (od‘𝐺) = ∅)
19		fvprc 6877	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
20	3, 19	eqtrid 2812	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → 𝑋 = ∅)
21	20	mpteq1d 5203	. . . . 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, < ))))
22		mpt0 6681	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅
23	21, 22	eqtrdi 2816	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅)
24	18, 23	eqtr4d 2803	. . 3 ⊢ (¬ 𝐺 ∈ V → (od‘𝐺) = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
25	17, 24	pm2.61i 184	. 2 ⊢ (od‘𝐺) = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
26	1, 25	eqtri 2788	1 ⊢ 𝑂 = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2146  {crab 3418  Vcvv 3457  ⦋csb 3854  ∅c0 4286  ifcif 4489   ↦ cmpt 5194  ‘cfv 6540  (class class class)co 7416  infcinf 9404  ℝcr 11110  0cc0 11111   < clt 11254  ℕcn 12244  Basecbs 17287  0_gc0g 17510  ._gcmg 19157  odcod 19618

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-od 19622

This theorem is used by: (None)