Theorem odfvalALT 19599

Description: Shorter proof of odfval 19598 using ax-rep 5239. (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 6879	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		odval.1	. . . . . 6 ⊢ 𝑋 = (Base‘𝐺)
4	2, 3	eqtr4di 2822	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
5		fveq2 6879	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺))
6		odval.2	. . . . . . . . . 10 ⊢ · = (.g‘𝐺)
7	5, 6	eqtr4di 2822	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · )
8	7	oveqd 7425	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑦(.g‘𝑔)𝑥) = (𝑦 · 𝑥))
9		fveq2 6879	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
10		odval.3	. . . . . . . . 9 ⊢ 0 = (0g‘𝐺)
11	9, 10	eqtr4di 2822	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
12	8, 11	eqeq12d 2785	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑦(.g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 · 𝑥) = 0 ))
13	12	rabbidv 3430	. . . . . 6 ⊢ (𝑔 = 𝐺 → {𝑦 ∈ ℕ ∣ (𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} = {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 })
14	13	csbeq1d 3865	. . . . 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 5199	. . . 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 19594	. . . 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 7224	. . 3 ⊢ (𝐺 ∈ V → (od‘𝐺) = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
18		fvprc 6871	. . . 4 ⊢ (¬ 𝐺 ∈ V → (od‘𝐺) = ∅)
19		fvprc 6871	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
20	3, 19	eqtrid 2816	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → 𝑋 = ∅)
21	20	mpteq1d 5202	. . . . 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 6675	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅
23	21, 22	eqtrdi 2820	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) = ∅)
24	18, 23	eqtr4d 2807	. . 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 2792	1 ⊢ 𝑂 = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))

Syntax hints:  ¬ wn 3   = wceq 1567   ∈ wcel 2149  {crab 3423  Vcvv 3463  ⦋csb 3861  ∅c0 4294  ifcif 4489   ↦ cmpt 5193  ‘cfv 6534  (class class class)co 7408  infcinf 9397  ℝcr 11095  0cc0 11096   < clt 11239  ℕcn 12229  Basecbs 17265  0_gc0g 17488  ._gcmg 19129  odcod 19590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7411  df-od 19594

This theorem is referenced by: (None)