Theorem bj-inftyexpiinv 37893

Description: Utility theorem for the inverse of +∞eⁱ. (Contributed by BJ, 22-Jun-2019.) This utility theorem is irrelevant and should generally not be used. (New usage is discouraged.)

Assertion

Ref	Expression
bj-inftyexpiinv	⊢ (𝐴 ∈ (-π(,]π) → (1st ‘(+∞ei‘𝐴)) = 𝐴)

Proof of Theorem bj-inftyexpiinv

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4843	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2		df-bj-inftyexpi 37892	. . . 4 ⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
3		opex 5450	. . . 4 ⊢ ⟨𝐴, ℂ⟩ ∈ V
4	1, 2, 3	fvmpt 6996	. . 3 ⊢ (𝐴 ∈ (-π(,]π) → (+∞ei‘𝐴) = ⟨𝐴, ℂ⟩)
5	4	fveq2d 6892	. 2 ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘(+∞ei‘𝐴)) = (1st ‘⟨𝐴, ℂ⟩))
6		cnex 11199	. . 3 ⊢ ℂ ∈ V
7		op1stg 8007	. . 3 ⊢ ((𝐴 ∈ (-π(,]π) ∧ ℂ ∈ V) → (1st ‘⟨𝐴, ℂ⟩) = 𝐴)
8	6, 7	mpan2 704	. 2 ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘⟨𝐴, ℂ⟩) = 𝐴)
9	5, 8	eqtrd 2801	1 ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘(+∞ei‘𝐴)) = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3458  ⟨cop 4600  ‘cfv 6543  (class class class)co 7423  1^st c1st 7993  ℂcc 11116  -cneg 11460  (,]cioc 13391  πcpi 16145  +∞eⁱcinftyexpi 37891

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 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745  ax-cnex 11174

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6499  df-fun 6545  df-fv 6551  df-1st 7995  df-bj-inftyexpi 37892

This theorem is used by:  bj-inftyexpiinj  37894