bj-inftyexpiinv - Mathbox for BJ

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Theorem bj-inftyexpiinv 38223

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) → (1^st ‘(+∞eⁱ‘𝐴)) = 𝐴)

Proof of Theorem bj-inftyexpiinv Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4839	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2		df-bj-inftyexpi 38222	. . . 4 ⊢ +∞eⁱ = (𝑥 ∈ ( -π(,]π) ↦ ⟨𝑥, ℂ⟩)
3		opex 5431	. . . 4 ⊢ ⟨𝐴, ℂ⟩ ∈ V
4	1, 2, 3	fvmpt 6994	. . 3 ⊢ (𝐴 ∈ ( -π(,]π) → (+∞eⁱ‘𝐴) = ⟨𝐴, ℂ⟩)
5	4	fveq2d 6890	. 2 ⊢ (𝐴 ∈ ( -π(,]π) → (1^st ‘(+∞eⁱ‘𝐴)) = (1^st ‘⟨𝐴, ℂ⟩))
6		cnex 11285	. . 3 ⊢ ℂ ∈ V
7		op1stg 8014	. . 3 ⊢ ((𝐴 ∈ ( -π(,]π) ∧ ℂ ∈ V) → (1^st ‘⟨𝐴, ℂ⟩) = 𝐴)
8	6, 7	mpan2 704	. 2 ⊢ (𝐴 ∈ ( -π(,]π) → (1^st ‘⟨𝐴, ℂ⟩) = 𝐴)
9	5, 8	eqtrd 2795	1 ⊢ (𝐴 ∈ ( -π(,]π) → (1^st ‘(+∞eⁱ‘𝐴)) = 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∈ wcel 2144 Vcvv 3450 ⟨cop 4834 ‘cfv 6539 (class class class)co 7421 1^st c1st 8000 ℂcc 11202 -cneg 11546 (,]cioc 13481 πcpi 16238 +∞eⁱcinftyexpi 38221

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5390 ax-un 7752 ax-cnex 11260

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-iota 6494 df-fun 6541 df-fv 6547 df-1st 8002 df-bj-inftyexpi 38222

This theorem is used by: bj-inftyexpiinj 38224

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