bj-inftyexpiinv - Mathbox for BJ

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Theorem bj-inftyexpiinv 38390

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 4843	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2		df-bj-inftyexpi 38389	. . . 4 ⊢ +∞eⁱ = (𝑥 ∈ ( -π(,]π) ↦ ⟨𝑥, ℂ⟩)
3		opex 5437	. . . 4 ⊢ ⟨𝐴, ℂ⟩ ∈ V
4	1, 2, 3	fvmpt 7022	. . 3 ⊢ (𝐴 ∈ ( -π(,]π) → (+∞eⁱ‘𝐴) = ⟨𝐴, ℂ⟩)
5	4	fveq2d 6918	. 2 ⊢ (𝐴 ∈ ( -π(,]π) → (1^st ‘(+∞eⁱ‘𝐴)) = (1^st ‘⟨𝐴, ℂ⟩))
6		cnex 11318	. . 3 ⊢ ℂ ∈ V
7		op1stg 8045	. . 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 2798	1 ⊢ (𝐴 ∈ ( -π(,]π) → (1^st ‘(+∞eⁱ‘𝐴)) = 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 Vcvv 3454 ⟨cop 4838 ‘cfv 6564 (class class class)co 7449 1^st c1st 8031 ℂcc 11235 -cneg 11579 (,]cioc 13515 πcpi 16273 +∞eⁱcinftyexpi 38388

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-nul 5264 ax-pr 5396 ax-un 7781 ax-cnex 11293

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5172 df-mpt 5191 df-id 5551 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-iota 6518 df-fun 6566 df-fv 6572 df-1st 8033 df-bj-inftyexpi 38389

This theorem is used by: bj-inftyexpiinj 38391

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