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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-inftyexpiinv | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
bj-inftyexpiinv | ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘(+∞ei‘𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 4863 | . . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩) | |
2 | df-bj-inftyexpi 35876 | . . . 4 ⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩) | |
3 | opex 5454 | . . . 4 ⊢ ⟨𝐴, ℂ⟩ ∈ V | |
4 | 1, 2, 3 | fvmpt 6981 | . . 3 ⊢ (𝐴 ∈ (-π(,]π) → (+∞ei‘𝐴) = ⟨𝐴, ℂ⟩) |
5 | 4 | fveq2d 6879 | . 2 ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘(+∞ei‘𝐴)) = (1st ‘⟨𝐴, ℂ⟩)) |
6 | cnex 11170 | . . 3 ⊢ ℂ ∈ V | |
7 | op1stg 7966 | . . 3 ⊢ ((𝐴 ∈ (-π(,]π) ∧ ℂ ∈ V) → (1st ‘⟨𝐴, ℂ⟩) = 𝐴) | |
8 | 6, 7 | mpan2 689 | . 2 ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘⟨𝐴, ℂ⟩) = 𝐴) |
9 | 5, 8 | eqtrd 2771 | 1 ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘(+∞ei‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3470 ⟨cop 4625 ‘cfv 6529 (class class class)co 7390 1st c1st 7952 ℂcc 11087 -cneg 11424 (,]cioc 13304 πcpi 15989 +∞eicinftyexpi 35875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7705 ax-cnex 11145 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3430 df-v 3472 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-iota 6481 df-fun 6531 df-fv 6537 df-1st 7954 df-bj-inftyexpi 35876 |
This theorem is referenced by: bj-inftyexpiinj 35878 |
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