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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elccinfty | Structured version Visualization version GIF version |
Description: A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.) |
Ref | Expression |
---|---|
bj-elccinfty | ⊢ (𝐴 ∈ (-π(,]π) → (+∞ei‘𝐴) ∈ ℂ∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-inftyexpi 37150 | . . . . 5 ⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ 〈𝑥, ℂ〉) | |
2 | 1 | funmpt2 6602 | . . . 4 ⊢ Fun +∞ei |
3 | 2 | jctl 523 | . . 3 ⊢ (𝐴 ∈ dom +∞ei → (Fun +∞ei ∧ 𝐴 ∈ dom +∞ei)) |
4 | opex 5467 | . . . . 5 ⊢ 〈𝑥, ℂ〉 ∈ V | |
5 | 4, 1 | dmmpti 6708 | . . . 4 ⊢ dom +∞ei = (-π(,]π) |
6 | 5 | eqcomi 2742 | . . 3 ⊢ (-π(,]π) = dom +∞ei |
7 | 3, 6 | eleq2s 2855 | . 2 ⊢ (𝐴 ∈ (-π(,]π) → (Fun +∞ei ∧ 𝐴 ∈ dom +∞ei)) |
8 | fvelrn 7090 | . 2 ⊢ ((Fun +∞ei ∧ 𝐴 ∈ dom +∞ei) → (+∞ei‘𝐴) ∈ ran +∞ei) | |
9 | df-bj-ccinfty 37155 | . . . . 5 ⊢ ℂ∞ = ran +∞ei | |
10 | 9 | eqcomi 2742 | . . . 4 ⊢ ran +∞ei = ℂ∞ |
11 | 10 | eleq2i 2829 | . . 3 ⊢ ((+∞ei‘𝐴) ∈ ran +∞ei ↔ (+∞ei‘𝐴) ∈ ℂ∞) |
12 | 11 | biimpi 216 | . 2 ⊢ ((+∞ei‘𝐴) ∈ ran +∞ei → (+∞ei‘𝐴) ∈ ℂ∞) |
13 | 7, 8, 12 | 3syl 18 | 1 ⊢ (𝐴 ∈ (-π(,]π) → (+∞ei‘𝐴) ∈ ℂ∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2104 〈cop 4636 dom cdm 5683 ran crn 5684 Fun wfun 6552 ‘cfv 6558 (class class class)co 7425 ℂcc 11144 -cneg 11484 (,]cioc 13378 πcpi 16088 +∞eicinftyexpi 37149 ℂ∞cccinfty 37154 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-iota 6510 df-fun 6560 df-fn 6561 df-fv 6566 df-bj-inftyexpi 37150 df-bj-ccinfty 37155 |
This theorem is referenced by: bj-pinftyccb 37164 |
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