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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 37186 | . . . . 5 ⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ 〈𝑥, ℂ〉) | |
| 2 | 1 | funmpt2 6603 | . . . 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 6710 | . . . 4 ⊢ dom +∞ei = (-π(,]π) |
| 6 | 5 | eqcomi 2745 | . . 3 ⊢ (-π(,]π) = dom +∞ei |
| 7 | 3, 6 | eleq2s 2858 | . 2 ⊢ (𝐴 ∈ (-π(,]π) → (Fun +∞ei ∧ 𝐴 ∈ dom +∞ei)) |
| 8 | fvelrn 7094 | . 2 ⊢ ((Fun +∞ei ∧ 𝐴 ∈ dom +∞ei) → (+∞ei‘𝐴) ∈ ran +∞ei) | |
| 9 | df-bj-ccinfty 37191 | . . . . 5 ⊢ ℂ∞ = ran +∞ei | |
| 10 | 9 | eqcomi 2745 | . . . 4 ⊢ ran +∞ei = ℂ∞ |
| 11 | 10 | eleq2i 2832 | . . 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 2108 〈cop 4630 dom cdm 5683 ran crn 5684 Fun wfun 6553 ‘cfv 6559 (class class class)co 7429 ℂcc 11149 -cneg 11489 (,]cioc 13384 πcpi 16098 +∞eicinftyexpi 37185 ℂ∞cccinfty 37190 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-iota 6512 df-fun 6561 df-fn 6562 df-fv 6567 df-bj-inftyexpi 37186 df-bj-ccinfty 37191 |
| This theorem is referenced by: bj-pinftyccb 37200 |
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