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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cndivrenred | Structured version Visualization version GIF version |
Description: The quotient of an imaginary number and a real number is not a real number. (Contributed by AV, 23-Jan-2023.) |
Ref | Expression |
---|---|
recnaddnred.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
recnaddnred.b | ⊢ (𝜑 → 𝐵 ∈ (ℂ ∖ ℝ)) |
cndivrenred.n | ⊢ (𝜑 → 𝐴 ≠ 0) |
Ref | Expression |
---|---|
cndivrenred | ⊢ (𝜑 → (𝐵 / 𝐴) ∉ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recnaddnred.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ (ℂ ∖ ℝ)) | |
2 | 1 | eldifbd 3960 | . 2 ⊢ (𝜑 → ¬ 𝐵 ∈ ℝ) |
3 | df-nel 3044 | . . 3 ⊢ ((𝐵 / 𝐴) ∉ ℝ ↔ ¬ (𝐵 / 𝐴) ∈ ℝ) | |
4 | 1 | eldifad 3959 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
5 | recnaddnred.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
6 | 5 | recnd 11272 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
7 | cndivrenred.n | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≠ 0) | |
8 | 4, 6, 7 | divcld 12020 | . . . . . 6 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ) |
9 | reim0b 15098 | . . . . . 6 ⊢ ((𝐵 / 𝐴) ∈ ℂ → ((𝐵 / 𝐴) ∈ ℝ ↔ (ℑ‘(𝐵 / 𝐴)) = 0)) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝐵 / 𝐴) ∈ ℝ ↔ (ℑ‘(𝐵 / 𝐴)) = 0)) |
11 | 4 | imcld 15174 | . . . . . . . 8 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℝ) |
12 | 11 | recnd 11272 | . . . . . . 7 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℂ) |
13 | 12, 6, 7 | diveq0ad 12030 | . . . . . 6 ⊢ (𝜑 → (((ℑ‘𝐵) / 𝐴) = 0 ↔ (ℑ‘𝐵) = 0)) |
14 | 5, 4, 7 | imdivd 15209 | . . . . . . 7 ⊢ (𝜑 → (ℑ‘(𝐵 / 𝐴)) = ((ℑ‘𝐵) / 𝐴)) |
15 | 14 | eqeq1d 2730 | . . . . . 6 ⊢ (𝜑 → ((ℑ‘(𝐵 / 𝐴)) = 0 ↔ ((ℑ‘𝐵) / 𝐴) = 0)) |
16 | reim0b 15098 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (𝐵 ∈ ℝ ↔ (ℑ‘𝐵) = 0)) | |
17 | 4, 16 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ (ℑ‘𝐵) = 0)) |
18 | 13, 15, 17 | 3bitr4d 311 | . . . . 5 ⊢ (𝜑 → ((ℑ‘(𝐵 / 𝐴)) = 0 ↔ 𝐵 ∈ ℝ)) |
19 | 10, 18 | bitrd 279 | . . . 4 ⊢ (𝜑 → ((𝐵 / 𝐴) ∈ ℝ ↔ 𝐵 ∈ ℝ)) |
20 | 19 | notbid 318 | . . 3 ⊢ (𝜑 → (¬ (𝐵 / 𝐴) ∈ ℝ ↔ ¬ 𝐵 ∈ ℝ)) |
21 | 3, 20 | bitrid 283 | . 2 ⊢ (𝜑 → ((𝐵 / 𝐴) ∉ ℝ ↔ ¬ 𝐵 ∈ ℝ)) |
22 | 2, 21 | mpbird 257 | 1 ⊢ (𝜑 → (𝐵 / 𝐴) ∉ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∉ wnel 3043 ∖ cdif 3944 ‘cfv 6548 (class class class)co 7420 ℂcc 11136 ℝcr 11137 0cc0 11138 / cdiv 11901 ℑcim 15077 |
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 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-2 12305 df-cj 15078 df-re 15079 df-im 15080 |
This theorem is referenced by: requad01 46961 |
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