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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 3837 | . 2 ⊢ (𝜑 → ¬ 𝐵 ∈ ℝ) |
3 | df-nel 3069 | . . 3 ⊢ ((𝐵 / 𝐴) ∉ ℝ ↔ ¬ (𝐵 / 𝐴) ∈ ℝ) | |
4 | 1 | eldifad 3836 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
5 | recnaddnred.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
6 | 5 | recnd 10467 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
7 | cndivrenred.n | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≠ 0) | |
8 | 4, 6, 7 | divcld 11216 | . . . . . 6 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ) |
9 | reim0b 14338 | . . . . . 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 14414 | . . . . . . . 8 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℝ) |
12 | 11 | recnd 10467 | . . . . . . 7 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℂ) |
13 | 12, 6, 7 | diveq0ad 11226 | . . . . . 6 ⊢ (𝜑 → (((ℑ‘𝐵) / 𝐴) = 0 ↔ (ℑ‘𝐵) = 0)) |
14 | 5, 4, 7 | imdivd 14449 | . . . . . . 7 ⊢ (𝜑 → (ℑ‘(𝐵 / 𝐴)) = ((ℑ‘𝐵) / 𝐴)) |
15 | 14 | eqeq1d 2775 | . . . . . 6 ⊢ (𝜑 → ((ℑ‘(𝐵 / 𝐴)) = 0 ↔ ((ℑ‘𝐵) / 𝐴) = 0)) |
16 | reim0b 14338 | . . . . . . 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 303 | . . . . 5 ⊢ (𝜑 → ((ℑ‘(𝐵 / 𝐴)) = 0 ↔ 𝐵 ∈ ℝ)) |
19 | 10, 18 | bitrd 271 | . . . 4 ⊢ (𝜑 → ((𝐵 / 𝐴) ∈ ℝ ↔ 𝐵 ∈ ℝ)) |
20 | 19 | notbid 310 | . . 3 ⊢ (𝜑 → (¬ (𝐵 / 𝐴) ∈ ℝ ↔ ¬ 𝐵 ∈ ℝ)) |
21 | 3, 20 | syl5bb 275 | . 2 ⊢ (𝜑 → ((𝐵 / 𝐴) ∉ ℝ ↔ ¬ 𝐵 ∈ ℝ)) |
22 | 2, 21 | mpbird 249 | 1 ⊢ (𝜑 → (𝐵 / 𝐴) ∉ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 = wceq 1508 ∈ wcel 2051 ≠ wne 2962 ∉ wnel 3068 ∖ cdif 3821 ‘cfv 6186 (class class class)co 6975 ℂcc 10332 ℝcr 10333 0cc0 10334 / cdiv 11097 ℑcim 14317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-po 5323 df-so 5324 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-2 11502 df-cj 14318 df-re 14319 df-im 14320 |
This theorem is referenced by: requad01 43184 |
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