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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > resubcnnred | Structured version Visualization version GIF version |
Description: The difference of a real number and an imaginary 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)) |
Ref | Expression |
---|---|
resubcnnred | ⊢ (𝜑 → (𝐴 − 𝐵) ∉ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recnaddnred.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ (ℂ ∖ ℝ)) | |
2 | 1 | eldifbd 3976 | . 2 ⊢ (𝜑 → ¬ 𝐵 ∈ ℝ) |
3 | df-nel 3045 | . . 3 ⊢ ((𝐴 − 𝐵) ∉ ℝ ↔ ¬ (𝐴 − 𝐵) ∈ ℝ) | |
4 | recnaddnred.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
5 | 4 | recnd 11287 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
6 | 1 | eldifad 3975 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
7 | 5, 6 | subcld 11618 | . . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
8 | reim0b 15155 | . . . . . 6 ⊢ ((𝐴 − 𝐵) ∈ ℂ → ((𝐴 − 𝐵) ∈ ℝ ↔ (ℑ‘(𝐴 − 𝐵)) = 0)) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 𝐵) ∈ ℝ ↔ (ℑ‘(𝐴 − 𝐵)) = 0)) |
10 | 4 | reim0d 15261 | . . . . . . . . 9 ⊢ (𝜑 → (ℑ‘𝐴) = 0) |
11 | 10 | oveq1d 7446 | . . . . . . . 8 ⊢ (𝜑 → ((ℑ‘𝐴) − (ℑ‘𝐵)) = (0 − (ℑ‘𝐵))) |
12 | df-neg 11493 | . . . . . . . 8 ⊢ -(ℑ‘𝐵) = (0 − (ℑ‘𝐵)) | |
13 | 11, 12 | eqtr4di 2793 | . . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐴) − (ℑ‘𝐵)) = -(ℑ‘𝐵)) |
14 | 13 | eqeq1d 2737 | . . . . . 6 ⊢ (𝜑 → (((ℑ‘𝐴) − (ℑ‘𝐵)) = 0 ↔ -(ℑ‘𝐵) = 0)) |
15 | 5, 6 | imsubd 15253 | . . . . . . 7 ⊢ (𝜑 → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵))) |
16 | 15 | eqeq1d 2737 | . . . . . 6 ⊢ (𝜑 → ((ℑ‘(𝐴 − 𝐵)) = 0 ↔ ((ℑ‘𝐴) − (ℑ‘𝐵)) = 0)) |
17 | reim0b 15155 | . . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (𝐵 ∈ ℝ ↔ (ℑ‘𝐵) = 0)) | |
18 | 6, 17 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ (ℑ‘𝐵) = 0)) |
19 | 6 | imcld 15231 | . . . . . . . . 9 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℝ) |
20 | 19 | recnd 11287 | . . . . . . . 8 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℂ) |
21 | 20 | negeq0d 11610 | . . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐵) = 0 ↔ -(ℑ‘𝐵) = 0)) |
22 | 18, 21 | bitrd 279 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ -(ℑ‘𝐵) = 0)) |
23 | 14, 16, 22 | 3bitr4d 311 | . . . . 5 ⊢ (𝜑 → ((ℑ‘(𝐴 − 𝐵)) = 0 ↔ 𝐵 ∈ ℝ)) |
24 | 9, 23 | bitrd 279 | . . . 4 ⊢ (𝜑 → ((𝐴 − 𝐵) ∈ ℝ ↔ 𝐵 ∈ ℝ)) |
25 | 24 | notbid 318 | . . 3 ⊢ (𝜑 → (¬ (𝐴 − 𝐵) ∈ ℝ ↔ ¬ 𝐵 ∈ ℝ)) |
26 | 3, 25 | bitrid 283 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) ∉ ℝ ↔ ¬ 𝐵 ∈ ℝ)) |
27 | 2, 26 | mpbird 257 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ∉ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ∉ wnel 3044 ∖ cdif 3960 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 − cmin 11490 -cneg 11491 ℑcim 15134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-2 12327 df-cj 15135 df-re 15136 df-im 15137 |
This theorem is referenced by: requad01 47546 |
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