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 3870 | . 2 ⊢ (𝜑 → ¬ 𝐵 ∈ ℝ) |
3 | df-nel 3040 | . . 3 ⊢ ((𝐴 − 𝐵) ∉ ℝ ↔ ¬ (𝐴 − 𝐵) ∈ ℝ) | |
4 | recnaddnred.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
5 | 4 | recnd 10844 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
6 | 1 | eldifad 3869 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
7 | 5, 6 | subcld 11172 | . . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
8 | reim0b 14665 | . . . . . 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 14771 | . . . . . . . . 9 ⊢ (𝜑 → (ℑ‘𝐴) = 0) |
11 | 10 | oveq1d 7217 | . . . . . . . 8 ⊢ (𝜑 → ((ℑ‘𝐴) − (ℑ‘𝐵)) = (0 − (ℑ‘𝐵))) |
12 | df-neg 11048 | . . . . . . . 8 ⊢ -(ℑ‘𝐵) = (0 − (ℑ‘𝐵)) | |
13 | 11, 12 | eqtr4di 2792 | . . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐴) − (ℑ‘𝐵)) = -(ℑ‘𝐵)) |
14 | 13 | eqeq1d 2736 | . . . . . 6 ⊢ (𝜑 → (((ℑ‘𝐴) − (ℑ‘𝐵)) = 0 ↔ -(ℑ‘𝐵) = 0)) |
15 | 5, 6 | imsubd 14763 | . . . . . . 7 ⊢ (𝜑 → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵))) |
16 | 15 | eqeq1d 2736 | . . . . . 6 ⊢ (𝜑 → ((ℑ‘(𝐴 − 𝐵)) = 0 ↔ ((ℑ‘𝐴) − (ℑ‘𝐵)) = 0)) |
17 | reim0b 14665 | . . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (𝐵 ∈ ℝ ↔ (ℑ‘𝐵) = 0)) | |
18 | 6, 17 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ (ℑ‘𝐵) = 0)) |
19 | 6 | imcld 14741 | . . . . . . . . 9 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℝ) |
20 | 19 | recnd 10844 | . . . . . . . 8 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℂ) |
21 | 20 | negeq0d 11164 | . . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐵) = 0 ↔ -(ℑ‘𝐵) = 0)) |
22 | 18, 21 | bitrd 282 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ -(ℑ‘𝐵) = 0)) |
23 | 14, 16, 22 | 3bitr4d 314 | . . . . 5 ⊢ (𝜑 → ((ℑ‘(𝐴 − 𝐵)) = 0 ↔ 𝐵 ∈ ℝ)) |
24 | 9, 23 | bitrd 282 | . . . 4 ⊢ (𝜑 → ((𝐴 − 𝐵) ∈ ℝ ↔ 𝐵 ∈ ℝ)) |
25 | 24 | notbid 321 | . . 3 ⊢ (𝜑 → (¬ (𝐴 − 𝐵) ∈ ℝ ↔ ¬ 𝐵 ∈ ℝ)) |
26 | 3, 25 | syl5bb 286 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) ∉ ℝ ↔ ¬ 𝐵 ∈ ℝ)) |
27 | 2, 26 | mpbird 260 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ∉ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 ∉ wnel 3039 ∖ cdif 3854 ‘cfv 6369 (class class class)co 7202 ℂcc 10710 ℝcr 10711 0cc0 10712 − cmin 11045 -cneg 11046 ℑcim 14644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-po 5457 df-so 5458 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-2 11876 df-cj 14645 df-re 14646 df-im 14647 |
This theorem is referenced by: requad01 44700 |
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