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 3949 | . 2 ⊢ (𝜑 → ¬ 𝐵 ∈ ℝ) |
3 | df-nel 3124 | . . 3 ⊢ ((𝐴 − 𝐵) ∉ ℝ ↔ ¬ (𝐴 − 𝐵) ∈ ℝ) | |
4 | recnaddnred.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
5 | 4 | recnd 10669 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
6 | 1 | eldifad 3948 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
7 | 5, 6 | subcld 10997 | . . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
8 | reim0b 14478 | . . . . . 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 14584 | . . . . . . . . 9 ⊢ (𝜑 → (ℑ‘𝐴) = 0) |
11 | 10 | oveq1d 7171 | . . . . . . . 8 ⊢ (𝜑 → ((ℑ‘𝐴) − (ℑ‘𝐵)) = (0 − (ℑ‘𝐵))) |
12 | df-neg 10873 | . . . . . . . 8 ⊢ -(ℑ‘𝐵) = (0 − (ℑ‘𝐵)) | |
13 | 11, 12 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐴) − (ℑ‘𝐵)) = -(ℑ‘𝐵)) |
14 | 13 | eqeq1d 2823 | . . . . . 6 ⊢ (𝜑 → (((ℑ‘𝐴) − (ℑ‘𝐵)) = 0 ↔ -(ℑ‘𝐵) = 0)) |
15 | 5, 6 | imsubd 14576 | . . . . . . 7 ⊢ (𝜑 → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵))) |
16 | 15 | eqeq1d 2823 | . . . . . 6 ⊢ (𝜑 → ((ℑ‘(𝐴 − 𝐵)) = 0 ↔ ((ℑ‘𝐴) − (ℑ‘𝐵)) = 0)) |
17 | reim0b 14478 | . . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (𝐵 ∈ ℝ ↔ (ℑ‘𝐵) = 0)) | |
18 | 6, 17 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ (ℑ‘𝐵) = 0)) |
19 | 6 | imcld 14554 | . . . . . . . . 9 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℝ) |
20 | 19 | recnd 10669 | . . . . . . . 8 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℂ) |
21 | 20 | negeq0d 10989 | . . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐵) = 0 ↔ -(ℑ‘𝐵) = 0)) |
22 | 18, 21 | bitrd 281 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ -(ℑ‘𝐵) = 0)) |
23 | 14, 16, 22 | 3bitr4d 313 | . . . . 5 ⊢ (𝜑 → ((ℑ‘(𝐴 − 𝐵)) = 0 ↔ 𝐵 ∈ ℝ)) |
24 | 9, 23 | bitrd 281 | . . . 4 ⊢ (𝜑 → ((𝐴 − 𝐵) ∈ ℝ ↔ 𝐵 ∈ ℝ)) |
25 | 24 | notbid 320 | . . 3 ⊢ (𝜑 → (¬ (𝐴 − 𝐵) ∈ ℝ ↔ ¬ 𝐵 ∈ ℝ)) |
26 | 3, 25 | syl5bb 285 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) ∉ ℝ ↔ ¬ 𝐵 ∈ ℝ)) |
27 | 2, 26 | mpbird 259 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ∉ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∉ wnel 3123 ∖ cdif 3933 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 − cmin 10870 -cneg 10871 ℑcim 14457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-2 11701 df-cj 14458 df-re 14459 df-im 14460 |
This theorem is referenced by: requad01 43806 |
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