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Mirrors > Home > ILE Home > Th. List > rimul | GIF version |
Description: A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
rimul | ⊢ ((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) → 𝐴 = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inelr 8539 | . . 3 ⊢ ¬ i ∈ ℝ | |
2 | recexre 8533 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 #ℝ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1) | |
3 | 2 | adantlr 477 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) ∧ 𝐴 #ℝ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1) |
4 | simplll 533 | . . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) ∧ 𝐴 #ℝ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → 𝐴 ∈ ℝ) | |
5 | 4 | recnd 7984 | . . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) ∧ 𝐴 #ℝ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → 𝐴 ∈ ℂ) |
6 | simprl 529 | . . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) ∧ 𝐴 #ℝ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → 𝑥 ∈ ℝ) | |
7 | 6 | recnd 7984 | . . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) ∧ 𝐴 #ℝ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → 𝑥 ∈ ℂ) |
8 | ax-icn 7905 | . . . . . . . . 9 ⊢ i ∈ ℂ | |
9 | mulass 7941 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((i · 𝐴) · 𝑥) = (i · (𝐴 · 𝑥))) | |
10 | 8, 9 | mp3an1 1324 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((i · 𝐴) · 𝑥) = (i · (𝐴 · 𝑥))) |
11 | 5, 7, 10 | syl2anc 411 | . . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) ∧ 𝐴 #ℝ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → ((i · 𝐴) · 𝑥) = (i · (𝐴 · 𝑥))) |
12 | oveq2 5882 | . . . . . . . . 9 ⊢ ((𝐴 · 𝑥) = 1 → (i · (𝐴 · 𝑥)) = (i · 1)) | |
13 | 8 | mulid1i 7958 | . . . . . . . . 9 ⊢ (i · 1) = i |
14 | 12, 13 | eqtrdi 2226 | . . . . . . . 8 ⊢ ((𝐴 · 𝑥) = 1 → (i · (𝐴 · 𝑥)) = i) |
15 | 14 | ad2antll 491 | . . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) ∧ 𝐴 #ℝ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → (i · (𝐴 · 𝑥)) = i) |
16 | 11, 15 | eqtrd 2210 | . . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) ∧ 𝐴 #ℝ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → ((i · 𝐴) · 𝑥) = i) |
17 | simpllr 534 | . . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) ∧ 𝐴 #ℝ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → (i · 𝐴) ∈ ℝ) | |
18 | 17, 6 | remulcld 7986 | . . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) ∧ 𝐴 #ℝ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → ((i · 𝐴) · 𝑥) ∈ ℝ) |
19 | 16, 18 | eqeltrrd 2255 | . . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) ∧ 𝐴 #ℝ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → i ∈ ℝ) |
20 | 3, 19 | rexlimddv 2599 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) ∧ 𝐴 #ℝ 0) → i ∈ ℝ) |
21 | 20 | ex 115 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) → (𝐴 #ℝ 0 → i ∈ ℝ)) |
22 | 1, 21 | mtoi 664 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) → ¬ 𝐴 #ℝ 0) |
23 | 0re 7956 | . . . 4 ⊢ 0 ∈ ℝ | |
24 | reapti 8534 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 = 0 ↔ ¬ 𝐴 #ℝ 0)) | |
25 | 23, 24 | mpan2 425 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 = 0 ↔ ¬ 𝐴 #ℝ 0)) |
26 | 25 | adantr 276 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) → (𝐴 = 0 ↔ ¬ 𝐴 #ℝ 0)) |
27 | 22, 26 | mpbird 167 | 1 ⊢ ((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) → 𝐴 = 0) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 class class class wbr 4003 (class class class)co 5874 ℂcc 7808 ℝcr 7809 0cc0 7810 1c1 7811 ici 7812 · cmul 7815 #ℝ creap 8529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-sub 8128 df-neg 8129 df-reap 8530 |
This theorem is referenced by: rereim 8541 cru 8557 cju 8916 crre 10861 |
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