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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 7368 | . . 3 ⊢ ¬ i ∈ ℝ | |
2 | recexre 7362 | . . . . . 6 ⊢ ((A ∈ ℝ ∧ A #ℝ 0) → ∃x ∈ ℝ (A · x) = 1) | |
3 | 2 | adantlr 446 | . . . . 5 ⊢ (((A ∈ ℝ ∧ (i · A) ∈ ℝ) ∧ A #ℝ 0) → ∃x ∈ ℝ (A · x) = 1) |
4 | simplll 485 | . . . . . . . . 9 ⊢ ((((A ∈ ℝ ∧ (i · A) ∈ ℝ) ∧ A #ℝ 0) ∧ (x ∈ ℝ ∧ (A · x) = 1)) → A ∈ ℝ) | |
5 | 4 | recnd 6851 | . . . . . . . 8 ⊢ ((((A ∈ ℝ ∧ (i · A) ∈ ℝ) ∧ A #ℝ 0) ∧ (x ∈ ℝ ∧ (A · x) = 1)) → A ∈ ℂ) |
6 | simprl 483 | . . . . . . . . 9 ⊢ ((((A ∈ ℝ ∧ (i · A) ∈ ℝ) ∧ A #ℝ 0) ∧ (x ∈ ℝ ∧ (A · x) = 1)) → x ∈ ℝ) | |
7 | 6 | recnd 6851 | . . . . . . . 8 ⊢ ((((A ∈ ℝ ∧ (i · A) ∈ ℝ) ∧ A #ℝ 0) ∧ (x ∈ ℝ ∧ (A · x) = 1)) → x ∈ ℂ) |
8 | ax-icn 6778 | . . . . . . . . 9 ⊢ i ∈ ℂ | |
9 | mulass 6810 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ A ∈ ℂ ∧ x ∈ ℂ) → ((i · A) · x) = (i · (A · x))) | |
10 | 8, 9 | mp3an1 1218 | . . . . . . . 8 ⊢ ((A ∈ ℂ ∧ x ∈ ℂ) → ((i · A) · x) = (i · (A · x))) |
11 | 5, 7, 10 | syl2anc 391 | . . . . . . 7 ⊢ ((((A ∈ ℝ ∧ (i · A) ∈ ℝ) ∧ A #ℝ 0) ∧ (x ∈ ℝ ∧ (A · x) = 1)) → ((i · A) · x) = (i · (A · x))) |
12 | oveq2 5463 | . . . . . . . . 9 ⊢ ((A · x) = 1 → (i · (A · x)) = (i · 1)) | |
13 | 8 | mulid1i 6827 | . . . . . . . . 9 ⊢ (i · 1) = i |
14 | 12, 13 | syl6eq 2085 | . . . . . . . 8 ⊢ ((A · x) = 1 → (i · (A · x)) = i) |
15 | 14 | ad2antll 460 | . . . . . . 7 ⊢ ((((A ∈ ℝ ∧ (i · A) ∈ ℝ) ∧ A #ℝ 0) ∧ (x ∈ ℝ ∧ (A · x) = 1)) → (i · (A · x)) = i) |
16 | 11, 15 | eqtrd 2069 | . . . . . 6 ⊢ ((((A ∈ ℝ ∧ (i · A) ∈ ℝ) ∧ A #ℝ 0) ∧ (x ∈ ℝ ∧ (A · x) = 1)) → ((i · A) · x) = i) |
17 | simpllr 486 | . . . . . . 7 ⊢ ((((A ∈ ℝ ∧ (i · A) ∈ ℝ) ∧ A #ℝ 0) ∧ (x ∈ ℝ ∧ (A · x) = 1)) → (i · A) ∈ ℝ) | |
18 | 17, 6 | remulcld 6853 | . . . . . 6 ⊢ ((((A ∈ ℝ ∧ (i · A) ∈ ℝ) ∧ A #ℝ 0) ∧ (x ∈ ℝ ∧ (A · x) = 1)) → ((i · A) · x) ∈ ℝ) |
19 | 16, 18 | eqeltrrd 2112 | . . . . 5 ⊢ ((((A ∈ ℝ ∧ (i · A) ∈ ℝ) ∧ A #ℝ 0) ∧ (x ∈ ℝ ∧ (A · x) = 1)) → i ∈ ℝ) |
20 | 3, 19 | rexlimddv 2431 | . . . 4 ⊢ (((A ∈ ℝ ∧ (i · A) ∈ ℝ) ∧ A #ℝ 0) → i ∈ ℝ) |
21 | 20 | ex 108 | . . 3 ⊢ ((A ∈ ℝ ∧ (i · A) ∈ ℝ) → (A #ℝ 0 → i ∈ ℝ)) |
22 | 1, 21 | mtoi 589 | . 2 ⊢ ((A ∈ ℝ ∧ (i · A) ∈ ℝ) → ¬ A #ℝ 0) |
23 | 0re 6825 | . . . 4 ⊢ 0 ∈ ℝ | |
24 | reapti 7363 | . . . 4 ⊢ ((A ∈ ℝ ∧ 0 ∈ ℝ) → (A = 0 ↔ ¬ A #ℝ 0)) | |
25 | 23, 24 | mpan2 401 | . . 3 ⊢ (A ∈ ℝ → (A = 0 ↔ ¬ A #ℝ 0)) |
26 | 25 | adantr 261 | . 2 ⊢ ((A ∈ ℝ ∧ (i · A) ∈ ℝ) → (A = 0 ↔ ¬ A #ℝ 0)) |
27 | 22, 26 | mpbird 156 | 1 ⊢ ((A ∈ ℝ ∧ (i · A) ∈ ℝ) → A = 0) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 ∃wrex 2301 class class class wbr 3755 (class class class)co 5455 ℂcc 6709 ℝcr 6710 0cc0 6711 1c1 6712 ici 6713 · cmul 6716 #ℝ creap 7358 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-mulrcl 6782 ax-addcom 6783 ax-mulcom 6784 ax-addass 6785 ax-mulass 6786 ax-distr 6787 ax-i2m1 6788 ax-1rid 6790 ax-0id 6791 ax-rnegex 6792 ax-precex 6793 ax-cnre 6794 ax-pre-ltirr 6795 ax-pre-lttrn 6797 ax-pre-apti 6798 ax-pre-ltadd 6799 ax-pre-mulgt0 6800 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-i1p 6450 df-iplp 6451 df-iltp 6453 df-enr 6654 df-nr 6655 df-ltr 6658 df-0r 6659 df-1r 6660 df-0 6718 df-1 6719 df-r 6721 df-lt 6724 df-pnf 6859 df-mnf 6860 df-ltxr 6862 df-sub 6981 df-neg 6982 df-reap 7359 |
This theorem is referenced by: rereim 7370 cru 7386 cju 7694 crre 9085 |
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