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Mirrors > Home > ILE Home > Th. List > iap0 | GIF version |
Description: The imaginary unit i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.) |
Ref | Expression |
---|---|
iap0 | ⊢ i # 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1ap0 8164 | . . . 4 ⊢ 1 # 0 | |
2 | 1 | olci 689 | . . 3 ⊢ (0 # 0 ∨ 1 # 0) |
3 | 0re 7585 | . . . 4 ⊢ 0 ∈ ℝ | |
4 | 1re 7584 | . . . 4 ⊢ 1 ∈ ℝ | |
5 | apreim 8177 | . . . 4 ⊢ (((0 ∈ ℝ ∧ 1 ∈ ℝ) ∧ (0 ∈ ℝ ∧ 0 ∈ ℝ)) → ((0 + (i · 1)) # (0 + (i · 0)) ↔ (0 # 0 ∨ 1 # 0))) | |
6 | 3, 4, 3, 3, 5 | mp4an 419 | . . 3 ⊢ ((0 + (i · 1)) # (0 + (i · 0)) ↔ (0 # 0 ∨ 1 # 0)) |
7 | 2, 6 | mpbir 145 | . 2 ⊢ (0 + (i · 1)) # (0 + (i · 0)) |
8 | ax-icn 7537 | . . . . 5 ⊢ i ∈ ℂ | |
9 | 8 | mulid1i 7587 | . . . 4 ⊢ (i · 1) = i |
10 | 9 | oveq2i 5701 | . . 3 ⊢ (0 + (i · 1)) = (0 + i) |
11 | 8 | addid2i 7722 | . . 3 ⊢ (0 + i) = i |
12 | 10, 11 | eqtri 2115 | . 2 ⊢ (0 + (i · 1)) = i |
13 | it0e0 8735 | . . . 4 ⊢ (i · 0) = 0 | |
14 | 13 | oveq2i 5701 | . . 3 ⊢ (0 + (i · 0)) = (0 + 0) |
15 | 00id 7720 | . . 3 ⊢ (0 + 0) = 0 | |
16 | 14, 15 | eqtri 2115 | . 2 ⊢ (0 + (i · 0)) = 0 |
17 | 7, 12, 16 | 3brtr3i 3894 | 1 ⊢ i # 0 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 667 ∈ wcel 1445 class class class wbr 3867 (class class class)co 5690 ℝcr 7446 0cc0 7447 1c1 7448 ici 7449 + caddc 7450 · cmul 7452 # cap 8155 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-ltxr 7624 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 |
This theorem is referenced by: 2muliap0 8738 irec 10185 iexpcyc 10190 imval 10415 imre 10416 reim 10417 crim 10423 cjreb 10431 tanval2ap 11169 tanval3ap 11170 efival 11188 |
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