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Mirrors > Home > ILE Home > Th. List > iap0 | Unicode version |
Description: The imaginary unit ![]() |
Ref | Expression |
---|---|
iap0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1ap0 8128 |
. . . 4
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2 | 1 | olci 687 |
. . 3
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3 | 0re 7549 |
. . . 4
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4 | 1re 7548 |
. . . 4
![]() ![]() ![]() ![]() | |
5 | apreim 8141 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 3, 4, 3, 3, 5 | mp4an 419 |
. . 3
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7 | 2, 6 | mpbir 145 |
. 2
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8 | ax-icn 7501 |
. . . . 5
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9 | 8 | mulid1i 7551 |
. . . 4
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10 | 9 | oveq2i 5677 |
. . 3
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11 | 8 | addid2i 7686 |
. . 3
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12 | 10, 11 | eqtri 2109 |
. 2
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13 | it0e0 8698 |
. . . 4
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14 | 13 | oveq2i 5677 |
. . 3
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15 | 00id 7684 |
. . 3
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16 | 14, 15 | eqtri 2109 |
. 2
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17 | 7, 12, 16 | 3brtr3i 3878 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-mulrcl 7505 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-precex 7516 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-lttrn 7520 ax-pre-apti 7521 ax-pre-ltadd 7522 ax-pre-mulgt0 7523 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-pnf 7585 df-mnf 7586 df-ltxr 7588 df-sub 7716 df-neg 7717 df-reap 8113 df-ap 8120 |
This theorem is referenced by: 2muliap0 8701 irec 10115 iexpcyc 10120 imval 10345 imre 10346 reim 10347 crim 10353 cjreb 10361 tanval2ap 11065 tanval3ap 11066 efival 11084 |
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