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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 8572 |
. . . 4
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2 | 1 | olci 733 |
. . 3
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3 | 0re 7982 |
. . . 4
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4 | 1re 7981 |
. . . 4
![]() ![]() ![]() ![]() | |
5 | apreim 8585 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 3, 4, 3, 3, 5 | mp4an 427 |
. . 3
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7 | 2, 6 | mpbir 146 |
. 2
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8 | ax-icn 7931 |
. . . . 5
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9 | 8 | mulid1i 7984 |
. . . 4
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10 | 9 | oveq2i 5903 |
. . 3
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11 | 8 | addid2i 8125 |
. . 3
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12 | 10, 11 | eqtri 2210 |
. 2
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13 | it0e0 9165 |
. . . 4
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14 | 13 | oveq2i 5903 |
. . 3
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15 | 00id 8123 |
. . 3
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16 | 14, 15 | eqtri 2210 |
. 2
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17 | 7, 12, 16 | 3brtr3i 4047 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-mulrcl 7935 ax-addcom 7936 ax-mulcom 7937 ax-addass 7938 ax-mulass 7939 ax-distr 7940 ax-i2m1 7941 ax-0lt1 7942 ax-1rid 7943 ax-0id 7944 ax-rnegex 7945 ax-precex 7946 ax-cnre 7947 ax-pre-ltirr 7948 ax-pre-lttrn 7950 ax-pre-apti 7951 ax-pre-ltadd 7952 ax-pre-mulgt0 7953 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5234 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-pnf 8019 df-mnf 8020 df-ltxr 8022 df-sub 8155 df-neg 8156 df-reap 8557 df-ap 8564 |
This theorem is referenced by: 2muliap0 9168 irec 10646 iexpcyc 10651 imval 10886 imre 10887 reim 10888 crim 10894 cjreb 10902 tanval2ap 11748 tanval3ap 11749 efival 11767 |
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