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Mirrors > Home > ILE Home > Th. List > cjdivap | Unicode version |
Description: Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.) |
Ref | Expression |
---|---|
cjdivap |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divclap 8653 |
. . . 4
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2 | cjcl 10875 |
. . . 4
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3 | 1, 2 | syl 14 |
. . 3
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4 | simp2 1000 |
. . . 4
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5 | cjcl 10875 |
. . . 4
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6 | 4, 5 | syl 14 |
. . 3
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7 | simp3 1001 |
. . . 4
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8 | cjap0 10934 |
. . . . 5
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9 | 4, 8 | syl 14 |
. . . 4
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10 | 7, 9 | mpbid 147 |
. . 3
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11 | 3, 6, 10 | divcanap4d 8771 |
. 2
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12 | cjmul 10912 |
. . . . 5
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13 | 1, 4, 12 | syl2anc 411 |
. . . 4
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14 | divcanap1 8656 |
. . . . 5
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15 | 14 | fveq2d 5534 |
. . . 4
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16 | 13, 15 | eqtr3d 2224 |
. . 3
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17 | 16 | oveq1d 5906 |
. 2
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18 | 11, 17 | eqtr3d 2224 |
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 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-mulrcl 7928 ax-addcom 7929 ax-mulcom 7930 ax-addass 7931 ax-mulass 7932 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-1rid 7936 ax-0id 7937 ax-rnegex 7938 ax-precex 7939 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-apti 7944 ax-pre-ltadd 7945 ax-pre-mulgt0 7946 ax-pre-mulext 7947 |
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-rmo 2476 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-mpt 4081 df-id 4308 df-po 4311 df-iso 4312 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-reap 8550 df-ap 8557 df-div 8648 df-2 8996 df-cj 10869 df-re 10870 df-im 10871 |
This theorem is referenced by: cjdivapi 10962 cjdivapd 10995 |
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