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Mirrors > Home > ILE Home > Th. List > cnfldmulg | Unicode version |
Description: The group multiple function in the field of complex numbers. (Contributed by Mario Carneiro, 14-Jun-2015.) |
Ref | Expression |
---|---|
cnfldmulg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5898 |
. . . 4
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2 | oveq1 5898 |
. . . 4
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3 | 1, 2 | eqeq12d 2204 |
. . 3
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4 | oveq1 5898 |
. . . 4
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5 | oveq1 5898 |
. . . 4
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6 | 4, 5 | eqeq12d 2204 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | oveq1 5898 |
. . . 4
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8 | oveq1 5898 |
. . . 4
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9 | 7, 8 | eqeq12d 2204 |
. . 3
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10 | oveq1 5898 |
. . . 4
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11 | oveq1 5898 |
. . . 4
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12 | 10, 11 | eqeq12d 2204 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | oveq1 5898 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | oveq1 5898 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 13, 14 | eqeq12d 2204 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | cnfldbas 13835 |
. . . . 5
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17 | cnfld0 13841 |
. . . . 5
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18 | eqid 2189 |
. . . . 5
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19 | 16, 17, 18 | mulg0 13039 |
. . . 4
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20 | mul02 8363 |
. . . 4
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21 | 19, 20 | eqtr4d 2225 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | oveq1 5898 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | cnring 13840 |
. . . . . . . 8
![]() ![]() ![]() | |
24 | ringmnd 13327 |
. . . . . . . 8
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25 | 23, 24 | ax-mp 5 |
. . . . . . 7
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26 | cnfldadd 13836 |
. . . . . . . 8
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27 | 16, 18, 26 | mulgnn0p1 13045 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 25, 27 | mp3an1 1335 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | nn0cn 9205 |
. . . . . . . 8
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30 | 29 | adantr 276 |
. . . . . . 7
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31 | simpr 110 |
. . . . . . 7
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32 | 30, 31 | adddirp1d 8003 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 28, 32 | eqeq12d 2204 |
. . . . 5
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34 | 22, 33 | imbitrrid 156 |
. . . 4
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35 | 34 | expcom 116 |
. . 3
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36 | fveq2 5530 |
. . . . 5
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37 | eqid 2189 |
. . . . . . 7
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38 | 16, 18, 37 | mulgnegnn 13044 |
. . . . . 6
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39 | nncn 8946 |
. . . . . . . 8
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40 | mulneg1 8371 |
. . . . . . . 8
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41 | 39, 40 | sylan 283 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | mulcl 7957 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
43 | 39, 42 | sylan 283 |
. . . . . . . 8
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44 | cnfldneg 13843 |
. . . . . . . 8
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45 | 43, 44 | syl 14 |
. . . . . . 7
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46 | 41, 45 | eqtr4d 2225 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
47 | 38, 46 | eqeq12d 2204 |
. . . . 5
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48 | 36, 47 | imbitrrid 156 |
. . . 4
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49 | 48 | expcom 116 |
. . 3
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50 | 3, 6, 9, 12, 15, 21, 35, 49 | zindd 9390 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
51 | 50 | impcom 125 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-mulrcl 7929 ax-addcom 7930 ax-mulcom 7931 ax-addass 7932 ax-mulass 7933 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-1rid 7937 ax-0id 7938 ax-rnegex 7939 ax-precex 7940 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 ax-pre-mulgt0 7947 ax-addf 7952 ax-mulf 7953 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 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-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-tp 3615 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 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-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-frec 6410 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-reap 8551 df-inn 8939 df-2 8997 df-3 8998 df-4 8999 df-5 9000 df-6 9001 df-7 9002 df-8 9003 df-9 9004 df-n0 9196 df-z 9273 df-dec 9404 df-uz 9548 df-fz 10028 df-seqfrec 10465 df-cj 10870 df-struct 12488 df-ndx 12489 df-slot 12490 df-base 12492 df-sets 12493 df-plusg 12574 df-mulr 12575 df-starv 12576 df-0g 12735 df-mgm 12804 df-sgrp 12837 df-mnd 12850 df-grp 12920 df-minusg 12921 df-mulg 13034 df-cmn 13192 df-mgp 13242 df-ring 13319 df-cring 13320 df-icnfld 13832 |
This theorem is referenced by: zsssubrg 13855 zringmulg 13864 mulgrhm2 13875 |
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