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Mirrors > Home > ILE Home > Th. List > axmulcom | Unicode version |
Description: Multiplication of complex numbers is commutative. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcom 7646 be used later. Instead, use mulcom 7673. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axmulcom |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcnqs 7576 |
. 2
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2 | mulcnsrec 7578 |
. 2
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3 | mulcnsrec 7578 |
. 2
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4 | simpll 501 |
. . . 4
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5 | simprl 503 |
. . . 4
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6 | mulcomsrg 7500 |
. . . 4
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7 | 4, 5, 6 | syl2anc 406 |
. . 3
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8 | simplr 502 |
. . . . 5
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9 | simprr 504 |
. . . . 5
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10 | mulcomsrg 7500 |
. . . . 5
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11 | 8, 9, 10 | syl2anc 406 |
. . . 4
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12 | 11 | oveq2d 5744 |
. . 3
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13 | 7, 12 | oveq12d 5746 |
. 2
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14 | mulcomsrg 7500 |
. . . . 5
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15 | 8, 5, 14 | syl2anc 406 |
. . . 4
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16 | mulcomsrg 7500 |
. . . . 5
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17 | 4, 9, 16 | syl2anc 406 |
. . . 4
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18 | 15, 17 | oveq12d 5746 |
. . 3
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19 | mulclsr 7497 |
. . . . 5
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20 | 5, 8, 19 | syl2anc 406 |
. . . 4
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21 | mulclsr 7497 |
. . . . 5
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22 | 9, 4, 21 | syl2anc 406 |
. . . 4
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23 | addcomsrg 7498 |
. . . 4
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24 | 20, 22, 23 | syl2anc 406 |
. . 3
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25 | 18, 24 | eqtrd 2147 |
. 2
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26 | 1, 2, 3, 13, 25 | ecovicom 6491 |
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 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-coll 4003 ax-sep 4006 ax-nul 4014 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-iinf 4462 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-iun 3781 df-br 3896 df-opab 3950 df-mpt 3951 df-tr 3987 df-eprel 4171 df-id 4175 df-po 4178 df-iso 4179 df-iord 4248 df-on 4250 df-suc 4253 df-iom 4465 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-f1 5086 df-fo 5087 df-f1o 5088 df-fv 5089 df-ov 5731 df-oprab 5732 df-mpo 5733 df-1st 5992 df-2nd 5993 df-recs 6156 df-irdg 6221 df-1o 6267 df-2o 6268 df-oadd 6271 df-omul 6272 df-er 6383 df-ec 6385 df-qs 6389 df-ni 7060 df-pli 7061 df-mi 7062 df-lti 7063 df-plpq 7100 df-mpq 7101 df-enq 7103 df-nqqs 7104 df-plqqs 7105 df-mqqs 7106 df-1nqqs 7107 df-rq 7108 df-ltnqqs 7109 df-enq0 7180 df-nq0 7181 df-0nq0 7182 df-plq0 7183 df-mq0 7184 df-inp 7222 df-i1p 7223 df-iplp 7224 df-imp 7225 df-enr 7469 df-nr 7470 df-plr 7471 df-mr 7472 df-m1r 7476 df-c 7553 df-mul 7559 |
This theorem is referenced by: rereceu 7624 recriota 7625 |
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