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Mirrors > Home > MPE Home > Th. List > axmulrcl | Structured version Unicode version |
Description: Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulrcl 9457 be used later. Instead, in most cases use remulcl 9479. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.) |
Ref | Expression |
---|---|
axmulrcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal 9410 |
. 2
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2 | elreal 9410 |
. 2
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3 | oveq1 6208 |
. . 3
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4 | 3 | eleq1d 2523 |
. 2
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5 | oveq2 6209 |
. . 3
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6 | 5 | eleq1d 2523 |
. 2
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7 | mulresr 9418 |
. . 3
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8 | mulclsr 9363 |
. . . 4
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9 | opelreal 9409 |
. . . 4
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10 | 8, 9 | sylibr 212 |
. . 3
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11 | 7, 10 | eqeltrd 2542 |
. 2
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12 | 1, 2, 4, 6, 11 | 2gencl 3109 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4522 ax-nul 4530 ax-pow 4579 ax-pr 4640 ax-un 6483 ax-inf2 7959 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3397 df-dif 3440 df-un 3442 df-in 3444 df-ss 3451 df-pss 3453 df-nul 3747 df-if 3901 df-pw 3971 df-sn 3987 df-pr 3989 df-tp 3991 df-op 3993 df-uni 4201 df-int 4238 df-iun 4282 df-br 4402 df-opab 4460 df-mpt 4461 df-tr 4495 df-eprel 4741 df-id 4745 df-po 4750 df-so 4751 df-fr 4788 df-we 4790 df-ord 4831 df-on 4832 df-lim 4833 df-suc 4834 df-xp 4955 df-rel 4956 df-cnv 4957 df-co 4958 df-dm 4959 df-rn 4960 df-res 4961 df-ima 4962 df-iota 5490 df-fun 5529 df-fn 5530 df-f 5531 df-f1 5532 df-fo 5533 df-f1o 5534 df-fv 5535 df-ov 6204 df-oprab 6205 df-mpt2 6206 df-om 6588 df-1st 6688 df-2nd 6689 df-recs 6943 df-rdg 6977 df-1o 7031 df-oadd 7035 df-omul 7036 df-er 7212 df-ec 7214 df-qs 7218 df-ni 9153 df-pli 9154 df-mi 9155 df-lti 9156 df-plpq 9189 df-mpq 9190 df-ltpq 9191 df-enq 9192 df-nq 9193 df-erq 9194 df-plq 9195 df-mq 9196 df-1nq 9197 df-rq 9198 df-ltnq 9199 df-np 9262 df-1p 9263 df-plp 9264 df-mp 9265 df-ltp 9266 df-plpr 9336 df-mpr 9337 df-enr 9338 df-nr 9339 df-plr 9340 df-mr 9341 df-0r 9343 df-m1r 9345 df-c 9400 df-r 9404 df-mul 9406 |
This theorem is referenced by: (None) |
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