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Mirrors > Home > MPE Home > Th. List > axmulcl | Structured version Visualization version GIF version |
Description: Closure law for multiplication of complex numbers. Axiom 6 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-mulcl 11174 be used later. Instead, in most cases use mulcl 11196. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axmulcl | β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axmulf 11143 | . 2 β’ Β· :(β Γ β)βΆβ | |
2 | 1 | fovcl 7539 | 1 β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β wcel 2104 (class class class)co 7411 βcc 11110 Β· cmul 11117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-omul 8473 df-er 8705 df-ec 8707 df-qs 8711 df-ni 10869 df-pli 10870 df-mi 10871 df-lti 10872 df-plpq 10905 df-mpq 10906 df-ltpq 10907 df-enq 10908 df-nq 10909 df-erq 10910 df-plq 10911 df-mq 10912 df-1nq 10913 df-rq 10914 df-ltnq 10915 df-np 10978 df-1p 10979 df-plp 10980 df-mp 10981 df-ltp 10982 df-enr 11052 df-nr 11053 df-plr 11054 df-mr 11055 df-m1r 11059 df-c 11118 df-mul 11124 |
This theorem is referenced by: (None) |
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