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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 11202 be used later. Instead, in most cases use mulcl 11224. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axmulcl | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axmulf 11171 | . 2 ⊢ · :(ℂ × ℂ)⟶ℂ | |
2 | 1 | fovcl 7549 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 (class class class)co 7419 ℂcc 11138 · cmul 11145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-omul 8492 df-er 8725 df-ec 8727 df-qs 8731 df-ni 10897 df-pli 10898 df-mi 10899 df-lti 10900 df-plpq 10933 df-mpq 10934 df-ltpq 10935 df-enq 10936 df-nq 10937 df-erq 10938 df-plq 10939 df-mq 10940 df-1nq 10941 df-rq 10942 df-ltnq 10943 df-np 11006 df-1p 11007 df-plp 11008 df-mp 11009 df-ltp 11010 df-enr 11080 df-nr 11081 df-plr 11082 df-mr 11083 df-m1r 11087 df-c 11146 df-mul 11152 |
This theorem is referenced by: (None) |
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