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Mirrors > Home > MPE Home > Th. List > axaddcl | Structured version Visualization version GIF version |
Description: Closure law for addition of complex numbers. Axiom 4 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-addcl 10932 be used later. Instead, in most cases use addcl 10954. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axaddcl | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axaddf 10902 | . 2 ⊢ + :(ℂ × ℂ)⟶ℂ | |
2 | 1 | fovcl 7396 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2110 (class class class)co 7271 ℂcc 10870 + caddc 10875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-oadd 8292 df-omul 8293 df-er 8481 df-ec 8483 df-qs 8487 df-ni 10629 df-pli 10630 df-mi 10631 df-lti 10632 df-plpq 10665 df-mpq 10666 df-ltpq 10667 df-enq 10668 df-nq 10669 df-erq 10670 df-plq 10671 df-mq 10672 df-1nq 10673 df-rq 10674 df-ltnq 10675 df-np 10738 df-plp 10740 df-ltp 10742 df-enr 10812 df-nr 10813 df-plr 10814 df-c 10878 df-add 10883 |
This theorem is referenced by: (None) |
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