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Mirrors > Home > MPE Home > Th. List > addcli | Structured version Visualization version GIF version |
Description: Closure law for addition. (Contributed by NM, 23-Nov-1994.) |
Ref | Expression |
---|---|
axi.1 | ⊢ 𝐴 ∈ ℂ |
axi.2 | ⊢ 𝐵 ∈ ℂ |
Ref | Expression |
---|---|
addcli | ⊢ (𝐴 + 𝐵) ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axi.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | axi.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | addcl 11266 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝐴 + 𝐵) ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2108 (class class class)co 7448 ℂcc 11182 + caddc 11187 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-addcl 11244 |
This theorem depends on definitions: df-bi 207 df-an 396 |
This theorem is referenced by: eqneg 12014 2cn 12368 3cn 12374 4cn 12378 5cn 12381 6cn 12384 7cn 12387 8cn 12390 9cn 12393 nummac 12803 binom2i 14261 sqeqori 14263 crreczi 14277 nn0opthlem1 14317 nn0opth2i 14320 3dvds2dec 16381 mod2xnegi 17118 karatsuba 17131 pige3ALT 26580 eff1o 26609 1cubrlem 26902 1cubr 26903 bposlem8 27353 ax5seglem7 28968 ipidsq 30742 ip1ilem 30858 pythi 30882 normlem2 31143 normlem3 31144 normlem7 31148 normlem9 31150 bcseqi 31152 norm-ii-i 31169 normpythi 31174 normpari 31186 polid2i 31189 lnopunilem1 32042 lnophmlem2 32049 dpmul100 32861 dpadd3 32876 dpmul4 32878 ballotlem2 34453 hgt750lem2 34629 quad3 35638 faclimlem1 35705 itg2addnclem3 37633 sqmid3api 42272 235t711 42293 sn-0tie0 42415 fltnltalem 42617 areaquad 43177 resqrtvalex 43607 imsqrtvalex 43608 fourierswlem 46151 fouriersw 46152 2t6m3t4e0 48073 |
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