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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 11262 | . 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 2103 (class class class)co 7445 ℂcc 11178 + caddc 11183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-addcl 11240 |
This theorem depends on definitions: df-bi 207 df-an 396 |
This theorem is referenced by: eqneg 12010 2cn 12364 3cn 12370 4cn 12374 5cn 12377 6cn 12380 7cn 12383 8cn 12386 9cn 12389 nummac 12799 binom2i 14257 sqeqori 14259 crreczi 14273 nn0opthlem1 14313 nn0opth2i 14316 3dvds2dec 16375 mod2xnegi 17113 karatsuba 17126 pige3ALT 26571 eff1o 26600 1cubrlem 26893 1cubr 26894 bposlem8 27344 ax5seglem7 28959 ipidsq 30733 ip1ilem 30849 pythi 30873 normlem2 31134 normlem3 31135 normlem7 31139 normlem9 31141 bcseqi 31143 norm-ii-i 31160 normpythi 31165 normpari 31177 polid2i 31180 lnopunilem1 32033 lnophmlem2 32040 dpmul100 32853 dpadd3 32868 dpmul4 32870 ballotlem2 34445 hgt750lem2 34621 quad3 35630 faclimlem1 35697 itg2addnclem3 37581 sqmid3api 42220 235t711 42230 sn-0tie0 42348 fltnltalem 42550 areaquad 43117 resqrtvalex 43547 imsqrtvalex 43548 fourierswlem 46085 fouriersw 46086 2t6m3t4e0 47991 |
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