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Mirrors > Home > ILE Home > Th. List > axaddcl | Unicode version |
Description: Closure law for addition of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcl 7882 be used later. Instead, in most cases use addcl 7911. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axaddcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxpi 4636 | . . . . 5 | |
2 | df-c 7792 | . . . . 5 | |
3 | 1, 2 | eleq2s 2270 | . . . 4 |
4 | elxpi 4636 | . . . . 5 | |
5 | 4, 2 | eleq2s 2270 | . . . 4 |
6 | 3, 5 | anim12i 338 | . . 3 |
7 | ee4anv 1932 | . . 3 | |
8 | 6, 7 | sylibr 134 | . 2 |
9 | simpll 527 | . . . . . . 7 | |
10 | simprl 529 | . . . . . . 7 | |
11 | 9, 10 | oveq12d 5883 | . . . . . 6 |
12 | addcnsr 7808 | . . . . . . 7 | |
13 | 12 | ad2ant2l 508 | . . . . . 6 |
14 | 11, 13 | eqtrd 2208 | . . . . 5 |
15 | addclsr 7727 | . . . . . . . . 9 | |
16 | 15 | ad2ant2r 509 | . . . . . . . 8 |
17 | addclsr 7727 | . . . . . . . . 9 | |
18 | 17 | ad2ant2l 508 | . . . . . . . 8 |
19 | opelxpi 4652 | . . . . . . . 8 | |
20 | 16, 18, 19 | syl2anc 411 | . . . . . . 7 |
21 | 20, 2 | eleqtrrdi 2269 | . . . . . 6 |
22 | 21 | ad2ant2l 508 | . . . . 5 |
23 | 14, 22 | eqeltrd 2252 | . . . 4 |
24 | 23 | exlimivv 1894 | . . 3 |
25 | 24 | exlimivv 1894 | . 2 |
26 | 8, 25 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wceq 1353 wex 1490 wcel 2146 cop 3592 cxp 4618 (class class class)co 5865 cnr 7271 cplr 7275 cc 7784 caddc 7789 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-eprel 4283 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-irdg 6361 df-1o 6407 df-2o 6408 df-oadd 6411 df-omul 6412 df-er 6525 df-ec 6527 df-qs 6531 df-ni 7278 df-pli 7279 df-mi 7280 df-lti 7281 df-plpq 7318 df-mpq 7319 df-enq 7321 df-nqqs 7322 df-plqqs 7323 df-mqqs 7324 df-1nqqs 7325 df-rq 7326 df-ltnqqs 7327 df-enq0 7398 df-nq0 7399 df-0nq0 7400 df-plq0 7401 df-mq0 7402 df-inp 7440 df-iplp 7442 df-enr 7700 df-nr 7701 df-plr 7702 df-c 7792 df-add 7797 |
This theorem is referenced by: axaddf 7842 |
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