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Mirrors > Home > ILE Home > Th. List > axaddf | Unicode version |
Description: Addition is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axaddcl 7854. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 7924. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axaddf |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moeq 2912 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | 1 | mosubop 4689 |
. . . . . . . 8
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3 | 2 | mosubop 4689 |
. . . . . . 7
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4 | anass 401 |
. . . . . . . . . . 11
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5 | 4 | 2exbii 1606 |
. . . . . . . . . 10
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6 | 19.42vv 1911 |
. . . . . . . . . 10
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7 | 5, 6 | bitri 184 |
. . . . . . . . 9
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8 | 7 | 2exbii 1606 |
. . . . . . . 8
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9 | 8 | mobii 2063 |
. . . . . . 7
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10 | 3, 9 | mpbir 146 |
. . . . . 6
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11 | 10 | moani 2096 |
. . . . 5
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12 | 11 | funoprab 5969 |
. . . 4
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13 | df-add 7813 |
. . . . 5
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14 | 13 | funeqi 5233 |
. . . 4
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15 | 12, 14 | mpbir 146 |
. . 3
![]() ![]() ![]() |
16 | 13 | dmeqi 4824 |
. . . . 5
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17 | dmoprabss 5951 |
. . . . 5
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18 | 16, 17 | eqsstri 3187 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | cnm 7822 |
. . . . . . 7
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20 | 19 | adantl 277 |
. . . . . 6
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21 | axaddcl 7854 |
. . . . . . 7
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22 | 21 | adantl 277 |
. . . . . 6
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23 | funrel 5229 |
. . . . . . 7
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24 | 15, 23 | mp1i 10 |
. . . . . 6
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25 | 20, 22, 24 | oprssdmm 6166 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 25 | mptru 1362 |
. . . 4
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27 | 18, 26 | eqssi 3171 |
. . 3
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28 | df-fn 5215 |
. . 3
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29 | 15, 27, 28 | mpbir2an 942 |
. 2
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30 | 21 | rgen2a 2531 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | ffnov 5973 |
. 2
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32 | 29, 30, 31 | mpbir2an 942 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 |
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 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-eprel 4286 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-1o 6411 df-2o 6412 df-oadd 6415 df-omul 6416 df-er 6529 df-ec 6531 df-qs 6535 df-ni 7294 df-pli 7295 df-mi 7296 df-lti 7297 df-plpq 7334 df-mpq 7335 df-enq 7337 df-nqqs 7338 df-plqqs 7339 df-mqqs 7340 df-1nqqs 7341 df-rq 7342 df-ltnqqs 7343 df-enq0 7414 df-nq0 7415 df-0nq0 7416 df-plq0 7417 df-mq0 7418 df-inp 7456 df-iplp 7458 df-enr 7716 df-nr 7717 df-plr 7718 df-c 7808 df-add 7813 |
This theorem is referenced by: (None) |
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