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Mirrors > Home > ILE Home > Th. List > axaddass | Unicode version |
Description: Addition of complex numbers is associative. This theorem transfers the associative laws for the real and imaginary signed real components of complex number pairs, to complex number addition itself. 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-addass 7901 be used later. Instead, use addass 7929. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axaddass |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcnqs 7828 |
. 2
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2 | addcnsrec 7829 |
. 2
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3 | addcnsrec 7829 |
. 2
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4 | addcnsrec 7829 |
. 2
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5 | addcnsrec 7829 |
. 2
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6 | addclsr 7740 |
. . . 4
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7 | addclsr 7740 |
. . . 4
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8 | 6, 7 | anim12i 338 |
. . 3
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9 | 8 | an4s 588 |
. 2
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10 | addclsr 7740 |
. . . 4
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11 | addclsr 7740 |
. . . 4
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12 | 10, 11 | anim12i 338 |
. . 3
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13 | 12 | an4s 588 |
. 2
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14 | addasssrg 7743 |
. . . . 5
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15 | 14 | 3adant3r 1235 |
. . . 4
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16 | 15 | 3adant2r 1233 |
. . 3
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17 | 16 | 3adant1r 1231 |
. 2
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18 | addasssrg 7743 |
. . . . 5
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19 | 18 | 3adant3l 1234 |
. . . 4
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20 | 19 | 3adant2l 1232 |
. . 3
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21 | 20 | 3adant1l 1230 |
. 2
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22 | 1, 2, 3, 4, 5, 9, 13, 17, 21 | ecoviass 6639 |
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 7291 df-pli 7292 df-mi 7293 df-lti 7294 df-plpq 7331 df-mpq 7332 df-enq 7334 df-nqqs 7335 df-plqqs 7336 df-mqqs 7337 df-1nqqs 7338 df-rq 7339 df-ltnqqs 7340 df-enq0 7411 df-nq0 7412 df-0nq0 7413 df-plq0 7414 df-mq0 7415 df-inp 7453 df-iplp 7455 df-enr 7713 df-nr 7714 df-plr 7715 df-c 7805 df-add 7810 |
This theorem is referenced by: (None) |
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