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Mirrors > Home > ILE Home > Th. List > muleqadd | Unicode version |
Description: Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.) |
Ref | Expression |
---|---|
muleqadd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 7501 |
. . . . 5
![]() ![]() ![]() ![]() | |
2 | mulsub 7942 |
. . . . . 6
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3 | 1, 2 | mpanr2 430 |
. . . . 5
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4 | 1, 3 | mpanl2 427 |
. . . 4
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5 | 1 | mulid1i 7553 |
. . . . . . 7
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6 | 5 | oveq2i 5679 |
. . . . . 6
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7 | 6 | a1i 9 |
. . . . 5
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8 | mulid1 7548 |
. . . . . 6
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9 | mulid1 7548 |
. . . . . 6
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10 | 8, 9 | oveqan12d 5687 |
. . . . 5
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11 | 7, 10 | oveq12d 5686 |
. . . 4
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12 | mulcl 7532 |
. . . . 5
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13 | addcl 7530 |
. . . . 5
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14 | addsub 7756 |
. . . . . 6
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15 | 1, 14 | mp3an2 1262 |
. . . . 5
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16 | 12, 13, 15 | syl2anc 404 |
. . . 4
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17 | 4, 11, 16 | 3eqtrd 2125 |
. . 3
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18 | 17 | eqeq1d 2097 |
. 2
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19 | 1 | addid2i 7688 |
. . . 4
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20 | 19 | eqeq2i 2099 |
. . 3
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21 | 12, 13 | subcld 7856 |
. . . 4
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22 | 0cn 7543 |
. . . . 5
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23 | addcan2 7726 |
. . . . 5
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24 | 22, 1, 23 | mp3an23 1266 |
. . . 4
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25 | 21, 24 | syl 14 |
. . 3
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26 | 20, 25 | syl5rbbr 194 |
. 2
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27 | 12, 13 | subeq0ad 7866 |
. 2
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28 | 18, 26, 27 | 3bitr2rd 216 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 ax-setind 4368 ax-resscn 7500 ax-1cn 7501 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-addcom 7508 ax-mulcom 7509 ax-addass 7510 ax-mulass 7511 ax-distr 7512 ax-i2m1 7513 ax-1rid 7515 ax-0id 7516 ax-rnegex 7517 ax-cnre 7519 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2624 df-sbc 2844 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-br 3854 df-opab 3908 df-id 4131 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-iota 4995 df-fun 5032 df-fv 5038 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-sub 7718 df-neg 7719 |
This theorem is referenced by: conjmulap 8259 |
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