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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 6776 |
. . . . 5
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2 | mulsub 7194 |
. . . . . 6
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3 | 1, 2 | mpanr2 414 |
. . . . 5
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4 | 1, 3 | mpanl2 411 |
. . . 4
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5 | 1 | mulid1i 6827 |
. . . . . . 7
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6 | 5 | oveq2i 5466 |
. . . . . 6
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7 | 6 | a1i 9 |
. . . . 5
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8 | mulid1 6822 |
. . . . . 6
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9 | mulid1 6822 |
. . . . . 6
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10 | 8, 9 | oveqan12d 5474 |
. . . . 5
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11 | 7, 10 | oveq12d 5473 |
. . . 4
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12 | mulcl 6806 |
. . . . 5
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13 | addcl 6804 |
. . . . 5
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14 | addsub 7019 |
. . . . . 6
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15 | 1, 14 | mp3an2 1219 |
. . . . 5
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16 | 12, 13, 15 | syl2anc 391 |
. . . 4
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17 | 4, 11, 16 | 3eqtrd 2073 |
. . 3
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18 | 17 | eqeq1d 2045 |
. 2
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19 | 1 | addid2i 6953 |
. . . 4
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20 | 19 | eqeq2i 2047 |
. . 3
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21 | 12, 13 | subcld 7118 |
. . . 4
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22 | 0cn 6817 |
. . . . 5
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23 | addcan2 6989 |
. . . . 5
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24 | 22, 1, 23 | mp3an23 1223 |
. . . 4
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25 | 21, 24 | syl 14 |
. . 3
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26 | 20, 25 | syl5rbbr 184 |
. 2
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27 | 12, 13 | subeq0ad 7128 |
. 2
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28 | 18, 26, 27 | 3bitr2rd 206 |
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 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-setind 4220 ax-resscn 6775 ax-1cn 6776 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-addcom 6783 ax-mulcom 6784 ax-addass 6785 ax-mulass 6786 ax-distr 6787 ax-i2m1 6788 ax-1rid 6790 ax-0id 6791 ax-rnegex 6792 ax-cnre 6794 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-sub 6981 df-neg 6982 |
This theorem is referenced by: conjmulap 7487 |
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