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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 7069 | . . . . 5 | |
2 | mulsub 7505 | . . . . . 6 | |
3 | 1, 2 | mpanr2 428 | . . . . 5 |
4 | 1, 3 | mpanl2 425 | . . . 4 |
5 | 1 | mulid1i 7121 | . . . . . . 7 |
6 | 5 | oveq2i 5543 | . . . . . 6 |
7 | 6 | a1i 9 | . . . . 5 |
8 | mulid1 7116 | . . . . . 6 | |
9 | mulid1 7116 | . . . . . 6 | |
10 | 8, 9 | oveqan12d 5551 | . . . . 5 |
11 | 7, 10 | oveq12d 5550 | . . . 4 |
12 | mulcl 7100 | . . . . 5 | |
13 | addcl 7098 | . . . . 5 | |
14 | addsub 7319 | . . . . . 6 | |
15 | 1, 14 | mp3an2 1256 | . . . . 5 |
16 | 12, 13, 15 | syl2anc 403 | . . . 4 |
17 | 4, 11, 16 | 3eqtrd 2117 | . . 3 |
18 | 17 | eqeq1d 2089 | . 2 |
19 | 1 | addid2i 7251 | . . . 4 |
20 | 19 | eqeq2i 2091 | . . 3 |
21 | 12, 13 | subcld 7419 | . . . 4 |
22 | 0cn 7111 | . . . . 5 | |
23 | addcan2 7289 | . . . . 5 | |
24 | 22, 1, 23 | mp3an23 1260 | . . . 4 |
25 | 21, 24 | syl 14 | . . 3 |
26 | 20, 25 | syl5rbbr 193 | . 2 |
27 | 12, 13 | subeq0ad 7429 | . 2 |
28 | 18, 26, 27 | 3bitr2rd 215 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 (class class class)co 5532 cc 6979 cc0 6981 c1 6982 caddc 6984 cmul 6986 cmin 7279 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-resscn 7068 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-neg 7282 |
This theorem is referenced by: conjmulap 7817 |
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