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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 6977 | . . . . 5 | |
2 | mulsub 7398 | . . . . . 6 | |
3 | 1, 2 | mpanr2 414 | . . . . 5 |
4 | 1, 3 | mpanl2 411 | . . . 4 |
5 | 1 | mulid1i 7029 | . . . . . . 7 |
6 | 5 | oveq2i 5523 | . . . . . 6 |
7 | 6 | a1i 9 | . . . . 5 |
8 | mulid1 7024 | . . . . . 6 | |
9 | mulid1 7024 | . . . . . 6 | |
10 | 8, 9 | oveqan12d 5531 | . . . . 5 |
11 | 7, 10 | oveq12d 5530 | . . . 4 |
12 | mulcl 7008 | . . . . 5 | |
13 | addcl 7006 | . . . . 5 | |
14 | addsub 7222 | . . . . . 6 | |
15 | 1, 14 | mp3an2 1220 | . . . . 5 |
16 | 12, 13, 15 | syl2anc 391 | . . . 4 |
17 | 4, 11, 16 | 3eqtrd 2076 | . . 3 |
18 | 17 | eqeq1d 2048 | . 2 |
19 | 1 | addid2i 7156 | . . . 4 |
20 | 19 | eqeq2i 2050 | . . 3 |
21 | 12, 13 | subcld 7322 | . . . 4 |
22 | 0cn 7019 | . . . . 5 | |
23 | addcan2 7192 | . . . . 5 | |
24 | 22, 1, 23 | mp3an23 1224 | . . . 4 |
25 | 21, 24 | syl 14 | . . 3 |
26 | 20, 25 | syl5rbbr 184 | . 2 |
27 | 12, 13 | subeq0ad 7332 | . 2 |
28 | 18, 26, 27 | 3bitr2rd 206 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 (class class class)co 5512 cc 6887 cc0 6889 c1 6890 caddc 6892 cmul 6894 cmin 7182 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-sub 7184 df-neg 7185 |
This theorem is referenced by: conjmulap 7705 |
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