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Mirrors > Home > ILE Home > Th. List > muladd | Unicode version |
Description: Product of two sums. (Contributed by NM, 14-Jan-2006.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Ref | Expression |
---|---|
muladd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcl 7098 | . . 3 | |
2 | adddi 7105 | . . . 4 | |
3 | 2 | 3expb 1139 | . . 3 |
4 | 1, 3 | sylan 277 | . 2 |
5 | adddir 7110 | . . . . 5 | |
6 | 5 | 3expa 1138 | . . . 4 |
7 | 6 | adantrr 462 | . . 3 |
8 | adddir 7110 | . . . . 5 | |
9 | 8 | 3expa 1138 | . . . 4 |
10 | 9 | adantrl 461 | . . 3 |
11 | 7, 10 | oveq12d 5550 | . 2 |
12 | mulcl 7100 | . . . . 5 | |
13 | 12 | ad2ant2r 492 | . . . 4 |
14 | mulcl 7100 | . . . . 5 | |
15 | 14 | ad2ant2lr 493 | . . . 4 |
16 | mulcl 7100 | . . . . . . 7 | |
17 | mulcl 7100 | . . . . . . 7 | |
18 | addcl 7098 | . . . . . . 7 | |
19 | 16, 17, 18 | syl2an 283 | . . . . . 6 |
20 | 19 | anandirs 557 | . . . . 5 |
21 | 20 | adantrl 461 | . . . 4 |
22 | 13, 15, 21 | add32d 7276 | . . 3 |
23 | mulcom 7102 | . . . . . . 7 | |
24 | 23 | ad2ant2l 491 | . . . . . 6 |
25 | 24 | oveq2d 5548 | . . . . 5 |
26 | 16 | ad2ant2rl 494 | . . . . . 6 |
27 | 17 | ad2ant2l 491 | . . . . . 6 |
28 | 13, 26, 27 | addassd 7141 | . . . . 5 |
29 | mulcl 7100 | . . . . . . . 8 | |
30 | 29 | ancoms 264 | . . . . . . 7 |
31 | 30 | ad2ant2l 491 | . . . . . 6 |
32 | 13, 26, 31 | add32d 7276 | . . . . 5 |
33 | 25, 28, 32 | 3eqtr3d 2121 | . . . 4 |
34 | mulcom 7102 | . . . . 5 | |
35 | 34 | ad2ant2lr 493 | . . . 4 |
36 | 33, 35 | oveq12d 5550 | . . 3 |
37 | addcl 7098 | . . . . . 6 | |
38 | 12, 30, 37 | syl2an 283 | . . . . 5 |
39 | 38 | an4s 552 | . . . 4 |
40 | mulcl 7100 | . . . . . 6 | |
41 | 40 | ancoms 264 | . . . . 5 |
42 | 41 | ad2ant2lr 493 | . . . 4 |
43 | 39, 26, 42 | addassd 7141 | . . 3 |
44 | 22, 36, 43 | 3eqtrd 2117 | . 2 |
45 | 4, 11, 44 | 3eqtrd 2117 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 (class class class)co 5532 cc 6979 caddc 6984 cmul 6986 |
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-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-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-addcl 7072 ax-mulcl 7074 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-distr 7080 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 |
This theorem is referenced by: mulsub 7505 muladdi 7513 muladdd 7520 sqabsadd 9941 |
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