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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 |
     ![/\](wedge.gif "/\"){kind=small}       
 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | addcl 7938 |
. . 3
| |
| 2 | adddi 7945 |
. . . 4
| |
| 3 | 2 | 3expb 1204 |
. . 3
|
| 4 | 1, 3 | sylan 283 |
. 2
|
| 5 | adddir 7950 |
. . . . 5
| |
| 6 | 5 | 3expa 1203 |
. . . 4
|
| 7 | 6 | adantrr 479 |
. . 3
|
| 8 | adddir 7950 |
. . . . 5
| |
| 9 | 8 | 3expa 1203 |
. . . 4
|
| 10 | 9 | adantrl 478 |
. . 3
|
| 11 | 7, 10 | oveq12d 5895 |
. 2
|
| 12 | mulcl 7940 |
. . . . 5
| |
| 13 | 12 | ad2ant2r 509 |
. . . 4
|
| 14 | mulcl 7940 |
. . . . 5
| |
| 15 | 14 | ad2ant2lr 510 |
. . . 4
|
| 16 | mulcl 7940 |
. . . . . . 7
| |
| 17 | mulcl 7940 |
. . . . . . 7
| |
| 18 | addcl 7938 |
. . . . . . 7
| |
| 19 | 16, 17, 18 | syl2an 289 |
. . . . . 6
|
| 20 | 19 | anandirs 593 |
. . . . 5
|
| 21 | 20 | adantrl 478 |
. . . 4
|
| 22 | 13, 15, 21 | add32d 8127 |
. . 3
|
| 23 | mulcom 7942 |
. . . . . . 7
| |
| 24 | 23 | ad2ant2l 508 |
. . . . . 6
|
| 25 | 24 | oveq2d 5893 |
. . . . 5
|
| 26 | 16 | ad2ant2rl 511 |
. . . . . 6
|
| 27 | 17 | ad2ant2l 508 |
. . . . . 6
|
| 28 | 13, 26, 27 | addassd 7982 |
. . . . 5
|
| 29 | mulcl 7940 |
. . . . . . . 8
| |
| 30 | 29 | ancoms 268 |
. . . . . . 7
|
| 31 | 30 | ad2ant2l 508 |
. . . . . 6
|
| 32 | 13, 26, 31 | add32d 8127 |
. . . . 5
|
| 33 | 25, 28, 32 | 3eqtr3d 2218 |
. . . 4
|
| 34 | mulcom 7942 |
. . . . 5
| |
| 35 | 34 | ad2ant2lr 510 |
. . . 4
|
| 36 | 33, 35 | oveq12d 5895 |
. . 3
|
| 37 | addcl 7938 |
. . . . . 6
| |
| 38 | 12, 30, 37 | syl2an 289 |
. . . . 5
|
| 39 | 38 | an4s 588 |
. . . 4
|
| 40 | mulcl 7940 |
. . . . . 6
| |
| 41 | 40 | ancoms 268 |
. . . . 5
|
| 42 | 41 | ad2ant2lr 510 |
. . . 4
|
| 43 | 39, 26, 42 | addassd 7982 |
. . 3
|
| 44 | 22, 36, 43 | 3eqtrd 2214 |
. 2
|
| 45 | 4, 11, 44 | 3eqtrd 2214 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wceq 1353
wcel 2148
(class class class)co 5877 cc 7811 caddc 7816 cmul 7818 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-addcl 7909 ax-mulcl 7911 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-distr 7917 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2741 df-un 3135 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-iota 5180 df-fv 5226 df-ov 5880 |
| This theorem is referenced by: mulsub 8360 muladdi 8368 muladdd 8375 sqabsadd 11066 demoivreALT 11783 |
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