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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 7878 |
. . 3
| |
| 2 | adddi 7885 |
. . . 4
| |
| 3 | 2 | 3expb 1194 |
. . 3
|
| 4 | 1, 3 | sylan 281 |
. 2
|
| 5 | adddir 7890 |
. . . . 5
| |
| 6 | 5 | 3expa 1193 |
. . . 4
|
| 7 | 6 | adantrr 471 |
. . 3
|
| 8 | adddir 7890 |
. . . . 5
| |
| 9 | 8 | 3expa 1193 |
. . . 4
|
| 10 | 9 | adantrl 470 |
. . 3
|
| 11 | 7, 10 | oveq12d 5860 |
. 2
|
| 12 | mulcl 7880 |
. . . . 5
| |
| 13 | 12 | ad2ant2r 501 |
. . . 4
|
| 14 | mulcl 7880 |
. . . . 5
| |
| 15 | 14 | ad2ant2lr 502 |
. . . 4
|
| 16 | mulcl 7880 |
. . . . . . 7
| |
| 17 | mulcl 7880 |
. . . . . . 7
| |
| 18 | addcl 7878 |
. . . . . . 7
| |
| 19 | 16, 17, 18 | syl2an 287 |
. . . . . 6
|
| 20 | 19 | anandirs 583 |
. . . . 5
|
| 21 | 20 | adantrl 470 |
. . . 4
|
| 22 | 13, 15, 21 | add32d 8066 |
. . 3
|
| 23 | mulcom 7882 |
. . . . . . 7
| |
| 24 | 23 | ad2ant2l 500 |
. . . . . 6
|
| 25 | 24 | oveq2d 5858 |
. . . . 5
|
| 26 | 16 | ad2ant2rl 503 |
. . . . . 6
|
| 27 | 17 | ad2ant2l 500 |
. . . . . 6
|
| 28 | 13, 26, 27 | addassd 7921 |
. . . . 5
|
| 29 | mulcl 7880 |
. . . . . . . 8
| |
| 30 | 29 | ancoms 266 |
. . . . . . 7
|
| 31 | 30 | ad2ant2l 500 |
. . . . . 6
|
| 32 | 13, 26, 31 | add32d 8066 |
. . . . 5
|
| 33 | 25, 28, 32 | 3eqtr3d 2206 |
. . . 4
|
| 34 | mulcom 7882 |
. . . . 5
| |
| 35 | 34 | ad2ant2lr 502 |
. . . 4
|
| 36 | 33, 35 | oveq12d 5860 |
. . 3
|
| 37 | addcl 7878 |
. . . . . 6
| |
| 38 | 12, 30, 37 | syl2an 287 |
. . . . 5
|
| 39 | 38 | an4s 578 |
. . . 4
|
| 40 | mulcl 7880 |
. . . . . 6
| |
| 41 | 40 | ancoms 266 |
. . . . 5
|
| 42 | 41 | ad2ant2lr 502 |
. . . 4
|
| 43 | 39, 26, 42 | addassd 7921 |
. . 3
|
| 44 | 22, 36, 43 | 3eqtrd 2202 |
. 2
|
| 45 | 4, 11, 44 | 3eqtrd 2202 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wceq 1343
wcel 2136
(class class class)co 5842 cc 7751 caddc 7756 cmul 7758 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-ext 2147 ax-addcl 7849 ax-mulcl 7851 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-distr 7857 |
| This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-rex 2450 df-v 2728 df-un 3120 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-iota 5153 df-fv 5196 df-ov 5845 |
| This theorem is referenced by: mulsub 8299 muladdi 8307 muladdd 8314 sqabsadd 10997 demoivreALT 11714 |
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