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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 8157 |
. . 3
| |
| 2 | adddi 8164 |
. . . 4
| |
| 3 | 2 | 3expb 1230 |
. . 3
|
| 4 | 1, 3 | sylan 283 |
. 2
|
| 5 | adddir 8170 |
. . . . 5
| |
| 6 | 5 | 3expa 1229 |
. . . 4
|
| 7 | 6 | adantrr 479 |
. . 3
|
| 8 | adddir 8170 |
. . . . 5
| |
| 9 | 8 | 3expa 1229 |
. . . 4
|
| 10 | 9 | adantrl 478 |
. . 3
|
| 11 | 7, 10 | oveq12d 6036 |
. 2
|
| 12 | mulcl 8159 |
. . . . 5
| |
| 13 | 12 | ad2ant2r 509 |
. . . 4
|
| 14 | mulcl 8159 |
. . . . 5
| |
| 15 | 14 | ad2ant2lr 510 |
. . . 4
|
| 16 | mulcl 8159 |
. . . . . . 7
| |
| 17 | mulcl 8159 |
. . . . . . 7
| |
| 18 | addcl 8157 |
. . . . . . 7
| |
| 19 | 16, 17, 18 | syl2an 289 |
. . . . . 6
|
| 20 | 19 | anandirs 597 |
. . . . 5
|
| 21 | 20 | adantrl 478 |
. . . 4
|
| 22 | 13, 15, 21 | add32d 8347 |
. . 3
|
| 23 | mulcom 8161 |
. . . . . . 7
| |
| 24 | 23 | ad2ant2l 508 |
. . . . . 6
|
| 25 | 24 | oveq2d 6034 |
. . . . 5
|
| 26 | 16 | ad2ant2rl 511 |
. . . . . 6
|
| 27 | 17 | ad2ant2l 508 |
. . . . . 6
|
| 28 | 13, 26, 27 | addassd 8202 |
. . . . 5
|
| 29 | mulcl 8159 |
. . . . . . . 8
| |
| 30 | 29 | ancoms 268 |
. . . . . . 7
|
| 31 | 30 | ad2ant2l 508 |
. . . . . 6
|
| 32 | 13, 26, 31 | add32d 8347 |
. . . . 5
|
| 33 | 25, 28, 32 | 3eqtr3d 2272 |
. . . 4
|
| 34 | mulcom 8161 |
. . . . 5
| |
| 35 | 34 | ad2ant2lr 510 |
. . . 4
|
| 36 | 33, 35 | oveq12d 6036 |
. . 3
|
| 37 | addcl 8157 |
. . . . . 6
| |
| 38 | 12, 30, 37 | syl2an 289 |
. . . . 5
|
| 39 | 38 | an4s 592 |
. . . 4
|
| 40 | mulcl 8159 |
. . . . . 6
| |
| 41 | 40 | ancoms 268 |
. . . . 5
|
| 42 | 41 | ad2ant2lr 510 |
. . . 4
|
| 43 | 39, 26, 42 | addassd 8202 |
. . 3
|
| 44 | 22, 36, 43 | 3eqtrd 2268 |
. 2
|
| 45 | 4, 11, 44 | 3eqtrd 2268 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wceq 1397
wcel 2202
(class class class)co 6018 cc 8030 caddc 8035 cmul 8037 |
| 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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-ext 2213 ax-addcl 8128 ax-mulcl 8130 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-distr 8136 |
| This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-nf 1509 df-sb 1811 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-rex 2516 df-v 2804 df-un 3204 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-iota 5286 df-fv 5334 df-ov 6021 |
| This theorem is referenced by: mulsub 8580 muladdi 8588 muladdd 8595 sqabsadd 11633 demoivreALT 12353 |
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