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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 7006 | . . 3 | |
2 | adddi 7013 | . . . 4 | |
3 | 2 | 3expb 1105 | . . 3 |
4 | 1, 3 | sylan 267 | . 2 |
5 | adddir 7018 | . . . . 5 | |
6 | 5 | 3expa 1104 | . . . 4 |
7 | 6 | adantrr 448 | . . 3 |
8 | adddir 7018 | . . . . 5 | |
9 | 8 | 3expa 1104 | . . . 4 |
10 | 9 | adantrl 447 | . . 3 |
11 | 7, 10 | oveq12d 5530 | . 2 |
12 | mulcl 7008 | . . . . 5 | |
13 | 12 | ad2ant2r 478 | . . . 4 |
14 | mulcl 7008 | . . . . 5 | |
15 | 14 | ad2ant2lr 479 | . . . 4 |
16 | mulcl 7008 | . . . . . . 7 | |
17 | mulcl 7008 | . . . . . . 7 | |
18 | addcl 7006 | . . . . . . 7 | |
19 | 16, 17, 18 | syl2an 273 | . . . . . 6 |
20 | 19 | anandirs 527 | . . . . 5 |
21 | 20 | adantrl 447 | . . . 4 |
22 | 13, 15, 21 | add32d 7179 | . . 3 |
23 | mulcom 7010 | . . . . . . 7 | |
24 | 23 | ad2ant2l 477 | . . . . . 6 |
25 | 24 | oveq2d 5528 | . . . . 5 |
26 | 16 | ad2ant2rl 480 | . . . . . 6 |
27 | 17 | ad2ant2l 477 | . . . . . 6 |
28 | 13, 26, 27 | addassd 7049 | . . . . 5 |
29 | mulcl 7008 | . . . . . . . 8 | |
30 | 29 | ancoms 255 | . . . . . . 7 |
31 | 30 | ad2ant2l 477 | . . . . . 6 |
32 | 13, 26, 31 | add32d 7179 | . . . . 5 |
33 | 25, 28, 32 | 3eqtr3d 2080 | . . . 4 |
34 | mulcom 7010 | . . . . 5 | |
35 | 34 | ad2ant2lr 479 | . . . 4 |
36 | 33, 35 | oveq12d 5530 | . . 3 |
37 | addcl 7006 | . . . . . 6 | |
38 | 12, 30, 37 | syl2an 273 | . . . . 5 |
39 | 38 | an4s 522 | . . . 4 |
40 | mulcl 7008 | . . . . . 6 | |
41 | 40 | ancoms 255 | . . . . 5 |
42 | 41 | ad2ant2lr 479 | . . . 4 |
43 | 39, 26, 42 | addassd 7049 | . . 3 |
44 | 22, 36, 43 | 3eqtrd 2076 | . 2 |
45 | 4, 11, 44 | 3eqtrd 2076 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 (class class class)co 5512 cc 6887 caddc 6892 cmul 6894 |
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-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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-addcl 6980 ax-mulcl 6982 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-distr 6988 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 |
This theorem is referenced by: mulsub 7398 muladdi 7406 muladdd 7413 sqabsadd 9653 |
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