Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > muladd11 | Unicode version |
Description: A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.) |
Ref | Expression |
---|---|
muladd11 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 7681 | . . . 4 | |
2 | addcl 7713 | . . . 4 | |
3 | 1, 2 | mpan 420 | . . 3 |
4 | adddi 7720 | . . . 4 | |
5 | 1, 4 | mp3an2 1288 | . . 3 |
6 | 3, 5 | sylan 281 | . 2 |
7 | 3 | mulid1d 7751 | . . . 4 |
8 | 7 | adantr 274 | . . 3 |
9 | adddir 7725 | . . . . 5 | |
10 | 1, 9 | mp3an1 1287 | . . . 4 |
11 | mulid2 7732 | . . . . . 6 | |
12 | 11 | adantl 275 | . . . . 5 |
13 | 12 | oveq1d 5757 | . . . 4 |
14 | 10, 13 | eqtrd 2150 | . . 3 |
15 | 8, 14 | oveq12d 5760 | . 2 |
16 | 6, 15 | eqtrd 2150 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 (class class class)co 5742 cc 7586 c1 7589 caddc 7591 cmul 7593 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-mulcl 7686 ax-mulcom 7689 ax-mulass 7691 ax-distr 7692 ax-1rid 7695 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-iota 5058 df-fv 5101 df-ov 5745 |
This theorem is referenced by: muladd11r 7886 bernneq 10380 |
Copyright terms: Public domain | W3C validator |