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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 7838 | . . . 4 | |
2 | addcl 7870 | . . . 4 | |
3 | 1, 2 | mpan 421 | . . 3 |
4 | adddi 7877 | . . . 4 | |
5 | 1, 4 | mp3an2 1314 | . . 3 |
6 | 3, 5 | sylan 281 | . 2 |
7 | 3 | mulid1d 7908 | . . . 4 |
8 | 7 | adantr 274 | . . 3 |
9 | adddir 7882 | . . . . 5 | |
10 | 1, 9 | mp3an1 1313 | . . . 4 |
11 | mulid2 7889 | . . . . . 6 | |
12 | 11 | adantl 275 | . . . . 5 |
13 | 12 | oveq1d 5852 | . . . 4 |
14 | 10, 13 | eqtrd 2197 | . . 3 |
15 | 8, 14 | oveq12d 5855 | . 2 |
16 | 6, 15 | eqtrd 2197 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 (class class class)co 5837 cc 7743 c1 7746 caddc 7748 cmul 7750 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-resscn 7837 ax-1cn 7838 ax-icn 7840 ax-addcl 7841 ax-mulcl 7843 ax-mulcom 7846 ax-mulass 7848 ax-distr 7849 ax-1rid 7852 ax-cnre 7856 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2724 df-un 3116 df-in 3118 df-ss 3125 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-br 3978 df-iota 5148 df-fv 5191 df-ov 5840 |
This theorem is referenced by: muladd11r 8046 bernneq 10565 |
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