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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 7737 |
. . . 4
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2 | addcl 7769 |
. . . 4
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3 | 1, 2 | mpan 421 |
. . 3
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4 | adddi 7776 |
. . . 4
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5 | 1, 4 | mp3an2 1304 |
. . 3
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6 | 3, 5 | sylan 281 |
. 2
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7 | 3 | mulid1d 7807 |
. . . 4
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8 | 7 | adantr 274 |
. . 3
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9 | adddir 7781 |
. . . . 5
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10 | 1, 9 | mp3an1 1303 |
. . . 4
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11 | mulid2 7788 |
. . . . . 6
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12 | 11 | adantl 275 |
. . . . 5
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13 | 12 | oveq1d 5797 |
. . . 4
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14 | 10, 13 | eqtrd 2173 |
. . 3
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15 | 8, 14 | oveq12d 5800 |
. 2
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16 | 6, 15 | eqtrd 2173 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-resscn 7736 ax-1cn 7737 ax-icn 7739 ax-addcl 7740 ax-mulcl 7742 ax-mulcom 7745 ax-mulass 7747 ax-distr 7748 ax-1rid 7751 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785 |
This theorem is referenced by: muladd11r 7942 bernneq 10443 |
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