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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 7918 |
. . . 4
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2 | addcl 7950 |
. . . 4
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3 | 1, 2 | mpan 424 |
. . 3
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4 | adddi 7957 |
. . . 4
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5 | 1, 4 | mp3an2 1335 |
. . 3
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6 | 3, 5 | sylan 283 |
. 2
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7 | 3 | mulridd 7988 |
. . . 4
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8 | 7 | adantr 276 |
. . 3
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9 | adddir 7962 |
. . . . 5
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10 | 1, 9 | mp3an1 1334 |
. . . 4
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11 | mullid 7969 |
. . . . . 6
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12 | 11 | adantl 277 |
. . . . 5
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13 | 12 | oveq1d 5903 |
. . . 4
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14 | 10, 13 | eqtrd 2220 |
. . 3
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15 | 8, 14 | oveq12d 5906 |
. 2
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16 | 6, 15 | eqtrd 2220 |
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 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 ax-resscn 7917 ax-1cn 7918 ax-icn 7920 ax-addcl 7921 ax-mulcl 7923 ax-mulcom 7926 ax-mulass 7928 ax-distr 7929 ax-1rid 7932 ax-cnre 7936 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-iota 5190 df-fv 5236 df-ov 5891 |
This theorem is referenced by: muladd11r 8127 bernneq 10655 |
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