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Mirrors > Home > ILE Home > Th. List > mul4 | Unicode version |
Description: Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.) |
Ref | Expression |
---|---|
mul4 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul32 8105 |
. . . . 5
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2 | 1 | oveq1d 5906 |
. . . 4
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3 | 2 | 3expa 1205 |
. . 3
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4 | 3 | adantrr 479 |
. 2
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5 | mulcl 7956 |
. . 3
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6 | mulass 7960 |
. . . 4
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7 | 6 | 3expb 1206 |
. . 3
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8 | 5, 7 | sylan 283 |
. 2
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9 | mulcl 7956 |
. . . 4
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10 | mulass 7960 |
. . . . 5
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11 | 10 | 3expb 1206 |
. . . 4
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12 | 9, 11 | sylan 283 |
. . 3
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13 | 12 | an4s 588 |
. 2
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14 | 4, 8, 13 | 3eqtr3d 2230 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-mulcl 7927 ax-mulcom 7930 ax-mulass 7932 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-iota 5193 df-fv 5239 df-ov 5894 |
This theorem is referenced by: mul4i 8123 mul4d 8130 recextlem1 8626 divmuldivap 8687 mulexp 10577 demoivreALT 11799 |
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