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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul32 8036 | . . . . 5 | |
2 | 1 | oveq1d 5865 | . . . 4 |
3 | 2 | 3expa 1198 | . . 3 |
4 | 3 | adantrr 476 | . 2 |
5 | mulcl 7888 | . . 3 | |
6 | mulass 7892 | . . . 4 | |
7 | 6 | 3expb 1199 | . . 3 |
8 | 5, 7 | sylan 281 | . 2 |
9 | mulcl 7888 | . . . 4 | |
10 | mulass 7892 | . . . . 5 | |
11 | 10 | 3expb 1199 | . . . 4 |
12 | 9, 11 | sylan 281 | . . 3 |
13 | 12 | an4s 583 | . 2 |
14 | 4, 8, 13 | 3eqtr3d 2211 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 973 wceq 1348 wcel 2141 (class class class)co 5850 cc 7759 cmul 7766 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-mulcl 7859 ax-mulcom 7862 ax-mulass 7864 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-v 2732 df-un 3125 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-iota 5158 df-fv 5204 df-ov 5853 |
This theorem is referenced by: mul4i 8054 mul4d 8061 recextlem1 8556 divmuldivap 8616 mulexp 10502 demoivreALT 11723 |
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