mul4 - Intuitionistic Logic Explorer

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Theorem mul4 7894

Description: Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)

**Assertion**
Ref	Expression
mul4	$/\$ $/\$ $/\$

**Proof of Theorem mul4**
Step	Hyp	Ref	Expression
1		mul32 7892	. . . . 5 $/\$ $/\$
2	1	oveq1d 5789	. . . 4 $/\$ $/\$
3	2	3expa 1181	. . 3 $/\$ $/\$
4	3	adantrr 470	. 2 $/\$ $/\$ $/\$
5		mulcl 7747	. . 3 $/\$
6		mulass 7751	. . . 4 $/\$ $/\$
7	6	3expb 1182	. . 3 $/\$ $/\$
8	5, 7	sylan 281	. 2 $/\$ $/\$ $/\$
9		mulcl 7747	. . . 4 $/\$
10		mulass 7751	. . . . 5 $/\$ $/\$
11	10	3expb 1182	. . . 4 $/\$ $/\$
12	9, 11	sylan 281	. . 3 $/\$ $/\$ $/\$
13	12	an4s 577	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2180	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $/\$ w3a 962

wceq 1331

wcel 1480 (class class class)co 5774

cc 7618

cmul 7625

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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-mulcl 7718 ax-mulcom 7721 ax-mulass 7723

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777

This theorem is referenced by: mul4i 7910 mul4d 7917 recextlem1 8412 divmuldivap 8472 mulexp 10332 demoivreALT 11480