mul4 - Intuitionistic Logic Explorer

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Theorem mul4 8175

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 8173	. . . . 5 $/\$ $/\$
2	1	oveq1d 5940	. . . 4 $/\$ $/\$
3	2	3expa 1205	. . 3 $/\$ $/\$
4	3	adantrr 479	. 2 $/\$ $/\$ $/\$
5		mulcl 8023	. . 3 $/\$
6		mulass 8027	. . . 4 $/\$ $/\$
7	6	3expb 1206	. . 3 $/\$ $/\$
8	5, 7	sylan 283	. 2 $/\$ $/\$ $/\$
9		mulcl 8023	. . . 4 $/\$
10		mulass 8027	. . . . 5 $/\$ $/\$
11	10	3expb 1206	. . . 4 $/\$ $/\$
12	9, 11	sylan 283	. . 3 $/\$ $/\$ $/\$
13	12	an4s 588	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2237	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 980

wceq 1364

wcel 2167 (class class class)co 5925

cc 7894

cmul 7901

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-ext 2178 ax-mulcl 7994 ax-mulcom 7997 ax-mulass 7999

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-rex 2481 df-v 2765 df-un 3161 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-br 4035 df-iota 5220 df-fv 5267 df-ov 5928

This theorem is referenced by: mul4i 8191 mul4d 8198 recextlem1 8695 divmuldivap 8756 mulexp 10687 demoivreALT 11956