mul4 - Intuitionistic Logic Explorer

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Theorem mul4 8158

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 8156	. . . . 5 $/\$ $/\$
2	1	oveq1d 5937	. . . 4 $/\$ $/\$
3	2	3expa 1205	. . 3 $/\$ $/\$
4	3	adantrr 479	. 2 $/\$ $/\$ $/\$
5		mulcl 8006	. . 3 $/\$
6		mulass 8010	. . . 4 $/\$ $/\$
7	6	3expb 1206	. . 3 $/\$ $/\$
8	5, 7	sylan 283	. 2 $/\$ $/\$ $/\$
9		mulcl 8006	. . . 4 $/\$
10		mulass 8010	. . . . 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 5922

cc 7877

cmul 7884

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 7977 ax-mulcom 7980 ax-mulass 7982

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 3628 df-pr 3629 df-op 3631 df-uni 3840 df-br 4034 df-iota 5219 df-fv 5266 df-ov 5925

This theorem is referenced by: mul4i 8174 mul4d 8181 recextlem1 8678 divmuldivap 8739 mulexp 10670 demoivreALT 11939