mul4 - Intuitionistic Logic Explorer

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Theorem mul4 8310

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 8308	. . . . 5 $/\$ $/\$
2	1	oveq1d 6032	. . . 4 $/\$ $/\$
3	2	3expa 1229	. . 3 $/\$ $/\$
4	3	adantrr 479	. 2 $/\$ $/\$ $/\$
5		mulcl 8158	. . 3 $/\$
6		mulass 8162	. . . 4 $/\$ $/\$
7	6	3expb 1230	. . 3 $/\$ $/\$
8	5, 7	sylan 283	. 2 $/\$ $/\$ $/\$
9		mulcl 8158	. . . 4 $/\$
10		mulass 8162	. . . . 5 $/\$ $/\$
11	10	3expb 1230	. . . 4 $/\$ $/\$
12	9, 11	sylan 283	. . 3 $/\$ $/\$ $/\$
13	12	an4s 592	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2272	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 1004

wceq 1397

wcel 2202 (class class class)co 6017

cc 8029

cmul 8036

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-ext 2213 ax-mulcl 8129 ax-mulcom 8132 ax-mulass 8134

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-nf 1509 df-sb 1811 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-rex 2516 df-v 2804 df-un 3204 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-iota 5286 df-fv 5334 df-ov 6020

This theorem is referenced by: mul4i 8326 mul4d 8333 recextlem1 8830 divmuldivap 8891 mulexp 10839 demoivreALT 12334