mul4 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > mul4		Unicode version

Theorem mul4 8448

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 8446	. . . . 5 $/\$ $/\$
2	1	oveq1d 6090	. . . 4 $/\$ $/\$
3	2	3expa 1234	. . 3 $/\$ $/\$
4	3	adantrr 483	. 2 $/\$ $/\$ $/\$
5		mulcl 8296	. . 3 $/\$
6		mulass 8300	. . . 4 $/\$ $/\$
7	6	3expb 1235	. . 3 $/\$ $/\$
8	5, 7	sylan 283	. 2 $/\$ $/\$ $/\$
9		mulcl 8296	. . . 4 $/\$
10		mulass 8300	. . . . 5 $/\$ $/\$
11	10	3expb 1235	. . . 4 $/\$ $/\$
12	9, 11	sylan 283	. . 3 $/\$ $/\$ $/\$
13	12	an4s 596	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2279	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 1009

wceq 1402

wcel 2209 (class class class)co 6075

cc 8167

cmul 8174

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 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-ext 2220 ax-mulcl 8267 ax-mulcom 8270 ax-mulass 8272

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-rex 2534 df-v 2823 df-un 3224 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-iota 5332 df-fv 5380 df-ov 6078

This theorem is referenced by: mul4i 8464 mul4d 8471 recextlem1 8969 divmuldivap 9032 mulexp 10993 demoivreALT 12519