mul4 - Intuitionistic Logic Explorer

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

Theorem mul4 8107

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 8105	. . . . 5 $/\$ $/\$
2	1	oveq1d 5906	. . . 4 $/\$ $/\$
3	2	3expa 1205	. . 3 $/\$ $/\$
4	3	adantrr 479	. 2 $/\$ $/\$ $/\$
5		mulcl 7956	. . 3 $/\$
6		mulass 7960	. . . 4 $/\$ $/\$
7	6	3expb 1206	. . 3 $/\$ $/\$
8	5, 7	sylan 283	. 2 $/\$ $/\$ $/\$
9		mulcl 7956	. . . 4 $/\$
10		mulass 7960	. . . . 5 $/\$ $/\$
11	10	3expb 1206	. . . 4 $/\$ $/\$
12	9, 11	sylan 283	. . 3 $/\$ $/\$ $/\$
13	12	an4s 588	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2230	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

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

wceq 1364

wcel 2160 (class class class)co 5891

cc 7827

cmul 7834

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-mulcl 7927 ax-mulcom 7930 ax-mulass 7932

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-iota 5193 df-fv 5239 df-ov 5894

This theorem is referenced by: mul4i 8123 mul4d 8130 recextlem1 8626 divmuldivap 8687 mulexp 10577 demoivreALT 11799