mul4 - Intuitionistic Logic Explorer

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Theorem mul4 8239

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 8237	. . . . 5 $/\$ $/\$
2	1	oveq1d 5982	. . . 4 $/\$ $/\$
3	2	3expa 1206	. . 3 $/\$ $/\$
4	3	adantrr 479	. 2 $/\$ $/\$ $/\$
5		mulcl 8087	. . 3 $/\$
6		mulass 8091	. . . 4 $/\$ $/\$
7	6	3expb 1207	. . 3 $/\$ $/\$
8	5, 7	sylan 283	. 2 $/\$ $/\$ $/\$
9		mulcl 8087	. . . 4 $/\$
10		mulass 8091	. . . . 5 $/\$ $/\$
11	10	3expb 1207	. . . 4 $/\$ $/\$
12	9, 11	sylan 283	. . 3 $/\$ $/\$ $/\$
13	12	an4s 588	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2248	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 981

wceq 1373

wcel 2178 (class class class)co 5967

cc 7958

cmul 7965

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-ext 2189 ax-mulcl 8058 ax-mulcom 8061 ax-mulass 8063

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1485 df-sb 1787 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-rex 2492 df-v 2778 df-un 3178 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-br 4060 df-iota 5251 df-fv 5298 df-ov 5970

This theorem is referenced by: mul4i 8255 mul4d 8262 recextlem1 8759 divmuldivap 8820 mulexp 10760 demoivreALT 12200