mul4 - Intuitionistic Logic Explorer

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Theorem mul4 7240

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 7238	. . . . 5 $/\$ $/\$
2	1	oveq1d 5547	. . . 4 $/\$ $/\$
3	2	3expa 1138	. . 3 $/\$ $/\$
4	3	adantrr 462	. 2 $/\$ $/\$ $/\$
5		mulcl 7100	. . 3 $/\$
6		mulass 7104	. . . 4 $/\$ $/\$
7	6	3expb 1139	. . 3 $/\$ $/\$
8	5, 7	sylan 277	. 2 $/\$ $/\$ $/\$
9		mulcl 7100	. . . 4 $/\$
10		mulass 7104	. . . . 5 $/\$ $/\$
11	10	3expb 1139	. . . 4 $/\$ $/\$
12	9, 11	sylan 277	. . 3 $/\$ $/\$ $/\$
13	12	an4s 552	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2121	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 102 $/\$ w3a 919

wceq 1284

wcel 1433 (class class class)co 5532

cc 6979

cmul 6986

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-mulcl 7074 ax-mulcom 7077 ax-mulass 7079

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535

This theorem is referenced by: mul4i 7256 mul4d 7263 recextlem1 7741 divmuldivap 7800 mulexp 9515