mul4 - Intuitionistic Logic Explorer

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Theorem mul4 7677

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 7675	. . . . 5 $/\$ $/\$
2	1	oveq1d 5683	. . . 4 $/\$ $/\$
3	2	3expa 1144	. . 3 $/\$ $/\$
4	3	adantrr 464	. 2 $/\$ $/\$ $/\$
5		mulcl 7532	. . 3 $/\$
6		mulass 7536	. . . 4 $/\$ $/\$
7	6	3expb 1145	. . 3 $/\$ $/\$
8	5, 7	sylan 278	. 2 $/\$ $/\$ $/\$
9		mulcl 7532	. . . 4 $/\$
10		mulass 7536	. . . . 5 $/\$ $/\$
11	10	3expb 1145	. . . 4 $/\$ $/\$
12	9, 11	sylan 278	. . 3 $/\$ $/\$ $/\$
13	12	an4s 556	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2129	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $/\$ w3a 925

wceq 1290

wcel 1439 (class class class)co 5668

cc 7411

cmul 7418

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-mulcl 7506 ax-mulcom 7509 ax-mulass 7511

This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-rex 2366 df-v 2624 df-un 3006 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-br 3854 df-iota 4995 df-fv 5038 df-ov 5671

This theorem is referenced by: mul4i 7693 mul4d 7700 recextlem1 8183 divmuldivap 8242 mulexp 10057 demoivreALT 11126