mul4 - Intuitionistic Logic Explorer

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Theorem mul4 8030

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 8028	. . . . 5 $/\$ $/\$
2	1	oveq1d 5857	. . . 4 $/\$ $/\$
3	2	3expa 1193	. . 3 $/\$ $/\$
4	3	adantrr 471	. 2 $/\$ $/\$ $/\$
5		mulcl 7880	. . 3 $/\$
6		mulass 7884	. . . 4 $/\$ $/\$
7	6	3expb 1194	. . 3 $/\$ $/\$
8	5, 7	sylan 281	. 2 $/\$ $/\$ $/\$
9		mulcl 7880	. . . 4 $/\$
10		mulass 7884	. . . . 5 $/\$ $/\$
11	10	3expb 1194	. . . 4 $/\$ $/\$
12	9, 11	sylan 281	. . 3 $/\$ $/\$ $/\$
13	12	an4s 578	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2206	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $/\$ w3a 968

wceq 1343

wcel 2136 (class class class)co 5842

cc 7751

cmul 7758

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-ext 2147 ax-mulcl 7851 ax-mulcom 7854 ax-mulass 7856

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-rex 2450 df-v 2728 df-un 3120 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-iota 5153 df-fv 5196 df-ov 5845

This theorem is referenced by: mul4i 8046 mul4d 8053 recextlem1 8548 divmuldivap 8608 mulexp 10494 demoivreALT 11714