mul4 - Intuitionistic Logic Explorer

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Theorem mul4 8274

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 8272	. . . . 5 $/\$ $/\$
2	1	oveq1d 6015	. . . 4 $/\$ $/\$
3	2	3expa 1227	. . 3 $/\$ $/\$
4	3	adantrr 479	. 2 $/\$ $/\$ $/\$
5		mulcl 8122	. . 3 $/\$
6		mulass 8126	. . . 4 $/\$ $/\$
7	6	3expb 1228	. . 3 $/\$ $/\$
8	5, 7	sylan 283	. 2 $/\$ $/\$ $/\$
9		mulcl 8122	. . . 4 $/\$
10		mulass 8126	. . . . 5 $/\$ $/\$
11	10	3expb 1228	. . . 4 $/\$ $/\$
12	9, 11	sylan 283	. . 3 $/\$ $/\$ $/\$
13	12	an4s 590	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2270	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 1002

wceq 1395

wcel 2200 (class class class)co 6000

cc 7993

cmul 8000

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 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-ext 2211 ax-mulcl 8093 ax-mulcom 8096 ax-mulass 8098

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-rex 2514 df-v 2801 df-un 3201 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-br 4083 df-iota 5277 df-fv 5325 df-ov 6003

This theorem is referenced by: mul4i 8290 mul4d 8297 recextlem1 8794 divmuldivap 8855 mulexp 10795 demoivreALT 12280