mul4 - Intuitionistic Logic Explorer

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Theorem mul4 8091

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 8089	. . . . 5 $/\$ $/\$
2	1	oveq1d 5892	. . . 4 $/\$ $/\$
3	2	3expa 1203	. . 3 $/\$ $/\$
4	3	adantrr 479	. 2 $/\$ $/\$ $/\$
5		mulcl 7940	. . 3 $/\$
6		mulass 7944	. . . 4 $/\$ $/\$
7	6	3expb 1204	. . 3 $/\$ $/\$
8	5, 7	sylan 283	. 2 $/\$ $/\$ $/\$
9		mulcl 7940	. . . 4 $/\$
10		mulass 7944	. . . . 5 $/\$ $/\$
11	10	3expb 1204	. . . 4 $/\$ $/\$
12	9, 11	sylan 283	. . 3 $/\$ $/\$ $/\$
13	12	an4s 588	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2218	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 978

wceq 1353

wcel 2148 (class class class)co 5877

cc 7811

cmul 7818

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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-mulcl 7911 ax-mulcom 7914 ax-mulass 7916

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2741 df-un 3135 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-iota 5180 df-fv 5226 df-ov 5880

This theorem is referenced by: mul4i 8107 mul4d 8114 recextlem1 8610 divmuldivap 8671 mulexp 10561 demoivreALT 11783