mul4 - Intuitionistic Logic Explorer

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Theorem mul4 8421

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 8419	. . . . 5 $/\$ $/\$
2	1	oveq1d 6073	. . . 4 $/\$ $/\$
3	2	3expa 1230	. . 3 $/\$ $/\$
4	3	adantrr 479	. 2 $/\$ $/\$ $/\$
5		mulcl 8270	. . 3 $/\$
6		mulass 8274	. . . 4 $/\$ $/\$
7	6	3expb 1231	. . 3 $/\$ $/\$
8	5, 7	sylan 283	. 2 $/\$ $/\$ $/\$
9		mulcl 8270	. . . 4 $/\$
10		mulass 8274	. . . . 5 $/\$ $/\$
11	10	3expb 1231	. . . 4 $/\$ $/\$
12	9, 11	sylan 283	. . 3 $/\$ $/\$ $/\$
13	12	an4s 592	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2275	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 1005

wceq 1398

wcel 2205 (class class class)co 6058

cc 8141

cmul 8148

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-ext 2216 ax-mulcl 8241 ax-mulcom 8244 ax-mulass 8246

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-rex 2528 df-v 2817 df-un 3218 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-br 4115 df-iota 5317 df-fv 5365 df-ov 6061

This theorem is referenced by: mul4i 8437 mul4d 8444 recextlem1 8942 divmuldivap 9003 mulexp 10964 demoivreALT 12485