mul4 - Intuitionistic Logic Explorer

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Theorem mul4 8151

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 8149	. . . . 5 $/\$ $/\$
2	1	oveq1d 5933	. . . 4 $/\$ $/\$
3	2	3expa 1205	. . 3 $/\$ $/\$
4	3	adantrr 479	. 2 $/\$ $/\$ $/\$
5		mulcl 7999	. . 3 $/\$
6		mulass 8003	. . . 4 $/\$ $/\$
7	6	3expb 1206	. . 3 $/\$ $/\$
8	5, 7	sylan 283	. 2 $/\$ $/\$ $/\$
9		mulcl 7999	. . . 4 $/\$
10		mulass 8003	. . . . 5 $/\$ $/\$
11	10	3expb 1206	. . . 4 $/\$ $/\$
12	9, 11	sylan 283	. . 3 $/\$ $/\$ $/\$
13	12	an4s 588	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2234	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 980

wceq 1364

wcel 2164 (class class class)co 5918

cc 7870

cmul 7877

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 ax-mulcl 7970 ax-mulcom 7973 ax-mulass 7975

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-rex 2478 df-v 2762 df-un 3157 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-iota 5215 df-fv 5262 df-ov 5921

This theorem is referenced by: mul4i 8167 mul4d 8174 recextlem1 8670 divmuldivap 8731 mulexp 10649 demoivreALT 11917