mul4 - Intuitionistic Logic Explorer

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Theorem mul4 8051

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 8049	. . . . 5 $/\$ $/\$
2	1	oveq1d 5868	. . . 4 $/\$ $/\$
3	2	3expa 1198	. . 3 $/\$ $/\$
4	3	adantrr 476	. 2 $/\$ $/\$ $/\$
5		mulcl 7901	. . 3 $/\$
6		mulass 7905	. . . 4 $/\$ $/\$
7	6	3expb 1199	. . 3 $/\$ $/\$
8	5, 7	sylan 281	. 2 $/\$ $/\$ $/\$
9		mulcl 7901	. . . 4 $/\$
10		mulass 7905	. . . . 5 $/\$ $/\$
11	10	3expb 1199	. . . 4 $/\$ $/\$
12	9, 11	sylan 281	. . 3 $/\$ $/\$ $/\$
13	12	an4s 583	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2211	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $/\$ w3a 973

wceq 1348

wcel 2141 (class class class)co 5853

cc 7772

cmul 7779

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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-mulcl 7872 ax-mulcom 7875 ax-mulass 7877

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-v 2732 df-un 3125 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856

This theorem is referenced by: mul4i 8067 mul4d 8074 recextlem1 8569 divmuldivap 8629 mulexp 10515 demoivreALT 11736