mul4 - Intuitionistic Logic Explorer

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Theorem mul4 8204

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 8202	. . . . 5 $/\$ $/\$
2	1	oveq1d 5959	. . . 4 $/\$ $/\$
3	2	3expa 1206	. . 3 $/\$ $/\$
4	3	adantrr 479	. 2 $/\$ $/\$ $/\$
5		mulcl 8052	. . 3 $/\$
6		mulass 8056	. . . 4 $/\$ $/\$
7	6	3expb 1207	. . 3 $/\$ $/\$
8	5, 7	sylan 283	. 2 $/\$ $/\$ $/\$
9		mulcl 8052	. . . 4 $/\$
10		mulass 8056	. . . . 5 $/\$ $/\$
11	10	3expb 1207	. . . 4 $/\$ $/\$
12	9, 11	sylan 283	. . 3 $/\$ $/\$ $/\$
13	12	an4s 588	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2246	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 981

wceq 1373

wcel 2176 (class class class)co 5944

cc 7923

cmul 7930

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-ext 2187 ax-mulcl 8023 ax-mulcom 8026 ax-mulass 8028

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-rex 2490 df-v 2774 df-un 3170 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-iota 5232 df-fv 5279 df-ov 5947

This theorem is referenced by: mul4i 8220 mul4d 8227 recextlem1 8724 divmuldivap 8785 mulexp 10723 demoivreALT 12085