mul4 - Intuitionistic Logic Explorer

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Theorem mul4 7918

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 7916	. . . . 5 $/\$ $/\$
2	1	oveq1d 5797	. . . 4 $/\$ $/\$
3	2	3expa 1182	. . 3 $/\$ $/\$
4	3	adantrr 471	. 2 $/\$ $/\$ $/\$
5		mulcl 7771	. . 3 $/\$
6		mulass 7775	. . . 4 $/\$ $/\$
7	6	3expb 1183	. . 3 $/\$ $/\$
8	5, 7	sylan 281	. 2 $/\$ $/\$ $/\$
9		mulcl 7771	. . . 4 $/\$
10		mulass 7775	. . . . 5 $/\$ $/\$
11	10	3expb 1183	. . . 4 $/\$ $/\$
12	9, 11	sylan 281	. . 3 $/\$ $/\$ $/\$
13	12	an4s 578	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2181	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $/\$ w3a 963

wceq 1332

wcel 1481 (class class class)co 5782

cc 7642

cmul 7649

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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-mulcl 7742 ax-mulcom 7745 ax-mulass 7747

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785

This theorem is referenced by: mul4i 7934 mul4d 7941 recextlem1 8436 divmuldivap 8496 mulexp 10363 demoivreALT 11516